Topics on Real and Complex Singularities

TOPICS ON REAL AND COMPLEX SINGULARITIES

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TOPICS ON REAL AND COMPLEX SINGULARITIES Proceedings of the 4th Japanese–Australian Workshop (JARCS4) Kobe, Japan

22 – 25 November 2011

Editors

Satoshi Koike Hyogo University of Teacher Education, Japan

Toshizumi Fukui Saitama University, Japan

Laurentiu Paunescu University of Sydney, Australia

Adam Harris University of New England, Australia

Alexander Isaev Australian National University, Australia

World Scientific NEW JERSEY

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

TOPICS â•›ON â•›REAL â•›AND â•›COMPLEX â•›SINGULARITIES Proceedings of the 4th Japanese–Australian Workshop (JARCS4) Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4596-03-9

Printed in Singapore

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Preface The fourth Japanese-Australian Workshop on Real and Complex Singularities (JARCS4 KOBE 2011) was held at the Kobe Satellite of Hyogo University of Teacher Education during the period 22-25 November, 2011. There were 31 participants from Australia and Japan. The Australian and Japanese Singularity Theory specialists have built up a strong research relationship in the past three decades. For instance, the blow-analytic theory introduced by Tzee-Char Kuo in Australia has been intensively developed in Japan. In addition, a lot of joint works by researchers of both countries have been established in several topics related to real and complex singularities. The present volume consists mainly of the texts of the invited talks of the workshop. Some of them are joint works of Australians and Japanese. This volume contains original articles on real and complex singularities, topology of differentiable maps, openings of differentiable map-germs, the relationship between free divisors and holonomic systems, effective computational method of invariants of singularities, the application of singularity theory to differential geometry, the deformation theory of CR structures and differential equations with singular points. In these articles some important new notions for characterizations of singularities are introduced, and several new results are presented. New approaches to classical topics and new computational methods of singularities are also presented. We would like to thank the contributors for their cooperation. All the articles in this volume have been very carefully refereed. We would also like to thank the referees for taking time to read and give a lot of comments again and again. The workshop was supported by the Grant-in-Aid for Scientific Researches No. No. No. No. No.

23244008 (Investigator: Osamu Saeki), 22340030 (Investigator: Goo Ishikawa), 23540099 (Investigator: Kimio Miyajima), 21540054 (Investigator: Shuzo Izumi), 23540087 (Investigator: Satoshi Koike)

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of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and Hyogo University of Teacher Education. We acknowledge the support. 18 September 2013 Satoshi Koike

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Organizing Committees of the fourth Japanese-Australian Workshop on Real and Complex Singularities

Satoshi Koike (Chairman) – Hyogo University of Teacher Education, Japan Toshizumi Fukui – Saitama University, Japan Laurentiu Paunescu – University of Sydney, Australia Adam Harris – University of New England, Australia Alexander Isaev – Australian National University, Australia

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List of Participants Takao Akahori (Hyogo University, Japan) Kana Ando (Chiba University, Japan) Takuo Fukuda (Nihon University, Japan) Toshizumi Fukui (Saitama University, Japan) Hirofumi Hagi (Hyogo University of Teacher Education, Japan) Adam Harris (University of New England, Australia) Jonathan Hillman (University of Sydney, Australia) Goo Ishikawa (Hokkaido University, Japan) Shuzo Izumi (Kinki University, Japan) Shigeyasu Kamiya (Okayama University of Science, Japan) Yoshihiro Kawazoe (Hyogo University of Teacher Education, Japan) Yumiko Kitagawa (Oita National College of Technology, Japan) Naoki Kitazawa (Tokyo Institute of Technology, Japan) Mahito Kobayashi (Akita University, Japan) Satoshi Koike (Hyogo University of Teacher Education, Japan) Yutaka Matsui (Kinki University, Japan) Kimio Miyajima (Kagoshima University, Japan) Masayuki Nishioka (Kyushu University, Japan) Masao Ogawa (Hyogo University of Teacher Education, Japan) Mutsuo Oka (Tokyo University of Science, Japan) Tomohiro Okuma (Yamagata University, Japan) Laurentiu Paunescu (University of Sydney, Australia) Osamu Saeki (Kyushu University, Japan) Jiro Sekiguchi (Tokyo University of Agriculture and Technology, Japan) Ryuhei Shigematsu (Hyogo University of Teacher Education, Japan) Masahiro Shiota (Nagoya University, Japan) Shinichi Tajima (Tsukuba University, Japan) Kiyoshi Takeuchi (Tsukuba University, Japan) Takahiro Yamamoto (Kyushu Sangyo University, Japan) Wataru Yukuno (Hokkaido University, Japan) Ruibin Zhang (University of Sydney, Australia)

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Contents

Preface

v

Organizing Committees

vii

List of Participants

viii

On the CR Hamiltonian flows and CR Yamabe problem T. Akahori An example of the reduction of a single ordinary differential equation to a system, and the restricted Fuchsian relation K. Ando

1

13

Fronts of weighted cones T. Fukui and M. Hasegawa

31

Involutive deformations of the regular part of a normal surface A. Harris and K. Miyajima

51

Connected components of regular fibers of differentiable maps J. T. Hiratuka and O. Saeki

61

The reconstruction and recognition problems for homogeneous hypersurface singularities A. V. Isaev Openings of differentiable map-germs and unfoldings G. Ishikawa Non concentration of curvature near singular points of two variable analytic functions S. Koike, T.-C. Kuo and L. Paunescu

75

87

115

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Saito free divisors in four dimensional affine space and reflection groups of rank four J. Sekiguchi

141

Holonomic systems of differential equations of rank two with singularities along Saito free divisors of simple type J. Sekiguchi

159

Parametric local cohomology classes and Tjurina stratifications for μ-constant deformations of quasi-homogeneous singularities S. Tajima Author Index

189

201

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On the CR Hamiltonian flows and CR Yamabe problem Takao Akahori Hyogo University, Kobe, Hyogo, 651-2197, Japan [email protected] CR Yamabe problem was treated by John M. Lee and several people. Their setting is that: by fixing the CR structure, they try to find a suitable contact form. Our new approach differs from theirs. We fix the contact form and change CR structures. Our approach is in progress. Keywords: CR Yamabe problem, deformation theory of CR structures AMS classification numbers: 32V20

The deformation theory of CR structures is initiated by Kuranishi, and developed by several authors and in several ways (for example, Miyajima, Lee, Cheng, and myself). Especially, in [A4], the notion of CR-Hamiltonian flows is found. Let {(M, 0 T  ), θ} be a CR structure with a contact form, embedded in a complex manifold N as a real hypersurface. Then, we have the Cartan connection, and have the scalar curvature R. Related with the Einstein Kaehler metric, CR Yamabe problem is of interest. This problem is already treated by J. M. Lee. His approach is that; for any given CR structure with a contact form {(M, 0 T  ), θ}, is there a real valued C ∞ non vanishing function g, which satisfies {(M, 0 T  ), gθ} that has a constant scalar curvature? Our approach differs from his. We fix the contact form and change the CR structure by the CR Hamiltonian flows. We set our approach. We assume that {(M, 0 T  ), θ} has a constant scalar curvature, and consider the deformation of the contact structure, {(M, 0 T  ), ρ12 θ}, where ρ is a C ∞ function, close to 1. Then, we cannot expect that our CR structure with the corresponding contact structure admits the constant scalar curvature. Let R(ρ) denote its scalar curvature. Our problem is that: if ρ is close enough to 1, is it possible to show that: there is a√real valued C ∞ function h which satisfies the scalar curvature of {(M, φ( −1h) T  ), ρ12 θ} is constant? Here √

(M, φ( −1h) T  ) means the CR structure, induced by the CR Hamiltonian √ flow, generated by −1h. We see the model case, the Heisenberg structure

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with real dimension 3. And let the contact structure be the standard one. Then, all curvature tensors vanish, and so the scalar curvature vanishes. But if we adopt ρ12 θ as a contact form, where θ is the standard contact form and ρ is a C ∞ function close to 1, then the scalar curvature may not vanish. For this structure, by changing the CR structure by the CRHamiltonian flow (the contact form is ρ12 θ and this is fixed), we would like to obtain the constant scalar curvature (in this case, 0 scalar curvature). This problem becomes an existence problem of the solution of the non-linear partial differential equation. For the linear part of this non-linear partial differential equation, we show the sub-ellipticity (the main theorem). This suggests that our problem would be solved affirmatively. And also, it might suggest the deformation of contact structures is something like a dual space with the deformation of CR structures, induced by the CR Hamiltonian flow, via the scalar curvature. While for our original problem, we have to see the non-linear part more carefully. In a future paper, with a wide scope, we discuss this problem. Finally, the author has to mention that this problem is inspired by the Kuranishi’s work for the Cartan connection. 1. Deformation theory of CR structures Let M a real 2n − 1 dimensional C ∞ manifold. And we assume that our M is embedded in a complex manifold N . Let 0 T  be a sub bundle of the complexfied tangent vector bundle of M . Now, pair (M, 0 T  ) is called a CR structure iff the following two conditions are satisfied. (1) 0 T  ∩ 0 T  = 0, dimC

C ⊗ TM

= 1, + 0 T  (2) [Γ(M, 0 T  ), Γ(M, 0 T  )] ⊂ Γ(M, 0 T  ) 0 T 

A pair (M, θ) is called a contact structure if a real 1-form on M , θ, which satisfies the following condition: θ ∧ (dθ)n−1 = 0 at any point of M is given. So a contact structure is nothing with a CR structure. Now let (V, o) be an isolated singularity in a complex euclidean space CN . Let M = V ∩ S2N −1 (o) where S2N −1 (o) is a hypersphere, centered at the origin with radius .

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Then, over M , naturally, a CR structure 0

T  := C ⊗ T (V − o) ∩ T  CN

is induced. In this case, our CR structure is strongly pseudo convex. Let r :=

N 

| zi |2 −

i=1

and θ :=

√

−1∂r.

Then, our θ is a real 1-form on M and (M, θ) is a contact structure. We note that if g is any positive C ∞ function on M , (M, gθ) is a different contact structure. In this sense, a contact structure is a completely different notion from a CR structure. Now we recall the deformation equation of CR structures. Let (M, 0 T  ) be a strongly pseudo convex CR structure. We fix a contact structure which is compatible with our CR structure. Here ”compatible” means that: θ |0 T  +0 T  = 0 where 0 T  = 0 T  . By using this θ, we determine a real vector ξ on M by  θ(ξ) = 1 dθ(W, ξ) = 0, W ∈ 0 T  Now by using this ξ, we give a C ∞ vector bundle decomposition of C ⊗ T M. C ⊗ T M = 0 T  + C ⊗ ξ + 0 T  . Set T  := C ⊗ ξ + 0 T  . Then, T  has the following meaning. Consider the following scheme. T  → C ⊗ T M → C ⊗ T (V −o) |M ↓ T  (V −o) |M By comparing the dimension of vector bundles T  and T  (V − o) the map: T  → T  (V − 0) is an isomorphism map. So our T  is a natural one. Later on, we write the vector bundle decomposition of C ⊗ T M as follows. C ⊗ T M = 0 T  + T  .

(1.1)

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With these preparations, we consider the deformation theory of CR structures of (M, 0 T  ). Proposition 1.1. Almost CR manifold with finite distance from (M, 0 T  ) is one to one correspondence to Γ(M, Hom(0 T  , T  )). The correspondence is that; φ ∈ Γ(M, Hom(0 T  , T  )) = Γ(M, T  ⊗ (0 T  )∗ ) φ

T  := {X  : X  = X + φ(X), X ∈ 0 T  }

By using this correspondence, we see that when our almost CR structure is actually a CR structure. Proposition 1.2. Almost CR manifold (M, φ T  ) is really CR structure iff our φ satisfies the following non-linear partial differential equation. (1)

∂ T  φ + R2 (φ) + R3 (φ) = 0.

(1.2)

(1)

We see ∂ T  more precisely. First, we set ∂ T  : Γ(M, T  ) → Γ(M, T  ⊗ 0  ∗ ( T ) ) by; for u ∈ Γ(M, T  ), ∂ T  u(Y ) = [Y, u]T  , Y ∈ 0 T  . And as for (i)

scalar valued differential forms, we introduce ∂ T  . Then, the case i = 1 is exactly the same as above operators. Now by the integrability condition, we have a differential complex. Standard deformation complex ∂

∂



(1) 

T 0 − −−−− → Γ(M, T  ) − −−− −→ Γ(M, T  ⊗ (0 T  )∗ ) − −−T−−→ Γ(M, T  ⊗ ∧2 (0 T  )∗ )...

This differential complex (M, 0 T  ) can be defined without strongly pseudo convexity. Especially, if our CR structure is strongly convexity and M is compact, our complex is sub-elliptic. and in the case dimR M = 2n − 1 ≥ 5, (1)

dimC H 1 (M, T  ) := Ker ∂ T  /Im ∂ T  < +∞ However, for the deformation theory, the standard complex is not enough, and so we introduce the new deformation complex. New deformation complex ∂



∂

(1) 

T −→ Γ(M, E2 )... H −−−T−→ Γ(M, E1 ) −−−

We have to explain notations. For j = 1, (1)

Γ(M, E1 ) = {u : u ∈ Γ(M, 0 T  ⊗ ∧1 (0 T  )∗ ), (∂ T  u)Cξ⊗∧2 (0 T  )∗ =0 }. And j ≥ 2, (j)

Γ(M, Ej ) = {v : v ∈ Γ(M, 0 T  ⊗ ∧j (0 T  )∗ ), (∂ T  u)Cξ⊗∧j+1 (0 T  )∗ =0 }.

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Here in the definition of Γ(M, Ej ), we note 0 T  -valued (not T  -valued). And H = {w : w ∈ Γ(M, T  ), (∂ T  w)Cξ⊗(0 T  )∗ = 0} This H is an infinite dimensional vector space (there is no vector bundles which satisfies ; H is the global sections). We discuss this differential in more details. This differential complex comes from D

∂

(1) 

T Γ(M, C) −−−−→ Γ(M, E1 ) −−− −→ Γ(M, E2 )...

Here g ∈ Γ(M, C), Dg := ∂ T  (gξ + Xg ) ∈ Γ(M, E1 ). We note that D is a second order partial differential operators. We explain gξ + Xg . Definition 1.3 (CR Hamilton vector). Let g be a C ∞ function on M . Let Xg be 0 T  valued section on M , defined by [Xg , Y ]Cξ = (∂ b g)(Y )ξ, for Y ∈ 0 T  We call this complex vector gξ + Xg a CR Hamilton vector. And also we call g a generating function. Proposition 1.4. Let gξ + Xg be a CR Hamilton vector. Then, we have ∂ T  (gξ + Xg ) is 0 T  ⊗ (0 T  )∗ − valued. more precisely, it holds that ; gξ + Xg ∈ H. 2. CR Hamiltonian flows In our former paper (see [A4]), we introduced the notion of CR Hamiltonian flows. Actually, there are two types. For any C ∞ function g, (type 1) gξ + Xg This is the kernel of our new complex. Furthermore, (type 2) gξ − Xg This vector field preserves the contact form θ. In fact, for Y ∈ 0 T  , Lgξ−Xg θ(Y ) = dθ(gξ − Xg , Y ) + d(θ(gξ − Xg ))(Y ) = −dθ(Xg , Y ) + dg(Y ) = 0. For any complex valued C ∞ function g, the existence of CR Hamiltonian flow is proved in the same way (in the former paper, the proof is given for

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type 2 vectors, but by the complete same way, we can obtain the same statement). Probably, more serious studies might be needed for these two type of vectors. Here, we see vectors which are of type 1. Let g be a complex-valued C ∞ function on M . We assume that our M is embedded in a complex manifold N as a real hypersurface. Roughly speaking, p ∈ M → fg (p) ∈ N . Here, in our case, for any real valued C ∞ function h, we solve that: for a sufficiently small positive number , for − < t < , √  d √ f −1h (t) = ( −1hξ + (Lh)L)f√−1ht dt f√−1h (0) = identity map It is not clear that we can put t = 1. Instead, if | h | is chosen sufficiently small, then we have √ z i · f√−1h = ( −1hξ + (Lh)L)z i + O(higher order term of h). j = 1, . . . , n and z j are complex coordinates of Cn And the corresponding deformation is √ √ φ( −1h) = ∂ T  (( −1h)ξ + X√−1h ) + O(h2 ). 3. The scalar curvature

√ As for a complex valued C ∞ fuction g (in Sect. 2), we take −1h, where h is a real valued C ∞ function. For this complex valued function, we consider √ the Hamiltonian flow, and φ( −1h) denotes the corresponding form by the induced CR structure by this Hamiltonian flow. As mentioned, √ √ φ( −1h) = ∂ T  (( −1h)ξ + X√−1h ) + O(h2 ). We consider the following map. h ∈ C ∞ (R) → Scalar curvature of the CR structure {(M, φ(

√ −1h)

T  ), θ}.

Here, we note that the contact form θ is fixed. By following Lee-Jerison (see [J-L1]), we recall the notion of scalar curvature. First, we set the TanakaWebster connection, ∇ : Γ(M, 0 T  ) → Γ(M, 0 T  ⊗ (C ⊗ T M )∗ ) satisfying: (1) compatible with the CR structure (this means that ∇W , for W ∈ 0  T , is determined), (2) preserving the Levi metric (so ∇W , for W ∈ 0 T  is determined),

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(3) and for the supplement vector, we set ∇ξ W = [ξ, W ]0 T  for W ∈ Γ(M, 0 T  ), where [ξ, W ]0 T  means the 0 T  part of [ξ, W ]. With these setting, we introduce the scalar curvature. Let {W1 , ., Wn−1 } an orthonormal system of bases of 0 T  with respect to the Levi metric, determined by the contact form θ. By these vectors, we determine a 1 form ωα β by  ωα β ⊗ Wβ . ∇Wα = β

The scalar curvature is determined by   {(dωα α − ωα γ ∧ ωγ α )(Wρ , Wρ )} α,ρ

γ

Needless to say, it does not depend on the choice of the orthonormal system. 4. The model case 1 (changing the contact structure) We see our Yamabe problem for the case Heisenberg structure with real dimension 3. In C2 , we consider {(z1 , z2 ) : (z1 , z2 ) ∈ C2 , Im z2 =

1 |z1 |2 }. 2

√ √ And we set L = ∂z∂ 1 − −1z1 ∂z∂ 2 . 0 T  is generated by L. Let θ = 2 −1∂r, where r = Im z2 − 12 | z1 |2 . In this case, all curvature tensors ωα β vanish. We change the contact form. For the contact form, we adopt ρ12 θ. Here, ρ is a non vanishing real valued C ∞ function. In this case, some curvature tensor ωα β may not vanish. In fact, our L is no more the unit vector. L := ρL is the unit vector. Our connection is compatible with the CR structure. So, ∇L L = ∇ρL ρL = [ρL, ρL]0 T  1 = (L ρ)L ρ We see ∇L L . Our connection preserves the metric. So, L L , L  = ∇L L , L  + L , ∇L L  Hence 0 = ∇L L , L  + L , ∇L L 

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So, 1 ∇L L = − (L ρ)L . ρ Set ∇L = ω1 1 ⊗ L . And we determine the connection 1-form ω1 1 . Then, 1  (L ρ), ρ 1 ω1 1 (L ) = − (L ρ). ρ ω1 1 (L ) =

(4.3) (4.4)

By the definition, the scalar curvature, here we write it by R(ρ), is dω1 1 (L , L ). We compute the principal part of this R(ρ). dω1 1 (L , L ) = L ω1 1 (L ) − L ω1 1 (L ) − dω1 1 ([L , L ]) = ρ(LLρ + LLρ) + lower order derivatives  (LLρ + LLρ) + lower order derivatives Here  means ”mod (1 − ρ)2 . This is a highly non-trivial term, and so may not vanish. 5. The model case 2 (changing the CR structure)

√ We displace this hypersurface by the Hamiltonian flow, generated by −1h, √ where h is a real valued C ∞ function. ωα β (ρ, −1h) denotes the corresponding curvature for this displacement (actually, in our case, there is √ only one component), ω1 1 (ρ, −1h). We would like to determine the prin√ cipal part of the liner term of ω1 1 (ρ, −1h) with respect to h. In this case, dimC 0 T  = 1. So, our CR structure is generated by L. √ [L, L]ξ = −1ξ, where ξ =

∂ ∂z2

+

∂ . ∂z2

So, X√−1h = (Lh)L. That is to say, √ Z√−1h = −1hξ + (Lh)L.

Now we consider the deformation of this CR structure. The case CR Hamiltonian preserves 0 T  + 0 T  (ξ part does not appear). This is not a trivial result. But, infinitesimally, it is obvious. And we are looking at the linear term. And so we can assume that our deformation is of type; Lψ = L + ψ · L .

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T  = {complex vectors generated by Lψ }

And consider the scalar curvature for (M, ψ T  ). Here, ψ is a C ∞ function and L is the same as in Section 4. As mentioned, our Lψ is still the unit vector with respect to mod | ψ |2 . And ψ

∇Lψ Lψ = ∇ρLψ ρL

ψ

= [ρLψ , ρL ]ψ T  = [ρ(L + ψL ), ρ(L + ψL )]ψ T   −(Lψ ψ)Lψ + lower order derivatives  −(L ψ)Lψ + lower order derivatives Here  means mod | ψ |2 . We see ∇Lψ Lψ . Similarly, ∇Lψ Lψ  (L ψ)Lψ Let ω1 1 (ρ, ψ) denote the corresponding the connection 1-form. By these computations, 

ω1 1 (ρ, ψ)(Lψ )  −(L ψ), ω1

1

(ρ, ψ)(Lψ )



 L ψ.

(5.5) (5.6)

(Compare with (4.3), (4.4).) While, ξ-direction, we have the following lemma. Lemma 5.1. The principal part of the linear term of ω1 1 (ρ, ψ)(ξ) with respect to ψ, is zero. Proof. By the definition of the connection 1-form, ω1 1 (ρ, ψ)(ξ)(L + ψL) = [ξ, L + ψL]ψ T  . Here [ξ, L + ψL]ψ T  means the ψ T  part of [ξ, L + ψL] according to the C ∞ vector bundle decomposition C ⊗ T M =ψ T  +ψ T  + C ⊗ ξ While, [ξ, L + ψL] = (ξψ)L. We determine ψ T  part. There are C ∞ functions A,B satisfying: (ξψ)L = A(L + ψL) + B(L + ψL).

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A(L + ψL) is the ψ T  part. While, by the type of vectors, Aψ + B = ξψ A + Bψ = 0. As A = −Bψ = −(ξψ − Aψ)ψ, A ≡ 0 mod {ξψψ, ψψ}. So in our case, this trouble-term does not appear in the linear term. Let R(ρ, h) denotes the scalar curvature for this CR structure (the contact form ρ12 θ is fixed). Then, in the real 3-dimensional case, the scalar curvature R(ρ, h) is dω(ρ, h)1 1 (L , L ) in mod | h |2 . Furthermore, by the definition of the CR Hamiltonian flow, ψ  ∂T  Z√−1h (L ) = L L h Proposition 5.2. The principal part of the linear part of R(ρ, h) with respect to h is −ρ(LLLL + LLLL)h. For this part, we have the following main theorem. Theorem 5.3. LLLL + LLLL is subelliptic. Proof. It suffices to show: (LLLL + LLLL)h ≥ cLLh, cLLh, cLLh2 , cLLh2 , cξh2 , for a real valued compactly supported C ∞ function h. Here c is a positive constant. √ We remember [L, L]ξ = −1ξ. For a real valued C ∞ function h, which is supported, compactly, we compute ((LLLL + LLLL)h, h). First, as for LLh2, LLh2 , this is obvious. For the other terms, we modify (LLLLu, u). (LLLLh, h) = (L[L, L]Lh, h) + (LLLLh, h) √ = (L −1ξLh, h) + LLh2 √ = ( −1ξLLh, h) + +LLh2 . For (LLLLh, h), (LLLLh, h) = (LLLLh, h) + (L[L, L]Lh, h)

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√ = LLh2 + (−L −1ξLh, h) √ = LLh2 + (− −1ξLLh, h) While

√ √ ( −1ξLLh, h) + (− −1ξLLh, h)

√ is ( −1ξ[L, L]u, u). That is to say √ √ ( −1ξ( −1ξ)h, h) = (−ξ 2 h, h) = (ξh, ξh). So, we have estimated LLh2 , LLh2 , ξh2 . Hence we have our theorem.

This theorem suggests that in our model case (the contact structure is ρ12 θ), if ρ is sufficiently close to 1, then there is a CR Hamiltonian flow, generated by a real valued C ∞ function h, satisfying that the corresponding scalar curvature vanishes, that is to say, R(ρ, h) = 0. For the proof, we have to see the non-linear part more carefully. But, I expect that in the more general setting, more complete theorem might be obtained, and so, in a future paper, we will discuss this problem. References T. Akahori, Complex analytic construction of the Kuranishi family on a normal strongly pseudoconvex manifold, Publ. Res. Inst. Math. Sci. 14 (1978), 789-847. A2. T. Akahori, The new Neumann operator associated with deformations of strongly pseudo convex domains and its application to deformation theory, Inventiones Mathematicae 68 (1982). 317-352, A3. T. Akahori, The new estimate for the subbundles Ej and its application to the deformation of the boundaries of strongly pseudo convex domains, Inventiones Mathematicae 63 (1981), 311–334. A4. T. Akahori, The notion of CR Hamiltonian flows and the local embedding problem of CR structures, Nova Publisher, 79–94, ISBN 978-1-60741-0119. AGL1. T. Akahori, P. M. Garfield, and J. M. Lee, Deformation theory of fivedimensional CR structures and the Rumin complex, Michigan Mathematical Journal, 50 (2002), 517–549. Mi. K. Miyajima, CR construction of the flat deformations of normal isolated singularities, J. Algebraic Geometry, 8 (1999), 403-470. A1.

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Ku.

Ku1. J-L1.

M. Kuranishi, Application of ∂ b to deformations of isolated singularities, Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975, 97–106, Amer. Math. Soc. , Providence, RI. 1977. M. Kuranishi, Local geometry of nondegenerate CR structures (Mexico, 1983), 1–36, Res. Notes in Math. 112, Pitman, Boston, MA 1985. D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Ymabe problem, J. Differential Geometry 29 (1989), 303-343.

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An example of the reduction of a single ordinary differential equation to a system, and the restricted Fuchsian relation Kana Ando Department of Mathematics, Chiba University, Chiba, 263-8522, Japan [email protected] In the paper [AK], we considered the reduction of a single linear differential equation which has a finite number of regular singular points and an irregular singular point to a system of linear differential equations, where all of the regular singular points are mutually distinct. Furthermore in the paper [A], we considered such a reduction problem, in which regular singular points are not necessarily distinct. In this paper, we shall review our method of reduction, and we shall describe an explicit example of our reduction, we shall compare and compute the restricted Fuchsian relations of the single differential equation and the system of differential equations. Keywords: regular singular point, irregular singularity, differential equation AMS classification numbers: 34M25

1. Introduction In this paper, we shall introduce an example of a reduction from a single differential equation with a finite number of regular singular points and one irregular singular point to a system of linear differential equations whose coefficients are polynomials of degree one. We also compute and compare the corresponding Fuchsian relations, which are an important step in understanding the Stokes phenomenon. In particular, the determination of a Fuchsian relation can be used to solve of the connection problem, which concerns the relationship between the asymptotic expansions of solutions of differential equations in different sectors. In the paper [AK], we considered the reduction from a linear differential equation which has a finite number of regular singular points and one irregular singular point of rank one at infinity in the complex projective line to a system of linear differential equations whose coefficients are polynomials of degree one. In the paper [A], we considered a more general reduction problem, where we relaxed the conditions on the regular singularities. In

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both papers, we showed that the system we obtain can be reduced to a generalized Schlesinger system. This result is promising, because it is intimately related with the connection problem. Indeed, in the last section of the paper [K2], Kohno sketched a method for finding the solutions of a multi-point connection problem for a generalized Schlesinger system. In order to solve the connection problem, we need a Fuchsian relation, and such a relation is found in [H]. In general, a Fuchsian relation is given by the sum of the characteristic exponents of all of the solutions for differential equations, but for the purposes of solving the connection problem, we are interested only in the exponents for nonholomorphic solutions. Therefore, we shall call this restricted sum the restricted Fuchsian relation. In the second section, we shall review the reduction that we proved in [AK] and [A]. In the third section, we shall describe an example of our reduction from a single differential equation to a generalized Schlesinger system. We believe that an explicit example is valuable for understanding our algorithm of the reduction. Furthermore, it has been helpful in our work on the multi-point connection problem, which should appear in the future. In the fourth section, we shall give an algebraic equation for determinating the characteristic exponents for the regular singular points of both a single differential equation and a system of differential equations. We find that the difference between the sum of the characteristic exponents of all nonholomorphic solutions at regular singular points for a single differential equation and a system of differential equations is always an integer. For that purpose, it is enough to know the algebraic equation which the characteristic exponents satisfy because of the relation of solutions and coefficients. This fact means that our reduction preserves monodromic properties of the solutions for differential equations at the regular singular points. We shall show that the nonholomorphic solutions for the single differential equation satisfy the restricted Fuchsian relation, by employing a method of Kohno in the last section. That is, we shall take a sum of the characteristic exponents obtained in the fourth section, and the characteristic exponents for irregular singular points. We recall the definition of characteristic exponents and constants, and the Fuchisan relation. For the rest of this paper, we assume that t is a complex variable. Consider an n-th order single linear differential equation, with unknown function y, of the form: dn y dn−1 y dy = Pn−1 (t) n−1 + · · · + P1 (t) (1.1) + P0 (t)y, n dt dt dt with a finite number of regular singular points at t = λν (ν = 1, 2, . . . , q for Pn (t)

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some 1 ≤ q ≤ n) and one irregular singular point of rank one at t = ∞ in the complex projective line, such that each coefficient Pj (t) (j = 0, 1, . . . , n − 1, n) is a complex polynomial of degree at most n. It is well-known that under these assumptions the roots of Pn (t) are the regular singular points. Furthermore, in a punctured disc 0 < |t − λν | < r := min{|λν − λi | : i = ν, i = 1, 2, . . . , q} (ν = 1, 2, . . . , q), there exists at least one solution of (1.1) of the form y(t) = (t − λν )ρ

∞ 

g(m) (t − λν )m ,

g(0) = 0,

(1.2)

m=0

and we can easily see that the t = λν (ν = 1, 2, . . . , q) are regular singular points of the fundamental solutions. We call ρ the characteristic exponent of the solution y. If ρ is a positive integer, the solution is holomorphic at λν and if not, the solution is non-holomorphic at λν . At the irregular singular point t = ∞, we can find formal solutions of (1.1) of the form μt η

y(t) = e t

∞ 

h(s)t−s ,

h(0) = 0.

(1.3)

s=0

We call η the characteristic exponent and μ the characteristic constant of this formal solution. We are interested in the sum of characteristic exponents over all non-holomorphic solutions. 2. Method of the reduction We review the method of our reduction of the papers [AK] and [A]. We recall the differential equation (1.1): dn y dn−1 y dy + P0 (t)y. = P (t) + · · · + P1 (t) n−1 dtn dtn−1 dt We now assume that Pn (t)

Pn (t) = (t − λ1 )n1 (t − λ2 )n2 · · · (t − λq )nq

(2.4)

with n1 + n2 + · · · + nq = n (1 ≤ nq ≤ nq−1 ≤ · · · ≤ n1 ≤ n). In order for t = λν (ν = 1, 2, . . . , q) to be a regular singularity of (1.1), it is necessary and sufficient for the functions (t − λν )i Pn−i (t) Pn (t)

(i = 1, 2, . . . , n)

to be holomorphic at t = λν . Hence, for each ν, the polynomials Pn−i (t) (1 ≤ i ≤ nν ) have the factor (t − λν )nν −i . It is therefore easy to see that

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the coefficients Pn−i (t) are written as: ⎧ 

q ⎪ nν −i  ⎪ Pn−i (t) = (t − λν ) Pn−i (t) (0 < i ≤ nq ), ⎪ ⎪ ⎪ ⎪ ν=1 ⎨

 k nν −i  ⎪ Pn−i (t) (nk+1 < i ≤ nk ; k = 1, 2, . . . , q), P (t) = (t − λ ) ν ⎪ ⎪ n−i ⎪ ⎪ ν=1 ⎪ ⎩ (n1 < i ≤ n) Pn−i (t) = Pn−i (t) (2.5) where Pn−i (t) is a polynomial for all i = 0, 1, . . . , n and nq+1 is equal to 1. The notation Pn−i (t) will appear again in the fourth section. In the paper [A], we considered the reduction of the single differential equation (1.1) to the system of differential equations, with unknown length n vector function Y dY = (A + C t) Y, (2.6) (tI − B) dt where I is the n by n identity matrix, A is an n by n constant matrix, C is an n by n constant lower triangular matrix, and n1

n2

nq

      B = diag( λ1 , · · · , λ1 , λ2 , · · · , λ2 , · · · , λq , · · · , λq ). In order to reduce (1.1) to (2.6), we apply the transformation: ⎧ ⎪ y1 = y, ⎪ ⎪ ⎪ ⎪ ⎪ y2 = ϕ1 y  + e2, 0 (t) y, ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎨ . ⎪ yj = ϕj−1 y (j−1) + ej, j−2 (t) y (j−2) + · · · + ej, 1 (t) y  + ej, 0 (t) y, ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩y = ϕ (n−1) + en, n−2 (t) y (n−2) + · · · + en, 1 (t) y  + en, 0 (t) y, n n−1 y (2.7) where we use the following notation: • We define {yj }nj=1 by using the unknown function y. • We define ϕj by the following: ϕj =

j

(t − λσ(k) ),

(j = 1, 2, . . . , n),

k=1

where σ : {1, 2, . . . , n} → {1, 2, . . . , q} is the unique weakly increasing function such that σ −1 (i) has cardinality ni for all 1 ≤ i ≤ q.

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• ej, j−i (t)(j = 2, . . . , n, i = 1, 2, . . . , j) is a polynomial in t. • y  and y (j) (j = 1, 2, . . . , n) mean the first and jth derivatives of y with respect to t, respectively. By using (2.7), we can derive the system of linear differential equations for the column vector Y = (y1 , y2 , . . . .yn )t : ⎞ ⎞ ⎛ ⎛ d1,1 1 t − λσ(1) ⎟ ⎟ ⎜ d2,1 d2, 2 1 ⎜ t − λσ(2) ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ . . . . . dY .. . . . . .. ⎟ ⎟ Y, ⎜ .. ⎜ = ⎟ dt ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ . ⎜ . . . .. .. 1 ⎠ .. ⎠ ⎝ .. ⎝ dn,1 dn,2 · · · · · · dn,n t − λσ(n)

0

0

0

where all dj,i = dj, i (t) are degree one polynomials, and Y = (y1 , y2 , . . . , yn)t means the transpose of the indicated row vector. This system is our candidate for the reduction from (1.1) to (2.6). In other words, we define dj,i by reducing (1.1) using (2.7). Details will appear in the paper [A]. Theorem 2.1. We define the entries dj,i (t)(j = 1, 2, . . . , n, i = 1, 2, . . . , j) of the matrix (A + Ct) in (2.6) by applying the transformation (2.7) to (1.1). Then we obtain the following relation: (t − λj ) (ej, j−−3 (t) + ej, j−−2 (t)) = ej+1, j−−2 (t) +

+1 

dj, j−h (t) ej−h, j−−2 (t)

(2.8)

h=0

(j = 1, 2, . . . , n ;  = −1, 0, . . . , j − 2) where • • • •

en+1, k (t) = −Pk (t) for k = 0, 1, . . . , n − 1, ej, −1 (t) ≡ 0, ej, j−−2 (t) means the first derivative with respect to t, all dj, i (t) are polynomials of degree one.

We will not describe the proof in detail, but we remark that it follows from comparing coefficients of t in (2.8). See [AK] and [A] for details. We showed that after multiplying by (tI − B)−1 on the left side of (2.6), the equation is reduced to the following generalized Schlesinger system:  q   A¯i dY = +C Y dt t − λi i=1

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where A¯i (i = 1, 2, . . . , q) are n by n constant matrices. In future work, we intend to use this result to find the solutions of a multi-point connection problem (see [K2]). 3. Example of the reduction In this section, we consider an example of the reduction of a fourth order linear differential equation. We apply our theory, with unknown function y to the following equation t (t − 1)3 y (4) = P3 (t) y (3) + P2 (t) y  + P1 (t) y  + P0 (t) y, where

⎧ ⎪ P3 (t) ⎪ ⎪ ⎪ ⎨P (t) 2

⎪ P1 (t) ⎪ ⎪ ⎪ ⎩ P0 (t)

(3.9)

= − (t − 1)2 (5 t + 1), = − (t − 1)2 (t − 4), = t4 − t3 − t2 + 21 t − 8, = 4 t3 − 3 t2 + 4 t + 3.

Obviously, this linear differential equation has regular singularities at t = 0, 1 and an irregular singular point of rank one at infinity. Now, we consider the reduction of (3.9) to a system of linear differential equations of the form (t I − B) Y  = (A + C t) Y by the transformation ⎧ ⎪ y1 = y, ⎪ ⎪ ⎪ ⎨y = ϕ y  + e (t) y, 2 1 2, 0  ⎪ y3 = ϕ2 y + e3, 1 (t) y  + e3, 0 (t) y, ⎪ ⎪ ⎪ ⎩ y4 = ϕ3 y (3) + e4, 2 (t) y  + e4, 1 (t) y  + e4, 0 (t) y, where I is the 4 × 4 identity matrix, A and C are 4 by 4 constant matrices with C lower triangular, B is a diagonal matrix: B = diag(1, 1, 1, 0), {yj }4j=1 are defined using the unknown function y from (3.9), ϕ1 = t − 1, ϕ2 = (t − 1)2 , ϕ3 = (t − 1)3 , and the coefficients ei, j (t) (i = 2, 3, 4, j = 0, . . . , i − 2) are polynomials in t. Then, we have the following relations: 1  2  3  4 

t (e4, 2 (t) + ϕ3 ) = −P3 + d4, 4 ϕ3 , t (e4, 1 (t) + e4, 2 (t)) = −P2 + d4, 4 e4, 2 (t) + d4, 3 ϕ2 , t (e4, 0 (t) + e4, 1 (t)) = −P1 + d4, 4 e4, 1 (t) + d4, 3 e3, 1 (t) + d4, 2 ϕ1 , t e4, 0 (t) = −P0 + d4, 4 e4, 0 (t) + d4, 3 e3, 0 (t) + d4, 2 e2, 0 (t) + d4, 1 ,

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(t − 1) (e3, 1 (t) + ϕ2 ) = e4, 2 (t) + d3, 3 ϕ2 , (t − 1) (e3, 0 (t) + e3, 1 (t)) = e4, 1 (t) + d3, 3 e3, 1 (t) + d3, 2 ϕ1 , (t − 1) e3, 0 (t) = e4, 0 (t) + d3, 3 e3, 0 (t) + d3, 2 e2, 0 (t) + d3, 1 , (t − 1) (e2, 0 (t) + ϕ1 ) = e3, 1 (t) + d2, 2 ϕ1 , (t − 1) e2, 0 (t) = e3, 0 (t) + d2, 2 e2, 0 (t) + d2, 1 , − e2, 0 (t) = d1, 1 .

In the above, dj, i are polynomials of degree 1 in t. Avoiding the difficult calculation, we shall show just the order of steps in our reduction algorithm. First, we calculate the principal diagonal elements di, i (t) (i = 4, 3, 2, 1). 1 5 8 → . 10 →  We follow the order of calculation → Next, we shall proceed to the calculation of the first subdiagonal ele2 6 9 ments di, i−1 (t) (i = 4, 3, 2) by following → → . In order to determine the second subdiagonal elements di, i−2 (t) (i = 3 7 4, 3), we follow the order of calculations → . 4 we obtain the value of d4, 1 (t). Lastly, from  We have thus determined all coefficients ei, j (t) of the transformation and the elements of dj,i (t) as follows: ⎧ ⎪ e2, 0 (t) = −ω 2 (t − 1), ⎪ ⎪ ⎪ ⎪ ⎪ e3, 0 (t) = (t − 1) (t − ω), ⎪ ⎪ ⎪ ⎨e (t) = (t − 1)2 , 3, 1

⎪ e4, 0 (t) ⎪ ⎪ ⎪ ⎪ ⎪e4, 1 (t) ⎪ ⎪ ⎪ ⎩ e4, 2 (t) ⎧ ⎪ ⎪ ⎪d1, 1 ⎪ ⎪ ⎪ d2, 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d2, 2 ⎪ ⎪ ⎪ ⎪ ⎪ d3, 1 ⎪ ⎪ ⎪ ⎨d 3, 2

⎪ d3, 3 ⎪ ⎪ ⎪ ⎪ ⎪ d4, 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d4, 2 ⎪ ⎪ ⎪ ⎪ ⎪ d4, 3 ⎪ ⎪ ⎪ ⎩d 4, 4

= − t3 + 3 t2 + (ω − 3) t − (ω + 5), = 3(t − 1)(t − 3), = 3(t − 1)2 , = ω 2 (t − 1), = (ω − 1) (t − 1), = ω (t − 1) + 1, = − (9 ω + 4) (t − 1) + 6, = − (ω − 1)(t − 1) + 6, = (t − 1) − 1, = − (17 ω + 12) t + (18 ω + 26), = (ω + 2) t + 18, = 2 t + 1, = 1,

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where ω is a non-real root of ω 3 − 1 = 0. Consequently, we can reduce the single linear differential equation (3.9) to a system of linear differential equations of the form dY ¯ = (A + C t) Y = {diag ( (t − 1)In1 , t ) C + A}Y, (3.10) dt where for the rest of this paper, Iν denotes the ν by ν identity matrix, and A, A¯ and C are the constant matrices given as follows: ⎞ ⎛ 1 0 0 − ω2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ − (ω − 1) − (ω − 1) 1 0 ⎟ ⎟ ⎜ ⎟, A = ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 9 ω + 10 ω + 5 − 2 1 ⎟ ⎟ ⎜ ⎠ ⎝ (t I − B)

18 ω + 26

18

1 1

⎛ ⎛

a ¯1 ⎜a ¯ 2, 1 A¯ = ⎜ ⎝a ¯3, 1 a ¯4, 1

1 a ¯2 a ¯3, 2 a ¯4, 3

0 1 a ¯3 a ¯4, 3

0

⎜ ⎞ ⎜ ⎜ 0 ⎜ ⎜ 0⎟ ⎟=⎜ 1⎠ ⎜ ⎜ ⎜ a ¯4 ⎜ ⎝

1

0 0

⎞

⎟ ⎟ ⎟ 1 1 0⎟ ⎟ ⎟, ⎟ ⎟ 6 −1 1⎟ ⎟ ⎠

0 6

18 ω + 26 18 1 1 and

⎛

ω2

0

00

− 17 ω − 12

ω +2

20

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ω 0 0⎟ ⎜ ω −1 ⎟ ⎜ ⎟. C = ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ − (9 ω + 4) − (ω − 1) 1 0 ⎟ ⎟ ⎜ ⎠ ⎝ We will describe how to compute the entries of A¯ in the next section. 4. Characteristic exponents and constants We shall investigate the characteristic exponents for the regular singular points and the characteristic constants for the irregular singular point for

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the differential equation (1.1) and the output (2.6) of the reduction. In order to show that our reduction preserves monodromic properties of the solutions for differential equations at the regular singular points, we will show that the difference between the sum of the characteristic exponents for the regular singular points of (1.1) and the corresponding sum for (2.6) is an integer. The calculation of this difference will appear in the proposition at the end of this section. In the reduction from (1.1) to (2.6), all entries ej, i (t) of the transformation matrix are polynomials in t. We shall show here that the characteristic exponents at each regular singular point of both (1.1) and (2.6) are invariant modulo integers. We recall the notation of [A] for obtaining the restricted Fuchsian relation of (1.1) and (2.6). We introduce the notation Nk , fki and ψk where k = 1, 2, · · · , q and i is an integer index (not an exponent). ⎧ ⎪ Nk = n1 + n2 + · · · + nk (k = 1, 2, . . . , q), ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪ ⎨f i = (t − λν )nν −i , k ν=1 ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ ψ = (t − λν ), ⎪ k ⎩ ν=1

where N0 ≡ 0, then we know that  Nq = fki

n, fki+1 ψk .

=

Then, we can rewrite the coefficients (2.5) of (1.1) as follows:  Pn−i (t) = fki Pn−i (t) (nk+1 < i ≤ nk ; k = 1, 2, . . . , q), Pn−i (t) = Pn−i (t) (n1 < i ≤ n) where nq+1 ≡ 0. As we saw in the introduction, in the punctured disc 0 < |t − λν | < r(ν = 1, 2, . . . , q), there exists at least one solution of (1.1) y(t) = (t − λν )

ρ

∞ 

g(m) (t − λν )m .

m=0

Substituting it into (1.1), we find that the characteristic exponent ρ is a root of the equation [ρ]n =

nν  i=1

γi [ρ]n−i ,

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where [ρ]k is the Pochhammer symbol for k = 0, 1, 2, . . . , defined by the following recursion: [ρ]k = ρ(ρ − 1) · · · (ρ − k + 1),

[ρ]0 ≡ 1,

and the coefficients γi are given by 

Pn−i (t) (t − λν )i γi = Pn (t) t=λν =

Pn−i (λν ) q k−1 i (λν − λ ) (λν − λ )n =1 =ν

(nk < i ≤ nk−1 ≤ nν ).

=k

Then ρ is a root of [ρ]n−nν = 0,

(4.11)

or a root of [ρ − n + nν ]nν =

nν 

γi [ρ − n + nν ]nν −i .

(4.12)

i=1

For (1.1), (4.11) and (4.12) imply that there exist n − nν holomorphic solutions and nν possibly nonholomorphic solutions in the punctured disc 0 < |t − λν | < r. Next, we shall consider the characteristic exponents and constants for (2.6). For (2.6), there also exist (n − nν )(ν = 1, 2, . . . , q) holomorphic solutions and nν possibly nonholomorphic solutions near each singular point ¯j := aj − λk cj , t = λν . For the calculation, we shall rewrite (2.6) by setting a and a ¯j,i := aj,i − λk cj,i where aj and aj,i are entries of A and cj and cj,i are entries of C of (2.6). aj,i and cj,i are the (j, i)-entries of A and C, respectively, and aj := aj,j and cj := cj,j are j-th diagonal entries. That is, we obtain the formula: ⎧ ¯j , ⎨ dj, j (t) = cj t + aj = cj (t − λk ) + a ¯j, i dj, i (t) = cj, i t + aj, i = cj, i (t − λk ) + a ⎩ (Nk−1 < j ≤ Nk ; k = q, q − 1, . . . , 1, i = 1, 2, . . . , j − 1). Then we rewrite the right hand side of (2.6) in the form ¯ {diag((t − λ1 )In1 , (t − λ2 )In2 , · · · , (t − λq )Inq )C + A}Y,

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where C is a lower triangular constant matrix, and A¯ is of the form ⎛ ⎞ a ¯1 1 ⎜a ⎟ ¯2 1 ⎜ ¯2, 1 a ⎟ ⎜ ⎟ .. .. ⎟. . . A¯ = ⎜ a ¯ a ¯ ⎜ 3,1 3,2 ⎟ ⎜ . ⎟ . . . .. 1 ⎠ .. .. ⎝ .. a ¯n, 1 a ¯n, 2 · · · a ¯n,n−1 a ¯n

0

In [A], for finding the order of ej, j−i (t), we showed that for k = 1, 2, . . . , q ej, j−i (t) = fki−1 (t − λk )j−Nk ej, j−i (t)

( Nk−1 < j ≤ Nk , i = 1, 2, . . . , j − 1)

where (t − λk ) ≡ 1 (p ≤ 0) and ej, j−i (t) is a polynomial of t, and that for 1 ≤ i ≤ nν we obtained the explicit form of A¯ as follows: ⎛ ⎞ 0 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . . ¯ ⎜ ⎟, .. .. Aν = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ nν − 2 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ p

0

0

¯ Nν , Nν − 1 + 2 · · · · · · a ¯ Nν , Nν − 1 a ¯N ν a ¯ Nν , Nν − 1 + 1 a ¯ and where A¯ν is the ν by ν block-diagonal matrix of A, a ¯Nν , Nν −i+1 = −

eNν +1, Nν −i (λν ) , (ψν−1 (λν ))i

(4.13)

a ¯Nν = nν − 1 + a ¯Nν ,Nν . The characteristic exponents of nonholomorphic solutions of (2.6) are given by eigenvalues of the constant matrix A¯ν . Since the matrix A¯ν is a companion matrix, the eigenvalues are roots of the equation [ ρ ]nν =

nν 

a ¯Nν , Nν − i + 1 [ ρ ]nν − i .

(4.14)

i=1

Now we shall show that the reduction described above preserves characteristic properties.

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Proposition 4.1. The sum of the characteristic exponents of nonholomorphic solutions of (2.6) differ from the sum of the characteristic exponents of nonholomorphic solutions of (1.1) at each regular singular point t = λν only by the integers nν Nν − 1 . Proof. We define ρ := ρ − n + Nν . Substituting ρ + n − Nν into ρ of (4.14), we find that from (4.13), for ν = 1, 2, . . . , q [ ρ ]nν =

nν 

γi [ ρ ]nν − i ,

(4.15)

i=1

but the proof is an induction argument that we omit. By the relation between the coefficients of a polynomial and its roots for (4.15), (4.12), and (4.14) we have thus verified that: nν 

ρν, −

=1

nν 

ρν, = nν (n − nν )

=1

and nν 

ρν, −

=1

nν 

ρν, − = nν (−n + Nν ).

=1

By adding the two formulas above, we obtain the desired equation: nν  =1

ρν, −

nq 

ρν, = nν Nν−1 .

=1

This proposition means that the transformation treated between (1.1) and (2.6) in this paper preserves the monodromic properties. We remark that there is a fundamental set of solutions to (2.6) with the form (1.2) in the punctured disk 0 < |t−λν | < r := min{|λν −λi | : i = ν, i = 1, 2, . . . , q} (ν = 1, 2, . . . , q) where g(m) is a nonzero n-entry column vector. There also exists a formal solution of (2.6) at the irregular singular point with the form (1.3) where h(s) is a nonzero n-entry column vector. Like the case of a single differential equation, we call the numbers ρ, η characteristic exponents and μ characteristic constants.

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5. Restricted Fuchsian relation Lastly, we shall explain an important identity, which necessarily exists among characteristic exponents for (1.1) and plays an essential role in the global analysis of linear differential equations with regular or irregular singularities. As in the previous section, in the punctured disc 0 < |t − λν | < r (ν = 1, 2, . . . , q), there exist nν non-holomorphic solutions of (1.1) yν,  (t) = (t − λν )ρν, 

∞ 

gν,  (m) (t − λν )m

( = 1, 2, . . . , nν ),

m=0

where the characteristic exponent ρν,  are roots of the characteristic equation [ρ − n + nν ]nν =

nν 

γν, i [ρ − n + nν ]nν −i .

(5.16)

i=1

The coefficients γν, i are given by

 Pn−i (t) (t − λν )i γν, i = . Pn (t) t=λν

(5.17)

We now consider the sum of all characteristic exponents ρν,  . From the characteristic equation (5.16), we immediately obtain nν 

ρν,  =

=1

nν 

(n − ) + γν, 1

=1

= n nν −

nν (nν + 1) + γν, 1 . 2

Then we have q nν   ν=1 =1

 q  q q  1  1 2 n− nν − nν + γν, 1 2 ν=1 2 ν=1 ν=1   q q  1 1 2 = n− n + γν, 1 . n− 2 2 ν=1 ν ν=1

ρν,  =

(5.18)

In order to calculate the last sum in the above formula, we apply the fact 

Pn−1 (t) (t − λν ) γν, 1 = . Pn (t) t=λν

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26 (t) The right hand side is the residue of PPn−1 dt at t = λν . We can then n (t) express the sum of γν, 1 in the form  q  Pn−1 (t) 1 dt, (5.19) γν, 1 = 2π i |t|=R Pn (t) ν=1

where all t = λν (ν = 1, 2, . . . , q) are included in the disk |t| < R for sufficiently large R, and the path of integration is oriented counterclockwise. According to the theory of residues for rational functions, the right hand side of (5.19) is equal to minus one times the value of the residue at infinity. If we denote the polynomial Pn− (t) of degree n by Pn− (t) =

n 

pn−, j tj

( = 0, 1, . . . , n),

j=0

then for sufficiently large values of t the integrand can be written as follows: Pn−1 (t) pn−1, n + pn−1, n−1 t−1 + · · · + pn−1, 0 t−n = Pn (t) 1 + pn,n−1 t−1 + · · · + pn, 0 t−n = (pn−1, n + pn−1, n−1 t−1 + · · · + pn−1, 0 t−n )(1 − pn, n−1 t−1 + · · · ) = pn−1, n + (pn−1, n−1 − pn−1, n pn, n−1 )t−1 + · · · . Consequently, we have q 

γν, 1 = pn−1, n−1 − pn−1, n pn, n−1 .

(5.20)

ν=1

Now we shall investigate the characteristic exponents of formal solutions for (1.1) at the irregular singularity t = ∞, which are expressed in the form y(t) = eμ t tη

∞ 

h(s) t−s ,

s=0

where we assume that h(0) = 0, and we define h(−s) ≡ 0 when s is a positive integer. We begin with some preparative calculations for finding the characteristic exponent η, following the method in the paper [K1]. We define yk (t)(k = 0, 1, . . . , n) to be the kth derivative of y(t) with respect to t: dk y(t) yk (t) = , dtk and we shall write the coefficients of the formal series hk (s), that is yk (t) = eμ t tη

∞  s=0

hk (s) t−s

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where y0 (t) ≡ y(t). Then, we have the following result:  Lemma 5.1. From yk (t) = yk−1 (t)(k = 1, 2, . . . , n), the relation

hk (s) = μ hk−1 (s) + (η − s + 1) hk−1 (s − 1)

(s = 0, 1, . . .)

(5.21)

holds, where hk (−s) = 0 for s > 0. Proof. yk (t) =

 yk−1 (t)

⇔e

μt η

t

∞ 

 −s

hk (s) t

μt η

=e t

s=0

+

∞ 

⇔

(η − s)hk−1 (s)t−s−1 

hk (s) t

−s

=

s=0

+

∞ 

hk−1 (s)t−s

s=0



s=0 ∞ 

μ

∞ 

μ

∞ 

hk−1 (s)t−s

s=0

(η − s)hk−1 (s)t



−s−1

.

s=0

Comparing the coefficients of t−s , we have the above formula. Moreover, substituting tj yk (t) = eμ t tη

∞ 

hk (s + j) t−s

s=−j

into (1.1), we have   n n−1   pk, j hk (s + j) = 0, pn, j hn (s + j) − j=0

(s = −j, −j + 1, . . .).

k=0

(5.22) With this preparation complete, we are now in a position to calculate the value of the characteristic constants μ and the characteristic exponents η of the formal solutions of (1.1). To this end, we iteratively apply (5.21) to get the k + 1-term sum (see [K1]): hk (s) = μk h(s) + {μk + (η − s + 1)k μk−1 } h(s − 1) + · · · .

(5.23)

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We then apply (5.22) with the substitution s = −n. Then, from (5.23), we obtain   n−1  μn − pk, n μk h(0) = 0. k=0

Hence, the equation J(μ) ≡ μn −

n−1 

pk, n μk = 0

k=0

determines the n characteristic constants, which we shall denote by μ ( = 1, 2, . . . , n). From here, we assume that they are mutually distinct, i.e., μ = μi ( = i). Without this assumption, the argument becomes more complicated, because we can no longer use the assumption that J  (μ ) = 0 to produce (5.24). Next, we substitute s = 1 into (5.23) to obtain hk (1) = μk h(1) + {μk + η k μk−1 } h(0). Then combining this with what we get from substituting s = −n + 1 into (5.22), we obtain J(μ) h(1) + {J(μ) + η J  (μ)} h(0)  +

pn, n−1 μ − n

n−1 

 pk, n−1 μ

k

h(0) = 0,

k=0

whence the characteristic exponent corresponding to μ is given by the formula n−1 pn, n−1 μn − k=0 pk, n−1 μk . (5.24) η = − J  (μ ) By exactly the same consideration as in the case (5.19), we can express the sum of the characteristic exponents (5.24) in the form of the integral n−1  n  pn, n−1 μn − k=0 pk, n−1 μk 1 dμ ηi = − 2π i |μ|=R J(μ) i=1 for sufficiently large R, with the path of integration oriented counterclockwise. From the residue theorem we obtain n  ηi = pn−1, n−1 − pn−1, n pn, n−1 . (5.25) i=1

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Combining this formula with (5.20) and (5.18), we consequently obtain the restricted Fuchs relation. Theorem 5.2. Consider (1.1), and set ρν, (ν = 1, 2, . . . , q,  = 1, 2, . . . , nν ) to be the characteristic exponents at regular singular points t = λν and ηi (i = 1, 2, . . . , n) to be the characteristic exponents at the irregular singular point at infinity. Assume the characteristic constants of the formal solutions at the irregular singular point are mutually distinct, i.e., μ = μi ( = i). Then the restricted Fuchs relation for non-holomorphic solutions for (1.1) is the following:   q q nν n    1 1 2 ρν,  − ηi = n − n . (5.26) n− 2 2 ν=1 ν ν=1 i=1 =1

Remark 5.3. For the special case q = 1, we have n1 = n and hence . In particular, the above the right hand side of (5.26) is equal to n(n−1) 2 restricted Fuchs relation is a generalization of the lemma 3.1 in [K1].

References K. Ando, On the reduction of a single differential equation to a system of first degree differential equations, (submitted). AK. K. Ando, M. Kohno, A certain reduction of a single differential equation to a system of differential equations, Kumamoto Journal of Mathematics 19 (2006), 99–114. H. M. Hukuhara, Sur la relation de Fuchs relative a l’equation differentielle lineaire, Proc. Japan Acad., 34 (1958), 102-106. K1. M. Kohno, A Two Point Connection Problem for General Linear Ordinary Differential Equations, Hiroshima Mathematical Journal 4, No.2, (1974), 293-338. K2. M. Kohno, Global Analysis in Linear Differential Equations, (Mathematics and Its Applications (Kluwer)) Vol.471 (1999).

A.

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Fronts of weighted cones Toshizumi Fukui and Masaru Hasegawa Department of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan. [email protected] Departamento de Matem´ atica, Instituto de Ciˆ encias Matem´ aticas e de Computa¸c˜ ao, Universidade de S˜ ao Paulo - Campus de S˜ ao Carlos, Caixa Postal 668, 13560-970 S˜ ao Carlos, SP, Brazil [email protected] We present an asymptotic formula for the principal curvatures and principal directions of weighted cones near the vertex, which is defined by (0.1). As a byproduct, we construct a singular C 3 -surface with bounded principal curvatures. We also remark that when the weights are (1, 2, 2) we have focal curves, which is a generalization of focal conics we discussed in [3]. Keywords: singularity of front, weighted cone, principal curvature, principal directions AMS classification numbers: 53A05, 35A18

Wave fronts (or simply, fronts) is a locus of points in space reached by a wave or vibration at the same instant as the wave travels through a medium. Fronts may have singularities at some moment, and to investigate the singularity types is one of the main topics in the application of singularity theory. The classification of singularity types of generic fronts is known ([1, page 336]), and the local classification of bifurcations in generic one parameter families of fronts in 3-dimensional spaces are also given in [1, page 348]. Fronts of a regular surface in the 3-dimensional Euclidean space is described by parallel surfaces and criteria of its singularity type in terms of differential geometric languages are given in [2]. We also gave ([3]) a similar criteria for fronts of Whitney umbrella, which is only stable singularities type from a plane to the 3-dimensional Euclidean space R3 . Since there are singularities of natural objects in the real world, we are motivated

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to study singularities of fronts of singular surfaces in R3 . In this paper, we investigate criteria of singularity types of fronts of weighted cones g : R × S 1 → R3 , (r, s) → (rw1 γ1 (s), rw2 γ2 (s), rw3 γ3 (s)),

(0.1)

which has singularities along the locus defined by r = 0. Here γ : S 1 → R3 − {0}, s → (γ1 (s), γ2 (s), γ3 (s)),

(0.2)

is a space curve and (w1 , w2 , w3 ) is a triple of three positive integers whose greatest common divisor is one. We assume that γ is an immersion. Observe that we have another space curve γˆ : S 1 → R3 − {0}, s → ((−1)w1 γ1 (s), (−1)w2 γ2 (s), (−1)w3 γ3 (s)). (0.3) Then we have two cases to consider. • If γ(S 1 ) = γˆ(S 1 ), then the map g is generically one to one, and we set g = 1. • If γ(S 1 ) = γˆ (S 1 ), we may assume that γˆ (s) = γ(s + π), after suitable re-parametrization of γ, and the map g is generically two to one. We set g = 2. We also remark that the map g factors through the natural obius strip obtained by identifying map R × S 1 → M where M is the M¨ (r, s) with (−r, s + π). The goal of the paper is to give criteria of singularity types of fronts if g is defined by (0.1). In §1, we start with the homogeneous case, that is, Case (w1 , w2 , w3 ) = (1, 1, 1), since this case is simple and it is better to treat separately. In §2, we treat Case (w1 , w2 , w3 ) = (1, 1, 1). To investigate criteria of singularity types of fronts of g, we show a formula of principal curvatures (Theorem 2.3) and describe the behaviors of principal directions near singular locus {r = 0} (Theorem 2.9). These information describe how swallowtail singularities of g t appear and how many of them degenerate to g as t → 0 (see Remark 2.6). In §3, we define focal curves when (w1 , w2 , w3 ) = (1, 2, 2). This should be considered as analogy of focal conics of Whitney umbrella which is discussed in [3]. In §4, we discuss some examples. We construct a C 3 - singular surface (Example 4.1) so that the principal curvatures are bounded. 1. Fronts of cones First we consider Case (w1 , w2 , w3 ) = (1, 1, 1), that is, a cone defined by g : R × S 1 → R3 − {0}, (r, s) → rγ(s).

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Without loss of generality we may assume that γ is a spherical curve and s is the arc-length parameter. We then have that γ  (s), γ  (s) ≡ 1,

γ(s), γ(s) ≡ 1,

and

γ  (s), γ(s) = 0.

a1 = γ  (s), a2 = γ(s) × γ  (s), a3 = γ(s) form a frame, and we have ⎛ ⎞ ⎛ ⎞⎛ ⎞ 0 κg −1 a1 a1 ⎝a2 ⎠ = ⎝−κg 0 0 ⎠ ⎝a2 ⎠ 1 0 0 a3 a3 where κg (s) is the geodesic curvature. Since gr = γ(s) = a3 ,

gs = rγ  (s) = ra1 ,

a2 is a unit normal to the cone, and the first fundamental form is given by I = dr2 + r2 ds2 . Since grr = 0,

grs = γ  (s) = a1 ,

gss = rγ  (s) = r(κg a2 − a3 ),

the second fundamental form is given by II = rκg ds2 , and the principal curvatures are 0 and respectively. Now we define fronts of the cone by

κg r

with principal directions ∂r , ∂s ,

g t (r, s) = g(r, s) + ta2 = ra3 + ta2 . Since ∂g t ∂g t = a3 , = (r − tκg )a1 , ∂r ∂s s → (tκg (s), s) is the singular curve and tκg (s)∂r +∂s is its tangent (singular direction) and the null direction is ∂s . Then we have the following criteria, by the criteria of the singularities of front in [4]. • The singularity type of g t is cuspidal edge at (r0 , s0 ), if r0 = tκg (s0 ), κg (s0 ) = 0. • The singularity type of g t is swallowtail at (r0 , s0 ), if r0 = tκg (s0 ), κg (s0 ) = 0, κg (s0 ) = 0. 2. Weighted cones We consider the weighted cone defined by (0.1) with (w1 , w2 , w3 ) = (1, 1, 1). We assume that w1 ≤ w2 ≤ w3 .

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2.1. Unit normals and fundamental forms s of the weighted cone Lemma 2.1. The unit normal vector n = ggrr ×g ×gs  g(r, s) has the unique extension except the points (r, s) = (0, s0 ) where   w1 γ1 γ1   = 0 at s = s0 if w2 < w3 , or  •  w2 γ2 γ2      w γ γ   w γ γ   •  1 1 1  ,  1 1 1  = (0, 0) at s = s0 if w2 = w3 . w3 γ2 γ2 w3 γ3 γ3

Proof. Set e1 = (1, 0, 0), e2 = (0, 1, 0) and e3 = (0, 0, 1). Since gr (s, r) =(w1 rw1 −1 γ1 (s), w2 rw2 −1 γ2 (s), w3 rw3 −1 γ3 (s)), gs (s, r) =(rw1 γ1 (s), rw2 γ2 (s), rw3 γ3 (s)), we obtain

  w1 rw1 −1 γ1 (s) rw1 γ1 (s) e1    gr × gs = w2 rw2 −1 γ2 (s) rw2 γ2 (s) e2  w rw3 −1 γ (s) rw3 γ  (s) e  3 3 3 3    w1 γ1 (s) γ1 (s) r−w1 e1    =rw1 +w2 +w3 −1 w2 γ2 (s) γ2 (s) r−w2 e2  w γ (s) γ  (s) r−w3 e  3 3 3 3   w1 γ1 (s) γ1 (s) rw3 −w1 e1    =rw1 +w2 −1 w2 γ2 (s) γ2 (s) rw3 −w2 e2  . w γ (s) γ  (s)  e3 3 3 3

Then the unit normal vector

  w1 γ1 γ1 rw3 −w1 e1    gr × gs 1  n= = w2 γ2 γ2 rw3 −w2 e2   A A1   w3 γ3 γ3 e3

is extensible near (r, s) = (0, s0 ) in R × S 1 under the assumptions. Here A is defined by A2 =EG − F 2  = det

w1 −1

w2 −1

w3 −1

γ1 w2 r γ2 w3 r γ3 w1 r rw1 γ1 rw2 γ2 rw3 γ3

=r2(w1 +w2 −1) A1 2 ,



⎛

⎞ w1 rw1 −1 γ1 rw1 γ1 ⎝w2 rw2 −1 γ2 rw2 γ2 ⎠ w3 rw3 −1 γ3 rw3 γ3

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and A1 is the non-negative function defined by     w γ (s) γ1 (s)2 w1 γ1 (s) γ1 (s)2 2(w −w ) 3 2  +  A1 2 =  1 1 w3 γ3 (s) γ  (s) r w2 γ2 (s) γ2 (s) 3   w γ (s) γ2 (s)2 2(w −w )  r 3 1 . +  2 2 w3 γ3 (s) γ3 (s) Lemma 2.2. The first and second fundamental forms I, II for g(r, s) are given by the following formulas: I =r2(w1 −1) (E1 dr2 + 2F1 r dr ds + G1 r2 ds2 ), II =

rw3 −2 (L0 dr2 + 2M0 r dr ds + N0 r2 ds2 ), A1

where E1 =(w1 γ1 )2 + r2(w2 −w1 ) (w2 γ2 )2 + r2(w3 −w1 ) (w3 γ3 )2 , F1 =w1 γ1 γ1 + r2(w2 −w1 ) w2 γ2 γ2 + r2(w3 −w1 ) w3 γ3 γ3 , G1 =(γ1 )2 + r2(w2 −w1 ) (γ2 )2 + r2(w3 −w1 ) (γ3 )2 ,   w1 γ1 γ1 w1 2 γ1    L0 = w2 γ2 γ2 w2 2 γ2  , w γ γ  w 2 γ  3 3 3 3 3

  w1 γ1 γ1 w1 γ1    M0 = w2 γ2 γ2 w2 γ2  , w γ γ  w γ   3 3 3 3 3

  w1 γ1 γ1 γ1    N0 = w2 γ2 γ2 γ2  . w γ γ  γ   3 3 3 3

Proof. The first fundamental form I is given by I = E dr2 + 2F dr ds + G ds2 where E =w12 r2w1 −2 (γ1 )2 + w22 r2w2 −2 (γ2 )2 + w32 r2w3 −2 (γ3 )2 = r2w1 −2 E1 , F =w1 r2w1 −1 γ1 γ1 + w2 r2w2 −1 γ2 γ2 + w3 r2w3 −1 γ3 γ3 = r2w1 −1 F1 , G =r2w1 γ12 r2w2 γ22 + r2w3 γ32 = r2w1 G1 . Since grr (r, s) =(w1 (w1 − 1)rw1 −2 γ1 (s), w2 (w2 − 1)rw2 −2 γ2 (s), w3 (w3 − 1)rw3 −2 γ3 (s)), grs (s, r) =(w1 rw1 −1 γ1 (s), w2 rw2 −1 γ2 (s), w3 rw3 −1 γ3 (s)), gss (s, r) =(rw1 γ1 (s), rw2 γ2 (s), rw3 γ3 (s)), we have L =n, grr  =

rw3 −2 L0 , A1

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rw3 −1 M0 , A1 rw3 N =n, gss  = N0 , A1

M =n, grs  =

and we obtain the formula for the second fundamental form. 2.2. Curvatures of weighted cones Theorem 2.3. The Gaussian curvature K and the mean curvature H of g are given by K=

L0 N0 − M0 2 2(w3 −w1 −w2 ) r , A1 4

H=

H1 w3 −2w1 r , 2A1 3

where H1 = E1 N0 − 2F1 M0 + G1 L0 . Moreover we have the following expansion for principal curvatures. If w1 < w2 , then A1 2 (L0 N0 − M0 2 ) 2w2 −2w1 L0 N0 − M0 2 w3 −2w1  κ1 = 1+ r r + ··· , A 1 H1 H1 2  A1 2 (L0 N0 − M0 2 ) 2w2 −2w1 H1 r + ··· , κ2 = 3 rw3 −2w2 1 − A1 H1 2 where “· · · ” denotes terms of 4(w2 − w1 ) and higher order in r. If w1 = w2 , then ! H1 − H1 2 − 4A1 2 (L0 N0 − M0 2 ) rw3 −2w1 , κ1 = 2A1 3 ! H1 + H1 2 − 4A1 2 (L0 N0 − M0 2 ) κ2 = rw3 −2w1 . 2A1 3 Proof. The Gaussian curvature K is expressed as K=

LN − M 2 L0 N0 − M0 2 2(w3 −1) L0 N0 − M0 2 2(w3 −w1 −w2 ) = r = r . EG − F 2 A2 A1 2 A1 4

L0 w3 −2 r , M = Since E = E1 r2w1 −2 , F = F1 r2w1 −1 , G = G1 r2w1 , L = A 1 2 M0 w3 −1 N0 w3 2 2w1 +2w2 −2 , N = A1 r , A = r A1 , we obtain that A1 r

H=

H1 w3 −2w2 EN − 2F M + GL = r . 2 2(EG − F ) 2A1 3

Therefore the principal curvatures are expressed as follows: " κ =H ± H 2 − K

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# H1 w3 −2w2 H1 2 2w3 −4w2 L0 N0 − M0 2 2w3 −2w2 −2w1 = 3r ± r − r 2A1 4A1 6 A1 4 ! rw3 −2w2  H = ± H1 2 − 4A1 2 (L0 N0 − M0 2 )r2w2 −2w1 1 2A1 3 #   rw3 −2w2 H1 4A1 2 (L0 N0 − M0 2 ) 2w2 −2w1 = r 1± 1− 2A1 3 H1 2 =

rw3 −2w2 H1 R 2A1 3 2

2

0 N0 −M0 ) 2w2 −2w1 where R = 1 ± (1 − 2A1 (LH r − 2 1 · · · ). We thus obtain the result for κi .

2A1 4 (L0 N0 −M0 2 )2 4w2 −4w1 r H1 4

+

2.3. Ridge points, subparabolic points and fronts of weighted cones For a regular surface with principal curvatures κi and principal directions vi , a ridge point relative to vi is defined by a point satisfying vi κi = 0. An m-th order ridge point relative to vi is defined by a point with vi κi = vi 2 κi = · · · = vi m κi = 0, vi m+1 κi = 0. A subparabolic point relative to vi is defined by a point with vi κj = 0 for i = j. We are able to generalize these definitions for weighted cones if κi is extensible over the singularity. Definition 2.4. Set i = 1, 2 and assume that w3 = 2wi . Let v˜i denote a vector field so that rd vi = v˜i and v˜i |r=0 ≡ 0. We say that a point (r, s) = (0, s0 ) is a ridge point relative to vi if v˜i κi = 0 at (r, s) = (0, s0 ). We also say a point (r, s) = (0, s0 ) is an m-th order ridge point relative to vi if v˜i κi = v˜i κi = v˜i2 κi = · · · = v˜im κi = 0 and v˜im+1 κi = 0 at (r, s) = (0, s0 ). We say a point (r, s) = (0, s0 ) is a subparabolic point relative to vi if v˜i κj = 0 at (r, s) = (0, s0 ) where j = 1, 2 with i = j. Now we define the fronts of weighted cone by g t : R × S → R3 , where

g t (r, s) = g(r, s) + tn(r, s),

⎧   $ w γ γ   ⎪ ⎪ 1  1 1 1 ⎪ (w2 < w3 ) ⎪ ⎨{s ∈ S : w γ γ   = 0 2 2 2 S=  .    $  ⎪ w1 γ1 γ1  w1 γ1 γ1  ⎪ 1 ⎪ ,   = (0, 0)  ⎪ (w2 = w3 ) ⎩ s∈S :  w2 γ2 γ2  w2 γ3 γ3 

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(2)

(i)

Set Σk = Σk ∪ Σk where Σk = {(r, s) ∈ R × S 1 : κi (r, s) = k}. It is well-known that Σk is the singular locus of g t for t = 1/k. Theorem 2.5. (i)

(1) If (r0 , s0 ) ∈ Σ1/t is not ridge relative to vi , then g t has cuspidal edge singularity at (r0 , s0 ). (i) (2) If (r, s) = (r0 , s0 ) ∈ Σ1/t is a first order ridge point relative to vi and not subparabolic relative to vj (j = i), then g t has swallowtail singularity at (r0 , s0 ). The proof is similar to that appeared in [3, Theorem 4.1], and we omit the details. Define Ri,0 (s) (i = 1, 2) by v˜i κi = rw3 −2wi (Ri,0 (s) + O(r2 )). Remark 2.6. If the equation Ri,0 (s) = 0 has ri simple solutions on the set S, then there is ri swallowtail singularities of g t for t = 1/k near r = 0. So we observe that • If w3 < 2w1 , g (r1 + r2 ) number of swallowtail singularities of the map g t tend to the origin as t → 0. • If 2w1 < w3 < 2w2 , g r2 number of swallowtail singularities of the map g t tend to the origin as t → 0. To show a closed formula for Ri,0 , we introduce the following notations:  w γ a0 =  1 1 w2 γ2  w γ  1 1  h0 = w2 γ2  w3 γ3  w γ  1 1  h1 = w2 γ2  w3 γ3  w γ  1 1  h2 = w2 γ2  w3 γ3

2 γ1  , γ2  γ1 γ2 γ3 γ1 γ2 γ3 γ1 γ2 γ3

  w γ γ  2 a1 =  1 1 1  , w3 γ3 γ3

  w γ γ  2 a2 =  2 2 2  , w3 γ3 γ3   2  2 w1 γ1 (γ1 γ1 − γ1 )   2 w2 2 γ2 γ1 − 2w1 w2 γ1 γ1 γ2 + w1 2 γ1 2 γ2  ,  2 w3 2 γ3 γ1 − 2w1 w3 γ1 γ1 γ3 + w1 2 γ1 2 γ3   2 w1 2 γ1 γ2 − 2w1 w2 γ2 γ1 γ2 + w2 2 γ2 2 γ1   w2 2 γ2 (γ2 γ2 − γ2 2 ) ,  2 2 2 2      w3 γ3 γ2 − 2w2 w3 γ2 γ2 γ3 + w2 γ2 γ3  2 w1 2 γ1 γ3 − 2w1 w3 γ3 γ1 γ3 + w3 2 γ3 2 γ1   2 w2 2 γ2 γ3 − 2w2 w3 γ3 γ2 γ3 + w3 2 γ3 2 γ2  .  2  w3 2 γ3 (γ3 γ3 − γ3 )

Proposition 2.7. We have the following closed formulas for Ri,0 .

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(1) If w1 < w2 < w3 , then we have R1,0 =(w3 − 2w1 )(N0 − − (M0 −

(γ1 )2 (L0 N0 −M0 2 ) L0 N0 −M0 2 ) a0 1/2 h0 h0

γ1 γ1 (L0 N0 −M0 2 ) L0 N0 −M0 2  )( a0 1/2 h0 ) , h0

R2,0 =(w3 − 2w2 )γ1

h0 h0 − w1 γ1 ( 3/2 ) . a0 3/2 a0

(2) If w1 = w2 < w3 , we have R1,0 =(w3 − 2w1 )(N0 −

((γ1 )2 +(γ2 )2 )(L0 N0 −M0 2 ) L0 N0 −M0 2 ) a 1/2 (h +h ) h0 +h1 0 0 1 w (γ  γ +γ  γ )(L N −M

R2,0

2

2

0 0 0 0 N0 −M0 2 2 − (M0 − 1 1 1 h )( aL1/2 ) , (h0 +h1 ) 0 +h1 0   ((γ1 )2 +(γ2 )2 )(L0 N0 −M0 2 ) h0 +h1 =(w3 − 2w1 ) N0 − h0 +h1 a 3/2

)

0

− (M0 −

w1 (γ1 γ1 +γ2 γ2 )(L0 N0 −M0 2 ) h0 +h1  )( a 3/2 ) . h0 +h1 0

(3) If w1 < w2 = w3 , then we have  w (w −w )(γ  )2 R1,0 =(w3 − 2w1 ) N0 + 3 3w1 γ11 1 R2,0 = − w3 (N0 − − (M0 −

  γ2 γ2    γ3 γ   3

h0 , (a0 +a1 )3/2

(γ1 )2 h0 h0 ) w1 2 (γ1 γ2 −γ1 γ2 )2 (a0 +a1 )3/2

γ1 γ1 h0 h0  w1 (γ1 γ2 −γ1 γ2 )2 )( (a0 +a1 )3/2 ) .

The proof of the proposition will be given in the next subsection. Corollary 2.8. Assume that (w1 , w2 , w3 ) = (1, 2, 2). Then the locus defined by r = 0 is in the ridge locus relative to v1 and the singularity types of g t on this locus are not cuspidal edge. Proof. By the previous proposition, we have R1,0 = 0. This implies the result. 2.4. Principal directions of weighted cones Theorem 2.9. Principal directions vi are given by (N − κi G)∂r − (M − κi F )∂s . The order of each components of vi in r is given by the following table: N − κ1 G

M − κ1 F

N − κ2 G

w2 < w3

w3

w3 − 1

w3 + 2w1 − 2w2

w2 = w3

w3

3w3 − 2w1 − 1

2w1 − w3

M − κ2 F

w3 + 2w1 −2w2 − 1

2w1 − w3 − 1

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This theorem asserts that the limit of principal direction which is represented by v1 is ∂r as r → 0 for generic s, when w1 < w2 = w3 . In the other cases, the limits of principal directions are ∂s for generic s. Proof. Since N − κ1 G = r w3

N0 A1

 A 2 (L0 N0 − M0 2 ) 2w2 −2w1 L0 N0 − M0 2 w3 −2w1  1+ 1 r r + · · · G1 r2w1 2 A1 H1 H1  G1 w3  N0 L0 N0 − M0 2 A1 2 (L0 N0 − M0 2 ) 2w2 −2w1 = r − (1 + r + ···) , 2 A1 G1 H1 H1 −

M − κ1 F = r w3 −1

M0 A1

 L0 N0 − M0 2 w3 −2w1  A 2 (L N −M 2 ) r 1 + 1 0H 02 0 r 2w2 −2w1 + · · · F1 r2w1 −1 1 A1 H1 M  F L N − M0 2 A 2 (L0 N0 − M0 2 ) 2w2 −2w1 0 = 1 rw3 −1 − 0 0 (1 + 1 r + ···) , 2 A1 F1 H1 H1 −r w3 −2w1

N − κ2 G = r w3

N0 A1 

 H1 w3 −2w2 A 2 (L0 N0 − M0 2 ) 2w2 −2w1 r r + · · · G1 r 2w1 1− 1 3 2 A1 H1  A 2 (L0 N0 − M0 2 ) 2w2 −2w1 N0 H1 G1 w3 +2w1 −2w2  1− 1 = r w3 − r r +··· , 3 2 A1 A1 H1 −

M0 A1  H1 w3 −2w2  A 2 (L0 N0 − M0 2 ) 2w2 −2w1 − 3r r + · · · F1 r2w1 −1 1− 1 2 A1 H1   M A 2 (L0 N0 − M0 2 ) 2w2 −2w1 H F = r w3 −1 0 − 1 31 r w3 +2w1 −2w2 −1 1 − 1 r + ··· , 2 A1 A1 H1

M − κ2 F = r w3 −1

we obtain that v1 = (N − κ1 G)∂r − (M − κ1 F )∂s % ' 2 w3 −1 & 0 −M0 ) N0 − G1 (L0 N = r A1 + O(r2 ) r∂r H1 & ' ( 2 − M0 − F1 (L0 NH01−M0 ) + O(r2 ) ∂s , v2 = (N − κ2 G)∂r − (M − κ2 F )∂s   ⎧  ⎨ HA1 γ31 r w3 +2w1 −2w2 −1 (γ1 + O(r2 ))r∂r − (w1 γ1 + O(r2 ))∂s (w1 < w2 ) 1 = .   ⎩ rw3 −1 N − G1 H1 + O(r 2 )r∂ − M − F1 H1 + O(r 2 )∂ (w = w ) r s 0 0 1 2 A1 A1 2 A1 2

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Since

        L 0 M0    = (w2 − w3 ) w1 γ1 γ1  w1 γ1 γ1  ,  w1 γ1 γ   w2 γ2 γ   w3 γ3 γ   1 2 3

we observe ∂r is a principal direction when w2 = w3 and r = 0. This suggests a cancellation of the coefficient of rw3 −1 in M − κ1 F . In fact, we have           M0 L0 N0 − M0 2   =(w3 − w2 ) w1 γ1 γ1  w1 γ1 γ1  w1 γ1 M0  ,        γ H1 |r=0 w2 γ2 γ w3 γ3 γ γ N0  1

2

3

1

w3 −1

in M − κ1 F , when and we see a cancellation of the coefficient of r w2 = w3 . When w1 < w2 = w3 ,    N0 L0 N0 − M0 2    G1 |r=0 H1 |r=0         M0 w1 γ1  M0 w1 γ1      2 γ2 γ2      = .      + (w1 − w3 )w3 (γ1 − γ1 )   N0 γ N0 γ γ γ3  1

1

3

This implies that the initial coefficient of N − κ1 F is not identically zero. In this way, we obtain the table desired. Remark 2.10. Set κ = re (k0 + k2 r2 + · · · ). • When v = (α1 r + α3 r3 + · · · )∂r + (β0 + β2 r2 + · · · )∂s , we have vκ =(α1 r + α3 r3 + · · · )(ek0 re−1 + (e + 2)k2 re+1 + · · · ) + (β0 + β2 r2 + · · · )re (k0 + k2 r2 + · · · ) =(eα1 k0 + β0 k0 )re + O(re+2 ), ' & v 2 κ = eα1 (eα1 k0 + β0 k0 ) + (eα1 k0 + β0 k0 ) re + O(re+2 ). If e = 0, then we obtain vκ =β0 k0 + O(r2 ), v 2 κ =(2β0 k0 + β0 k0 ) + O(r2 ). • When v = (α0 + α2 r2 + · · · )∂r + (β1 r + β3 r3 + · · · )∂s , we have vκ =(α0 + α2 r2 + · · · )(ek0 re−1 + (e + 2)k2 re+1 + · · · + (β1 r + β3 r3 + · · · )re (k0 + k2 r2 + · · · ) =eα0 k0 re−1 + (α0 (e + 2)k2 + α2 ek0 + β1 k0 )re+1 + O(re+2 ),

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v2 κ =α0 (e(e − 1)α0 k0 re−2 + (e + 1)(α0 (e + 2)k2 + α2 ek0 + β1 k0 )re + · · · ) + β1 (e(α0 k0 ) re + · · · ) =e(e − 1)α0 2 k0 re−2 + ((e + 1)α0 (α0 (e + 2)k2 + α2 ek0 + β1 k0 ) + β1 e(α0 k0 ) )re + · · · . If e = 0, then we obtain vκ =(2α0 k2 + β1 k0 )r + O(r2 ), v 2 κ =α0 (2α0 k2 + β1 k0 ) + O(r2 ). We describe below the expression of v˜i κj for i, j = 1, 2. These complete the proof of Proposition 2.7, since they contain the expansions of v˜i κi (i = 1, 2). • When w1 < w2 < w3 , we have L0 N0 − M0 2 w3 −2w1 r (1 + O(r2 )), a0 1/2 h0 h0 κ2 = 3/2 rw3 −2w2 (1 + O(r2 )), a0

κ1 =

v˜1 =(N0 −

(γ1 )2 (L0 N0 −M0 2 ) h0

− (M0 −

+ O(r2 ))r∂r

γ1 γ1 (L0 N0 −M0 2 ) h0

+ O(r2 ))∂s ,

v˜2 =(γ1 + O(r2 ))r∂r − (w1 γ1 + O(r2 ))∂s . Then we conclude that % v˜1 κ1 =rw3 −2w1 (w3 − 2w1 )(N0 −

(γ1 )2 (L0 N0 −M0 2 ) L0 N0 − M0 ) h0 a0 1/2 h0 2 ( γ1 γ1 (L0 N0 −M0 2 ) L0 N0 − M0  2 )( ) + O(r ) , h0 a0 1/2 h0

2

− (M0 − % ( 2 2 0 −M0 0 −M0 v˜2 κ1 =rw3 −2w2 (w3 − 2w2 )γ1 L0aN0 1/2 − w1 γ1 ( L0aN0 1/2 ) + O(r2 ) , h0 h0 % h0 (γ  )2 (L0 N0 −M0 2 ) ) 3/2 v˜1 κ2 =rw3 −2w1 (w3 − 2w1 )(N0 − 1 h0 a0 ( h0 γ  γ (L N −M 2 ) − (M0 − 1 1 0h00 0 )( 3/2 ) + O(r2 ) , a0 ( % h0 h 0 v˜2 κ2 =rw3 −2w2 (w3 − 2w2 )γ1 3/2 − w1 γ1 ( 3/2 ) + O(r2 ) . a0 a0 • When w1 = w2 < w3 , we have E1 =w1 2 (γ1 2 + γ2 2 ) + r2w3 −2w1 w3 2 γ3 2 , F1 =w1 (γ1 γ1 + γ2 γ2 ) + r2w3 −2w1 w3 γ3 γ3 ,

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G1 =(γ1 )2 + (γ2 )2 + r2w3 −2w1 (γ3 )2 , A1 2 =w1 2 (γ1 γ2 − γ1 γ2 )2 + r2w3 −2w1 ((w1 γ1 γ3 − w3 γ1 γ3 )2 + (w1 γ2 γ3 − w3 γ2 γ3 )2 ),    γ γ1  L0 =w1 w3 (w1 − w3 )γ3  1  , γ2 γ2    γ γ1  M0 =w1 (w3 − w1 )γ3  1  , γ2 γ2   w1 γ1 γ1 γ1    N0 = w1 γ2 γ2 γ2  . w3 γ3 γ  γ   3 3 So we obtain that L0 N0 − M0 2 w3 −2w1 r κ1 = 1/2 (1 + O(r2 )), a0 (h0 + h1 ) h0 + h1 w3 −2w2 r (1 + O(r2 )), κ2 = a0 3/2 & ' ((γ  )2 +(γ2 )2 )(L0 N0 −M0 2 ) v˜1 = N0 − 1 + O(r2 ) r∂r h0 +h1 − (M0 − v˜2 =(N0 −

w1 (γ1 γ1 +γ2 γ2 )(L0 N0 −M0 2 ) h0 +h1

((γ1 )2 +(γ2 )2 )h0 w1 2 (γ1 γ2 −γ1 γ2 )2

− (M0 −

+ O(r2 ))r∂r

(γ1 γ1 +γ2 γ2 )h0 w1 (γ1 γ2 −γ1 γ2 )2

Then we conclude that % v˜1 κ1 =rw3 −2w2 (w3 − 2w1 )(N0 − −(M0 −

+ O(r2 ))∂s . ((γ1 )2 +(γ2 )2 )(L0 N0 −M0 2 ) L0 N0 −M0 2 ) a0 1/2 (h0 +h1 ) h0 +h1

w1 (γ1 γ1 +γ2 γ2 )(L0 N0 −M0 2 ) & ) h0 +h1

% v˜2 κ1 =rw3 −2w2 (w3 − 2w1 )(N0 − − (M0 −

+ O(r2 ))∂s ,

( L0 N0 − M0 2 ' + O(r2 ) , 1/2 a0 (h0 + h1 )

((γ1 )2 +(γ2 )2 )h0 h0 +h1 w1 2 (γ1 γ2 −γ1 γ2 )2 ) a0 3/2

(γ1 γ1 +γ2 γ2 )h0 1  )( ha00+h 3/2 ) ∂s w1 (γ1 γ2 −γ1 γ2 )2

( + O(r2 ) ,

% v˜1 κ2 =rw3 −2w2 (w3 − 2w1 )(N0 −

((γ1 )2 +(γ2 )2 )(L0 N0 −M0 2 ) L0 N0 − M0 ) h0 +h1 a0 1/2 h0 ( 2 w (γ  γ +γ  γ )(L N −M 2 ) L0 N0 − M0  2 − (M0 − 1 1 1 h2 02+h1 0 0 0 )( ) + O(r ) , a0 1/2 h0 % ((γ  )2 +(γ2 )2 )(L0 N0 −M0 2 ) h0 +h1 ) a0 3/2 v˜2 κ2 =rw3 −2w2 (w3 − 2w1 )(N0 − 1 h0 +h1

− (M0 −

w1 (γ1 γ1 +γ2 γ2 )(L0 N0 −M0 2 ) h0 +h1  )( a0 3/2 ) h0 +h1

( + O(r2 ) .

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• When w1 < w2 = w3 , we have E1 =(w1 γ1 )2 + r2w3 −2w1 w3 2 (γ2 2 + γ3 2 ), F1 =w1 γ1 γ1 + r2w3 −2w1 w3 (γ2 γ2 + γ3 γ3 ), G1 =(γ1 )2 + r2w3 −2w1 ((γ2 )2 + (γ3 )2 ), A1 2 =(w1 γ1 γ2 − w3 γ1 γ2 )2 + (w1 γ1 γ3 − w3 γ1 γ3 )2 + r2w3 −2w1 w3 2 (γ2 γ3 − γ2 γ3 )2 ,    γ γ  L0 =w1 w3 (w3 − w1 )γ1  2 2  , γ3 γ3    γ γ  M0 =w3 (w3 − w1 )γ1  2 2  , γ3 γ3   w1 γ1 γ1 γ1    N0 = w3 γ2 γ2 γ2  . w γ γ  γ   3 3 3 3 Observing that H1 = h0 + r2w3 −2w1 (h1 + h2 ) and A1 2 = a0 + a1 + r2w3 −2w1 a2 , we have L0 N0 − M0 2 A1 2 (L0 N0 − M0 2 ) (1 + + ···) A 1 H1 H1 2   w3 (w3 − w1 ) γ2 γ2  =− (1 + O(r2w3 −2w1 )). w1 γ1 (a0 + a1 )1/2 γ3 γ  

κ1 =

3

We also conclude that h0 r−w3 (1 + O(r2 )), κ2 = (a0 + a1 )3/2 v˜1 =(N0 −

(γ1 )2 (L0 N0 −M0 2 ) h0

v˜2 =(N0 −

(γ1 )2 h0 w1 2 (γ1 γ2 −γ1 γ2 )2

− (M0 −

+ O(r2 ))r∂r

γ1 γ1 h0 w1 (γ1 γ2 −γ1 γ2 )2

Finally we obtain that ) v˜1 κ1 =rw3 −2w1 −1 (w3 − 2w1 )(N0 − * + O(r) , ) v˜2 κ1 =rw3 −2w1 (w3 − 2w1 )(N0 − − (M0 −

+ O(r2 ))∂r + O(r)∂s ,

+ O(r2 ))∂s .

(γ1 )2 (L0 N0 −M0 2 ) L0 N0 −M0 2 ) (a0 +a1 )1/2 h0 h0

(γ1 )2 h0 w1 2 (γ1 γ2 −γ1 γ2 )2

)

L0 N 0 − M 0 2 (a0 + a1 )1/2 h0

* γ1 γ1 h0 L0 N0 − M0 2  )( ) + O(r) ,   2 1/2 w1 (γ1 γ2 − γ1 γ2 ) (a0 + a1 ) h0

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) * (γ  )2 (L0 N0 − M0 2 ) h0 v˜1 κ2 = (w3 − 2w1 )(N0 − 1 ) + O(r) , 3/2 h0 (a0 + a1 ) %  2 (γ ) h 0 1 h0 v˜2 κ2 =r−w3 −w3 (N0 − 3/2 ) w1 2 (γ1 γ2 − γ1 γ2 )2 (a0 +a1 ) ( γ1 γ1 h0 h0  2 )( ) + O(r ) . − (M0 − w1 (γ1 γ2 − γ1 γ2 )2 (a0 + a1 )3/2 3. Focal curves: Case (w1 , w2 , w3 ) = (1, 2, 2) Assume that (w1 , w2 , w3 ) = (1, 2, 2). Then we have     γ1 γ1  1 ,  n(0, s) = a where a = 0, −  2γ3 γ3  |a|

   γ1 γ1    2γ2 γ   . 2

We assume a is not 0 for all s. The tangent plane at the origin degenerates into a line or point. We call the plane orthogonal to such a line at the origin the normal plane. The focal set (or caustic) is given by Σ = {p ∈ R3 : p = g t (S(g t ))}, where S(g t ) is the set of singular points of g t . Since n(0, s) lies in the normal plane, Σ|r=0 also lies in the normal plane. The set Σ|r=0 is the image of the map      γ1 γ1  γ1   γ1 γ1  1 1 3  ,   0,  n(0, s) =  φ : S → R , s → , − γ2 γ2  2γ3 γ3  2γ2 γ2  κ1 (0, s)  2  γ3 γ   3

which we call by the focal curve. Focal curves are analogous to focal conics of Whitney umbrella (cf. [3, Lemma 3.3]). Proposition 3.1. A point φ(s0 ) of the focal curve, which is not the origin, is subparabolic relative to v˜2 if and only if it is a critical point of |φ(s)|2 . This condition is equivalent to one of the following conditions. (1) φ is regular at s = s0 and there is a circle centered at the origin which tangents to the focal curve at φ(s0 ). (2) φ is singular at s = s0 . Proof. We observe that d 2γ1 P Q , |φ(s)|2 = − ds (γ2 γ3 − γ3 γ2 )3 where

     γ1 γ1 γ1      2 γ2 γ2      P = 2γ1    − γ1 2γ2 γ2 γ2  γ3 γ3 2γ γ  γ   3 3 3

(˜ v2 κ1 )(0, s) =

and

2P Q + a1 )3/2

γ1 2 (a0

   0 γ1 γ1    Q = −γ3 2γ2 γ2  .  γ 2γ γ   2 3 3

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Since φ =

(γ2 γ3

P (0, γ3 , −γ2 ), − γ3 γ2 )2

Q(s0 ) = 0 if and only if there is a circle centered at the origin which tangents to the focal curve at φ(s0 ) whenever φ (s0 ) = 0. We complete the proof. Example 3.2. Here we show some pictures of focal curves for several γ(s)’s.

γ(s) = (cos 2s, cos 3s, sin 3s) γ(s) = (cos 3s, cos 2s, sin 2s) γ(s) = (cos s, cos 3s, sin s)

4. Examples Example 4.1. Let γ be a curve defined by γ(s) = (cos 2s, sin 2s, cos 3s). Setting (w1 , w2 , w3 ) = (1, 1, 2), we consider the weighted cone: g(r, s) = (r cos 2s, r sin 2s, r2 cos 3s). 1 (7 cos 2s − 11), and H|r=0 = (−1/4) cos 3s. So Then we obtain K|r=0 = 128 both principal curvatures are bounded near the locus defined by r = 0. The image of g is the same as the image of the map " R2 → R3 , (u, v) → (u2 − v 2 , 2uv, u2 + v 2 (u3 − 3uv 2 )).

Let gi (u, v) (i = 1, 2, 3) be a homogeneous polynomial of degree wi and set g(u, v) = (g1 (u, v), g2 (u, v), g3 (u, v)). Considering γ(s) a parametrization of a connected component of S 2 ∩g(R2 ), we obtain many examples of weighted cones. Example 4.2. Consider the curve defined by γ(s) = (cos s, cos s sin s, sin2 s). The map g(r, s) = (r cos s, r2 cos s sin s, r2 sin2 s) is a weighted cone with (w1 , w2 , w3 ) = (1, 2, 2) and its image is known as Whitney umbrella. Remark that the focal curve of g coincides with the focal conic of the standard

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Whitney umbrella defined by (u, v) → (u, uv, v2 ). We then have E = cos2 s + 4r2 sin2 s, F =r sin s cos s(−1 + 2r2 ), G =r2 sin2 s + r4 , EG − F 2 =r4 (cos2 s + 4 sin2 s) + 4r6 sin4 s, 1 (2r sin2 s, 2 sin s, cos s), n =" 2 cos s + 4 sin2 s + 4r2 sin4 s 2 sin2 s cos s , L=− " cos2 s + 4 sin2 s + 4r2 sin4 s 2r sin3 s M =" , cos2 s + 4 sin2 s + 4r2 sin4 s 2r2 cos s(1 + sin2 s) N =" , cos2 s + 4 sin2 s + 4r2 sin4 s 4 sin2 s K =− 2 , r (cos2 s + 4 sin2 s + r2 sin4 s)2 H=

cos s(1 + 3r2 sin2 s) 3

r2 (cos2 s + 4 sin2 s + 4r2 sin4 s) 2

.

We have κ1 = − " −

2 sin s tan s cos2 s + 4 sin2 s

+

4r2 sin4 s(2 cos s + sin s tan s(11 + 8 tan2 s)) 3

(cos2 s + 4 sin2 s) 2

4r 4 sin6 s(10 cos3 s+108 cos s sin2 s+sin3 s tan s(391+528 tan2 s+256 tan4 s)) 5

(cos2 s+4 sin2 s) 2

+ O(r6 ), 1 κ2 = 2 √ 22 cos s 2 + cos s+4 sin s r

4r 2 (2 cos3 s sin2 s+7 cos s sin4 s+8 sin5 s tan s) 5

(cos2 s+4 sin2 s) 2

+ O(r4 ) .

Observe cancellation of the coefficients of r0 , r2 of L − κ1 E and the coefficient of r of M − κ1 F : L − κ1 E = M − κ1 F =

(33 − 28 cos 2s + 3 cos 4s)2 sin3 s tan3 s (cos2

2

3 2

s + 4 sin s) (33 − 28 cos 2s + 3 cos 4s) sin s tan2 s 3

2(cos2 s + 4 sin2 s) 2 2 " N − κ1 G = r2 + O(r4 ). cos s cos2 s + 4 sin2 s

r4 + O(r6 ),

r3 + O(r5 ),

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Moreover, we have

  1 2 cos2 s sin s 2 + O(r ) , M − κ2 F = r (cos2 s + 4 sin2 s) 32 N − κ2 G = −

2 cos s sin2 s 2 3 + O(r ). (cos2 s + 4 sin2 s) 2

Then we have   2 2 " + O(r ) ∂r v˜1 = cos s cos2 s + 4 sin2 s   (33 − 28 cos 2s + 3 cos 4s) sin s tan2 s 3 + − r + O(r ) ∂s , 3 2(cos2 s + 4 sin2 s) 2   2 cos s sin2 s 3 v˜2 = − 3 r + O(r ) ∂r (cos2 s + 4 sin2 s) 2   2 cos2 s sin s 2 + − + O(r ) ∂s . 3 (cos2 s + 4 sin2 s) 2 We conclude that v˜1 κ1 |r=0 =0, v˜12 κ1 |r=0 =

96(−4 + 3 cos2 s)2 tan4 s 7 , cos s(cos2 s + 4 sin2 s) 2

v˜2 κ1 |r=0 =

8(2 − cos2 s) sin2 s . (cos2 s + 4 sin2 s)3

Hence, from Theorem 2.5 it follows that if cos s0 = 0 or sin s0 = 0 then g 1/κ1 (0,s0 ) has a swallowtail singularity at g 1/κ1 (0,s0 ) (0, s0 ).

References 1. V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps I, Birkh¨ auser, 1986. 2. T. Fukui and M. Hasegawa, Singularities of parallel surfaces, to appear in Tohoku Magthematical Journal 64 (2012). 3. T. Fukui and M. Hasegawa, Fronts of Whitney umbrella, Journal of Singularities 4 (2012), 35–67. 4. M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic 3-space, Pacific J. of Math. 221 (2005), no. 2, 303–351.

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Fig. 1. Examples of a weighted cone, its focal curve, and its front: g(r, s) = (r cos 2s, r2 cos 3s, r2 sin 3s) (left), g t (r, s) = g(r, s) + n(r, s)/3 (right).

Fig. 2. Examples of a weighted cone and its front: g(r, s) = (r cos 2s, r sin 2s, r2 cos 3s) (left), g t (r, s) = g(r, s) + 3/2n(r, s) (right).

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Involutive deformations of the regular part of a normal surface Adam Harris and Kimio Miyajima Department of Mathematics and Statistics, School of Science and Technology, University of New England Armidale, NSW 2351 Australia [email protected] Department of Mathematics and Computer Science, Faculty of Science, Kagoshima University, Kagoshima 890, Japan [email protected] We define the property of involutivity for deformations of complex structure on a manifold X, with particular reference to the regular part of a normal surface. ¯ Our main result is a sufficient condition for involutivity in terms of a “∂Cartan formula”, previously examined in [3] in the more special context of cone singularities. By way of examples we show that some involutive deformations of the regular part determine a subspace, if not the entire versal space, of flat deformations of normal surface singularities, while others may determine Stein surfaces which lie outside the versal space of flat deformations of a given normal surface. Keywords: normal singularity, complex deformation, Stein completion AMS classification numbers: 32S30, 32G05, 32C15

1. Introduction 1,0 0,1 ∗ ¯ A ∂-closed form ψ ∈ C ∞ (X, TX ⊗ (TX ) ) on any complex manifold X will be said to represent an involutive deformation of the complex structure if the Fr¨ olicher-Nijenhuis bracket [ψ, ψ]F N vanishes identically. Such deformations automatically satisfy the Kodaira-Spencer integrability equation independently of their analytic properties, and are therefore well-suited to the context of non-compact spaces, provided they can be shown to represent a significant subspace of the tangent cohomology H 1 (X, TX ). It was shown in [3] that this subspace is non-trivial for X = V \ {0}, where V ⊆ CN is a complex variety with isolated cone-singularity at the origin, and is in fact infinite-dimensional when V is a surface of this type. Given a potentially large space of integrable complex deformations of V \ {0}, the following should be noted:

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• On the “link” defined by V ∩ S2N −1 each complex deformation induces a corresponding deformation of the CR-structure. If dimC (V ) ≥ 3, and if strict pseudoconvexity of the initial CR-structure is maintained by a given deformation, then a well-known theorem of Rossi [5] ensures that the ambient complex structure may be completed to a normal Stein space V  which is biholomorphically distinct from V . • There is no a priori reason to expect that the Stein embedding dimen sion of V  , i.e., V  ⊂ CN should be such that N  = N . In fact, it was shown in [4] that only stably embeddable CR-deformations of the link of an isolated singularity are in one-to-one correspondence with the finite-dimensional versal space of flat algebraic deformations of V . In light of these remarks, the question to be addressed in this article is whether there exist non-trivial deformations [ψ] ∈ H 1 (V \ {0}, TV ) such that • [ψ] is involutive • there exists a normal Stein completion V  of the associated deformation of V \ {0} • V  does not belong to the versal space of flat deformations of V . When dimC (V ) = 2 the existence of a Stein completion is guaranteed if and only if the induced strongly pseudoconvex CR-structure of an integrable deformation (involutive or otherwise) is embeddable in some complex Euclidean space [6]. Equivalently, it has been shown in [1] that a two-dimensional 1-corona Y (cf. section 3) admits a Stein compactification (unique after normalization) if and only if the Cauchy-Riemann operator ∂¯Y : C ∞ (Y, C) → C ∞ (Y, (TY0,1 )∗ ) has closed range with respect to the C ∞ -topology. Our first result, in section 2, treats the existence of involutive deformations on a complex surface. Theorem: Let X be a smooth complex surface, and σ ∈ H 0 (X, TX ) a holomorphic vector field. Let 1,0 0,1 ∗ 1,0 0,1 ∗ ⊗ (TX ) ) → C ∞ (X, TX ⊗ (TX ) ) Lσ¯ : C ∞ (X, TX

represent the natural first-order operator defined by Lie differentiation along the anti-holomorphic vector field σ ¯ . Then a sufficient condition for any class

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[ψ] ∈ H 1 (X, TX ) to be an involutive deformation is that it belong to the quotient ¯ im(Lσ¯ ) ∩ ker(∂) . Γ1σ¯ := ¯ im(∂) This is an easy generalization of the surface-case of cone-singularities, X = V \ {0}, as treated in [3]. We recall briefly that these singularities are obtained specifically by blowing down the zero-section of a negative hermitian-holomorphic line bundle defined over a Riemann surface. The infinite-dimensionality of Γ1σ¯ was demonstrated with respect to a particular vector field σ ¯ , the intrinsic definition of which is reviewed in section 3. In the final section we present two very different examples in which involutive deformation of the regular part of an isolated surface singularity can admit a Stein completion. The first follows the approach to constructing the versal space of flat deformations of an isolated singularity developed in [4], and is applied specifically to the case of the orbifold surface singularity of multiplicity two. The second example is of a complex-analytic family of involutive deformations, each fibre of which blows-down to a normal Stein surface in such a way that they are not stably embedded in complex Euclidean space. It is easily derived from a construction of unstable perturbations of an embedded CR-structure of dimension three, due to Catlin and Lempert [2]. From this we conclude that the versal space of flat deformations of an isolated singularity, although it may contain a subspace of involutive deformations of the regular part, does not in general contain all Stein-complete involutive deformations. 2. Involutive deformations of surfaces 1,0 At first, let X be an arbitrary complex manifold, with ϕ ∈ C ∞ (X, TX ⊗ +q 0,1 ∗ (TX ) ) written locally in the form

ϕ = Σ(λ) Σ1≤α≤n+1 ϕα ¯(λ) . (λ) ∂α ⊗ dw 1,0 , as usual, denotes the holomorphic tangent bundle, with local Here TX basis {∂α }nα=1 , (λ) is a multi-index (λi1 , ...., λiq ) and ϕα (λ) is smooth in any local chart. The following calculation, reproduced for the reader’s convenience from [3], is carried out for q = 1, though the general case is essentially well-known. A vector field ξ 0,1 will be referred to as “anti-holomorphic” if the Lie bracket [ξ 0,1 , ϑ] = 0 for any holomorphic vector field ϑ, and will be written locally in the form ξ 0,1 = Σn ξ ν ∂¯ν . ν=1

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Now consider the contraction λ ιξ0,1 ϕα = Σnλ=1 ϕα λξ

' & ¯ ξ0,1 ϕα = Σλ ∂ι ν



such that

∂ϕα ∂ξ λ λ λ ξ + ϕα λ ∂w ¯ν ∂w ¯ν

On the other hand,

 ¯ λ,ν = Σλ<ν (∂ϕ)

∂ϕα ∂ϕα ν λ − ∂w ¯λ ∂w ¯ν

 .



so that  α   α  & ' ∂ϕα ∂ϕα ∂ϕν ∂ϕλ ν λ λ λ ¯ − − ιξ0,1 ∂ϕ ν = Σλ<ν ξ − Σλ>ν ξ ∂w ¯λ ∂w ¯ν ∂w ¯ν ∂w ¯λ = Σλ =ν ξ λ

∂ϕα ∂ϕα ν − Σλ =ν ξ λ λ , ∂w ¯λ ∂w ¯ν

and therefore α λ α & & ' ' ¯ ξ0,1 ϕα + ιξ0,1 ∂ϕ ¯ α = ∂ϕν ξ ν + Σλ ϕα ∂ξ + Σλ =ν ξ ν ∂ϕν ∂ι λ ν ν ∂w ¯ν ∂w ¯ν ∂w ¯λ α = ξ 0,1 (ϕα ν ) + Σλ ϕλ

∂ξ λ . ∂w ¯ν

0,1 Writing ϕν = Σα ϕα is antiν ∂α , we note that the assumption ξ holomorphic implies specifically that

[ξ 0,1 , ϕν ]α = ξ 0,1 (ϕα ν) . In addition, ∂ξ λ ¯ [ξ 0,1 , ∂¯ν ] = −Σλ ∂λ , ∂w ¯ν so that Σλ ϕα λ

∂ξ λ = −ϕα ([ξ 0,1 , ∂¯ν ]) ∂w ¯ν

implies & & ' ' ' & ¯ α = ξ 0,1 (ϕα ) − ϕα ([ξ 0,1 , ∂¯ν ]) = Lξ0,1 ϕα , ¯ ξ0,1 ϕα + ιξ0,1 ∂ϕ ∂ι ν ν ν ν which recalls the famous formula for the Lie derivative due to E. Cartan. 1,0 In particular, let σ ∈ H 0 (X, TX ) be a holomorphic vector field on X, and consider 1,0 0,1 ∗ ψ ∈ C ∞ (X, TX ⊗ (TX ) )

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¯ = 0 and ψ = Lσ¯ ϕ, for some smooth ϕ ∈ C ∞ (X, T 1,0 ⊗(T 0,1)∗ ). such that ∂ψ X X ¯ It follows from the formula above that ψ is cohomologous to ισ¯ ∂ϕ. ¯λ we recall the specific formula Writing ψ in local form as Σα,λ ψλα ∂α ⊗dw for the Fr¨olicher-Nijenhuis bracket ' & [ψ, ψ]F N = Σnα,β=1 2 ψ α ∧ ∂α ψ β ∂β = Σ1≤λ<ν≤n [ψλ , ψν ]dw ¯λ ∧ dw ¯ν , where [ψλ , ψν ] denotes the standard Lie bracket of vector fields. Let η ∈ 1,0 +2 0,1 ∗ γ ⊗ (TX ) ) be represented locally as ηα,β ∂γ ⊗dw ¯α ∧dw ¯β . Now C ∞ (X, TX γ γ (ισ¯ η)γλ = Σα<λ σ ¯ α ηα,λ − Σβ>λ σ ¯ β ηλ,β ,

and in particular, n = 2 implies γ (ισ¯ η)γ1 = −¯ σ2 η12 ,

γ (ισ¯ η)γ2 = σ ¯ 1 η12 ,

so that σ 2 η12 (¯ σ1 ) + σ ¯ 1 η12 (¯ σ 2 ))η12 = 0 , [ισ¯ η, ισ¯ η]F N = (−¯ given σ ¯ is anti-holomorphic. We summarize with the following Theorem 2.1. Let X be a smooth complex surface, and σ ∈ H 0 (X, TX ) a holomorphic vector field. Let 1,0 0,1 ∗ 1,0 0,1 ∗ Lσ¯ : C ∞ (X, TX ⊗ (TX ) ) → C ∞ (X, TX ⊗ (TX ) )

represent the natural first-order operator defined by Lie differentiation along the anti-holomorphic vector field σ ¯ . Then a sufficient condition for any class [ψ] ∈ H 1 (X, TX ) to be an involutive deformation is that it belong to the quotient Γ1σ¯ :=

¯ im(Lσ¯ ) ∩ ker(∂) . ¯ im(∂)

¯ Proof. If ψ = Lσ¯ ϕ then [ψ] ∈ H 1 (X, TX ) is also represented by ισ¯ ∂ϕ, which is clearly involutive when n = 2. 3. Some remarks on Stein completion Our primary interest is in deformations of the regular part of a normal complex surface, in particular X = V \ {0}, where V ⊂ CN is a surface with isolated singularity at the origin. When V corresponds to the neighbourhood of a “cone” singularity, i.e., obtained by blowing-down the zero section

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of a negative holomorphic line bundle over a Riemann surface, then it was shown in [3] that Γ1σ¯ is in fact infinite-dimensional as a complex vector space. The specific vector field σ ¯ used in [3] was defined intrinsically as the K¨ ahler metric-dual of ∂ρ, where ρ is a strongly plurisubharmonic function on the complement of the zero section, defined relative to the hermitian metric of the line bundle in the standard way. In general small deformations, when restricted to the link Σ := V ∩ S2N −1 ⊂ CN induce distinct CR-structures which preserve the strict pseudoconvexity of the initial structure. If ρ is a strongly plurisubharmonic function on X = V \ {0}, such that ρ(0) = 0 and Σ = ρ−1 (1), then the same conclusion can be drawn simultaneously for all level sets ρ = ε, 0 < ε0 ≤ ε ≤ 1. A small involutive deformation ψ therefore induces a smooth function ρ on a surface X  which is strongly plurisubharmonic on (the “1-corona”) Yε0 = {ε0 ≤ ρ ≤ 1}. Let σ ¯ ψ denote a vector field on Yε0 which is anti-holomorphic with respect to the complex structure induced by an involutive ψ, i.e., σ ¯ψ = σ ¯ + ψ(¯ σ) , where σ ¯ is anti-holomorphic with respect to the initial complex structure on X. Hence, for u ∈ CC∞ (Yε0 ), we have ¯ σ ) + ∂u(ψ(¯ ∂¯ψ u(¯ σ ψ ) = du(¯ σψ ) = ∂u(¯ σ )) . Theorem 1.1 of [1] states that for the existence of a Stein compactification of Yε0 it is necessary and sufficient that ∂¯ψ have closed range relative to the C ∞ -topology on C ∞ (Yε0 , (TY0,1 )∗ ). ε0 In the following examples we present two very different ways in which involutive deformation of the regular part of an isolated surface singularity can admit a Stein completion. The first follows the approach to constructing the versal space of flat deformations of an isolated singularity developed in [4]. It is applied specifically to the case of the orbifold surface singularity of multiplicity two as follows. Let X := C2 \{(0, 0)} with coordinate (z, w), and denote r := |z|2 +|w|2 . Fix a global frame {Z, ξ} of T 1,0X such that Z := w∂ ¯ z − z¯∂w , and ξ := z∂z + w∂w . (While the above is a convenient definition for ξ, it plays no specific role in the following calculation.) Now define & '∗ 1 1,0 0,1 ∗ ψ := 2 Z ⊗ Z¯ ∈ C ∞ (X, TX ⊗ (TX ) ). r ¯ = 0 and [ψ, ψ]F N = 0 since, in the first instance, Note that ∂ψ ¯ ψ(Z)] ¯ , ¯ Z, ¯ = [Z, ¯ T 1,0 X − [ξ, ¯ T 1,0 X − ψ([Z, ¯ ξ]) ¯ ξ) ¯ ψ(ξ)] ∂ψ(

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where the subscript T 1,0 X implies taking the (1,0)-part of the Lie bracket, and by a direct calculation this is seen to vanish. On the other hand, it is clear that ¯ = [ψ(Z), ¯ =0. ¯ ξ) ¯ ψ(ξ)] [ψ, ψ]F N (Z, Next consider a Z2 -action (z, w) → (−z, −w) on X. Since ψ is invariant under this Z2 -action, it is considered as a form on X1 , say ψ1 ∈ 1,0 0,1 ∗ C ∞ (X1 , TX ⊗ (TX ) ), where X1 := X/Z2 . (X1 , T 0,1 X1 ) is realized as 1 1 the regular part of a Stein space V1 defined by an equation ζ1 ζ3 − ζ22 = 0 in C3 , with embedding X1  (z, w) → (ζ1 , ζ2 , ζ3 ) = (z 2 , zw, w2 ) ∈ C3 . Now tψ1 defines an involutive deformation of complex structure on X1 , and by a specific application of the theory of [4] it is possible to compute a family of embeddings w ¯2 z¯w ¯ 2 z¯2 , zw + t , w − t ) ∈ C3 r2 r2 r2 which realizes (X1 , tψ1 T 0,1 X1 ) as a stable family of submanifolds of C3 . It follows that (X1 , tψ1 T 0,1 X1 ) admits a simultaneous family of Stein completions in C3 defined by X1  (z, w) → (z 2 − t

ζ1 ζ3 − ζ32 = −t. This is clearly recognizable as the versal family of flat deformations of the orbifold singularity. By contrast, we now present an example of an involutive deformation of the regular part of a normal surface which is not stably embedded, based on the construction of unstable CR-perturbations introduced in [2]. While existence of a Stein completion for each individual member of this family is guaranteed, it is not simultaneously so, as will be seen below. Let C be a compact Riemann surface of genus g, and P ick (C) one of the connected components, i.e., a coset of the Jacobi variety, inside P ic(C). A natural holomorphic family of ruled surfaces π : F → P ick (C) is defined, such that for all λ ∈ P ick (C), π −1 (λ) corresponds to the total space of the associated line bundle of degree k over C. If Λ → C ×P ick (C) is the relative line bundle of this family, it is clear that the direct image with respect to projection onto P ick (C) of the sheaf of sections of Λ is not locally free a fact which lies at the heart of the following construction. Following [2], we assume C contains a Weierstrass point p0 such that the line bundle λ0 := [kp0 ] is very ample for an appropriate k. A suitable example of C is given by the locus of Z0k = Z1k + Z2k inside P2 ,

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for k ≥ 5. Note that the genus is of course related to k by the formula g=

1 (k − 1)(k − 2) . 2

Let Δ ⊂ C be a small disc, and ϕ : Δ → C a holomorphic map such that ϕ(0) = p0 . For k ≥ 5 the line bundle λ0 is very ample (in fact h0 (C, λ0 ) = 3), while neighbouring bundles of the form λt := [kϕ(t)] are not very ample (in fact h0 (C, λt ) = 1 , t = 0). Now consider the (holomorphic) map ψ : Δ → P ick (C) , ψ(t) = [kϕ(t)] . If  : F ∗ → P ick (C) denotes the “dual family”, i.e., each fibre −1 (λ) denotes the total space of the dual line bundle λ−1 , then let ψ ∗ F ∗ denote its pullback over the disc. The fibres of this restricted family will “blowdown”, via collapsing of the zero-section, to normal Stein surfaces provided each zero-section has a system of relatively compact, strictly pseudoconvex neighbourhoods. From the very-ampleness of λ0 it is clear that a holomorphic map which collapses the zero-section of  −1 (λ0 ) also embeds the resulting normal surface in C3 . Hence there is a smooth function on −1 (λ0 ) which is strongly plurisubharmonic on the complement of its zero-section, and by the previous consideration of strictly pseudoconvex coronas being preserved under small deformations of the regular part of a singular surface, we arrive at an individual-fibre-wise reduction of ψ ∗ F ∗ to normal Stein surfaces. Since the natural embedding of the central fibre does not extend stably to its neighbours the assumption that such a blowing-down of members of this specific family is “simultaneous”, i.e., that we have thereby defined an analytically fibred space, is unwarranted. The required condition for simultaneous blow-down is known to be the t-independence of dim H 1 (Xt , OXt ), which assures the extendability to neighbouring fibres of all holomorphic functions on the central fibre. Since all holomorphic functions on an analytic space are restrictions of holomorphic functions on an ambient CN , such extendability is equivalent to the stability of holomorphic embedding into CN . It remains to see that the family of deformations of the central fibre is moreover involutive for all t. Given P ick (C) ∼ = Cg /Z2g it follows that a small neighbourhood of p0 ∈ P ick (C) may be identified with any one of its translates via the integer lattice in H 1 (C, OC ). Letting L0 := −1 (λ0 ), and EC denote the divisor corresponding to the embedded

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image of C as the zero-section of L0 , we recall the natural exact sequence of sheaves 0 → OC → TL0 (− log(EC )) |C → TC → 0 , from which it follows there is a natural injection H 1 (C, OC ) → Γ1σ¯ (cf. [3]). In this way, ψ(t) is seen to be involutive for all t ∈ Δ. References 1. M. Brumberg and J. Leiterer, On the Compactification of Concave Ends. Math. Ann. 347 (2010), pp. 235-244. 2. D. Catlin and L. Lempert, A Note on the Instability of Embeddings of Cauchy-Riemann Manifolds, J. Geom. Anal. 2, No. 2 (1992) pp. 99-104. 3. A. Harris and M. Kolar, On Infinitesimal Deformations of the Regular Part of a Complex Cone Singularity, Kyushu J. Math. 65, No. 1 (2011), pp. 25-38. 4. K. Miyajima, CR Construction of the Flat Deformations of Normal Isolated Singularities, J. Alg. Geom. 8 (1999), pp. 403-470. 5. H. Rossi, Attaching Analytic Spaces to an Analytic Space along a Pseudoconcave Boundary, Proc. Conf. on Complex Analysis (Minneapolis, 1964), Springer, Berlin (1965), pp. 242-256. 6. S. S. T. Yau : Kohn-Rossi Cohomology and its application to the Complex Plateau Problem. Ann. Math. 113, No. 1 (1981), pp. 67-110.

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Connected components of regular fibers of differentiable maps

Dedicated to Professors Satoshi Koike and Laurentiu Paunescu on the occasion of their sixtieth birthdays

Jorge T. Hiratuka and Osamu Saeki Departamento de Matem´ atica, Instituto de Matem´ atica e Estat´ıstica, Universidade de S˜ ao Paulo, Caixa Postal 66.281, CEP: 05389-970, S˜ ao Paulo, SP, Brazil [email protected] Institute of Mathematics for Industry, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan [email protected] For a map between smooth manifolds, the space of the connected components of its fibers is called the Stein factorization. In our previous paper, we showed that for generic smooth maps, the Stein factorizations are triangulable. As an application, we show that every connected component of a regular fiber is null-cobordant if the top dimensional homology of the Stein factorization vanishes. Keywords: cobordism, regular fiber component, Stein factorization, generic map, triangulation AMS classification numbers: 57R45, 57R75, 57R20, 57R05

1. Introduction Let f : M → N be a generic C ∞ map between smooth manifolds. The space of the connected components of fibers of f is denoted by Wf . Then, we have the canonical quotient map qf : M → Wf and the natural map f¯: Wf → N such that f = f¯ ◦ qf . Such a decomposition of f into the composition of qf and f¯ is called the Stein factorization of f . Sometimes the quotient space Wf is also called the Stein factorization of f . It is known that when dim M > dim N , the Stein factorization of f : M → N , or the quotient space Wf , is a very important tool in studying

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the topological properties of the map f . Refer to [1,7–11,15], for example. In our previous paper [5], the authors have shown that the Stein factorization, and in particular the quotient space Wf , is triangulable for a large class of generic smooth maps f . In this paper, we use the triangulation of the Stein factorization in order to study the cobordism classes of the components of regular fibers of generic smooth maps. It is known that if the target manifold N of a smooth map f : M → N is connected, then the regular fibers of f are all cobordant. However, the components of regular fibers may not be cobordant to each other. We show that for a generic smooth map f : M → N , we can associate a top dimensional homology class γf ∈ Hn (Wf ), n = dim N , in such a way that if f has a regular fiber component that is not null-cobordant, then γf does not vanish, where the coefficient group is the cobordism group of manifolds of dimension m − n, m = dim M . The paper is organized as follows. In §2 we give a precise definition of the Stein factorization of a continuous map between topological spaces and its triangulation. We also recall the cobordism group of manifolds, and state our main theorem. In §3 we define the homology class γf and prove our main theorem. We also give some enlightening examples. In §4, we show that the above homology class γf gives a bordism invariant for maps whose fibers are connected. We also give some observations which show that the cobordism classes of regular fibers have little relation to the cobordism class of the source manifold in general. Finally we give some related problems. Throughout the paper, we will often abuse the terminology “simplicial complex” (or “simplicial map”) to indicate the corresponding polyhedron (resp. PL map). The symbol “≈” denotes a homeomorphism between topological spaces. 2. Preliminaries In this section, we define the notion of a triangulation of the Stein factorization of a map and state our main theorem. Definition 2.1. Let g : X → Y be a continuous map between topological spaces X and Y . Two points x, x ∈ X are g-equivalent if g(x) = g(x ) and the points x and x are in the same connected component of g −1 (g(x)) = g −1 (g(x )). We denote by Wg the quotient space with respect to the g-equivalence, endowed with the quotient topology. The quotient map is denoted by qg : X → Wg . Then there exists a unique continuous map g¯ : Wg → Y such that g = g¯ ◦ qg . The quotient space Wg or the

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commutative diagram g

X .... .... ... .... ... .... qg ........... ......... .

.. .........

Y

........... ..... ... .... . . . .... ¯ .... g .... ....



Wg is called the Stein factorization of g. There is a one-to-one correspondence between the quotient space and the space of the connected components of the fibers of g. Note that each fiber of the quotient map qg is connected. Remark 2.2. The space Wg is often called the quotient space or the Reeb space (or the Reeb complex ) of g. Let g : X → Y be a continuous map between topological spaces. Then, g is said to be triangulable if there exist simplicial complexes K and L, a simplicial map s : K → L, and homeomorphisms λ : |K| → X and μ : |L| → Y such that the following diagram is commutative: g

X −−−−→ , ⏐ λ⏐

Y , ⏐μ ⏐

|s|

|K| −−−−→ |L|, where |K| and |L| are polyhedrons associated with K and L, respectively, and |s| is the PL map associated with s. In [5], the authors have proved the following. Theorem 2.3. Let g : X → Y be a proper continuous map between locally compact topological spaces X and Y . If g is triangulable, then so is the Stein factorization of g. More precisely, we have the commutative diagram

X

.... .....

g ... .... .... .... .... . qg .......... ........ .

.......... .

Wg

Y

........ ... ........ ..... ..... ...... ..... . . . . ..... ¯ ..... g ...... .....

.. .....

λ

μ Θ

|ϕ|....................

... ...... ..... ..... ......

|K  |

|V | |s |

..... ..... |ψ| ..... ..... ..... ...... ........... .......... .

|L |

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for some finite simplicial complexes K  , L and V , some simplicial maps s : K  → L , ϕ : K  → V and ψ : V → L , and some homeomorphisms λ, μ and Θ. Let us recall the notion of a cobordism of manifolds. Let M0 and M1 be closed oriented manifolds with dim M0 = dim M1 (= m). We say that M0 and M1 are oriented cobordant if there exists a compact oriented (m + 1)dimensional manifold Q such that ∂Q = (−M0 ) ∪ M1 , where −M0 denotes the manifold M0 with the orientation reversed. Such a manifold Q is often called an oriented cobordism between M0 and M1 . The above relation clearly defines an equivalence relation. The equivalence class of a manifold M will be denoted by [M ]. We can define [M ] + [M  ] = [M ∪ M  ], in such a way that the set Ωm of all oriented cobordism classes of closed oriented m-dimensional manifolds forms an additive group. This is called the m-dimensional oriented cobordism group. In the above definition, if we ignore the orientations of the manifolds, then we get the m-dimensional (unoriented ) cobordism group, denoted by Nm . The groups Ωm and Nm have been extensively studied and their structures have been completely determined (see [16,17]). For example, the following is known. • • • • •

Ωm is a finitely generated abelian group. Nm is a finitely generated Z2 -module. Ωm is a finite group unless m is a multiple of four. Ω0 ∼ = Z, Ω1 = Ω2 = Ω3 = 0, Ω4 ∼ = Z, Ω5 ∼ = Z2 , . . . ∼ ∼ N0 = Z2 , N1 = 0, N2 = Z2 , N3 = 0, N4 ∼ = Z22 , N5 ∼ = Z2 , . . .

A closed (oriented) manifold M with [M ] = 0 is said to be (oriented ) null-cobordant. Our main theorem of this paper is the following. Theorem 2.4. Let M be a closed manifold and f : M → N a smooth map into a manifold N with m = dim M ≥ dim N = n. Assume that f is triangulable (e.g. a topologically stable map). Then, we have the following. (1) If there exists a regular fiber component of f which is not nullcobordant, then Hn (Wf ; Z2 ) = 0. (2) Suppose that both M and N are oriented (note that then the regular fibers are naturally oriented ). If there exists a regular fiber component of f which is not oriented null-cobordant, then Hn (Wf ; Ωm−n ) = 0.

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3. Proof of Theorem 2.4 Proof. Let s : K → L be a triangulation of f : M → N . Then, for the barycentric subdivision L of L, there exist a subdivision K  of K and a simplicial map s : K  → L such that |s | = |s| (for example, see [6]). By Theorem 2.3 (see also [5]), we have a triangulation of the Stein factorization as in the commutative diagram

M

.. .....

f .... ... .... .... .... . qf .......... ..........

.......... .

Wf

N

........ ... ....... ..... ...... ..... . . . . .... ..... ¯ ...... f ..... ......

... ....

λ

μ Θ

|ϕ|..................

... ...... ...... ...... .....

|K  |

|V | |s |

..... ..... |ψ| ..... ..... ..... ....... .......... .......... .

|L |,

where V is a finite simplicial complex of dimension n, ϕ : K  → V and ψ : V → L are simplicial maps with ψ being non-degenerate, and λ, Θ and μ are homeomorphisms. Here, a simplicial map is non-degenerate if it preserves the dimension of each simplex. For each n-simplex σ ∈ V , λ maps |ϕ|−1 (bσ ) homeomorphically onto −1 qf (Θ(bσ )), which is a component of a fiber of f , where bσ is a point in the interior of σ. By the Sard theorem, we may assume that f¯(Θ(bσ )) is a regular value of f . Then, define ωσ = [qf−1 (Θ(bσ ))] ∈ Nm−n , which is the cobordism class of the regular fiber component corresponding to σ ⊂ |V | ≈ Wf . Lemma 3.1. The cobordism class [qf−1 (Θ(bσ ))] does not depend on a choice of bσ ∈ Int σ. Proof. Let bσ be another point in Int σ such that f¯(Θ(bσ )) is a regular value of f . We can choose an embedded arc γ in Int σ connecting bσ and bσ in such a way that f¯(Θ(γ)) is transverse to f (for example, see [3, §4.3]). Then, λ(|ϕ|−1 (γ)) gives a smooth cobordism between qf−1 (Θ(bσ )) and qf−1 (Θ(bσ )).

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|K  | ≈ M |ϕ|

|s |

τ

τ¯ α ¯

Θ

|V | ≈ Wf

σr

σ1

|ψ| σ ¯2

σ2

σ ¯1

α Fig. 1.

Set cf =



μ

|L | ≈ N

The small arc α ¯

ωσ σ ∈ Cn (V ; Nm−n ),

σ

where σ runs over all n-simplices of V , and Cn (V ; Nm−n ) denotes the n-th chain group of V with coefficients in Nm−n . Lemma 3.2. We have ∂cf = 0 in Cn−1 (V ; Nm−n ), i.e. cf is an n-cycle. Proof. Let τ be an arbitrary (n − 1)-simplex of V , and let σ1 , σ2 , . . . , σr be the n-simplices of V containing τ as a face (see Fig. 1). We have only to show r  ωσj = 0 j=1

in Nm−n , i.e. the vanishing of the coefficient of τ in ∂cf . Let α ¯ be a small arc in |L | which intersects τ¯ = ψ(τ ) transversely in one ¯2 be the n-simplices of L adjacent to τ¯. point (see Fig. 1), and let σ ¯1 and σ Note that ψ(σi ) coincides with either σ ¯1 or σ ¯2 . We take α ¯ so that μ(¯ α) is a smooth arc in N transverse to f . Let α be the component of |ψ|−1 (¯ α) that intersects τ . Then, Q = qf−1 (Θ(α)) is an (m − n + 1)-dimensional compact manifold and r . ∂Q = λ(|ϕ|−1 (bσj )). j=1

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Therefore, we have r  j=1

ωσj =

r  * ) λ(|ϕ|−1 (bσj )) = 0 j=1

in Nm−n . Thus, cf defines a homology class γf ∈ Hn (Wf ; Nm−n ).

(3.1)

Furthermore, since dim Wf = n, we have γf = 0 if and only if cf = 0. Moreover, cf = 0 if and only if there exists a component of a regular fiber of f which is not null-cobordant. Therefore, if such a regular fiber component exists, we have Hn (Wf ; Z2 ) = 0, since Nm−n is isomorphic to Z2 ⊕ Z2 ⊕ · · · ⊕ Z2 , the direct sum of a finite number of copies of Z2 . The case where both M and N are oriented can be treated similarly. (In this case, the orientation of N induces an orientation of each n-simplex of V . Therefore, the argument also works with coefficients in the abelian group Ωm−n .) This completes the proof of Theorem 2.4. Example 3.1. Let us consider a tree T . Then, since H1 (T ; Z2 ) = 0, there exists no Morse function f1 : M15 → R on a closed 5-dimensional manifold M15 such that the quotient space Wf1 is homeomorphic to T and that f1 has CP 2 as a component of a regular fiber. (Recall that CP 2 is not nullcobordant.) Example 3.2. There exists a Morse function f2 : M25 → R on a closed 5-dimensional manifold M25 whose quotient space is as depicted in Fig. 2. The integer at each vertex denotes the index of the corresponding critical point, and the 4-manifold attached to each edge denotes the corresponding regular fiber component. Note that H1 (Wf2 ; Z) ∼ = H1 (Wf2 ; Ω4 ) ∼ = Z is generated by the homology class γf2 of (3.1). We note that the 5-dimensional manifold M25 can be chosen to be diffeomorphic to S 1 × CP 2 , which is null-cobordant. Example 3.3. There exists a Morse function f3 : M35 → R on a closed 5dimensional manifold M35 whose quotient space is as depicted in Fig. 3. Note that the quotient space Wf3 is homeomorphic to Wf2 ; however, γf3 = 0 in H1 (Wf3 ; Z) ∼ = Z, while γf2 = 0 in H1 (Wf2 ; Z). This means that even if the

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CP 2 CP 2

CP 2 CP 2 S4

S4 CP 2 0

2

5

3

1

4

CP 2 Fig. 2.

An example with non-vanishing γf2

S2 × S2 S

S 2 × S2 S4

4

S2 × S2 0

2

1

4

3

5

S4 Fig. 3.

An example with vanishing γf3

top dimensional homology group of the quotient space does not vanish, the map may not have a regular fiber component that is not null-cobordant. We note that the 5-dimensional manifold M35 can be chosen to be diffeomorphic to S 1 × S 4 , which is null-cobordant. By considering the product maps f/i = fi × idS k : Mi5 × S k → R × S k , k ≥ 1, i = 1, 2, 3, where idS k denotes the identity map of S k , we can construct examples of higher dimensional quotient spaces as well. Remark 3.1. For a smooth map f : M → N as in Theorem 2.4 with N being a closed manifold, we see that f¯∗ γf ∈ Hn (N ; Nm−n ) ∼ = Hn (N ; Z2 ) ⊗ ¯ Nm−n coincides with [N ] ⊗ Ff , where f : Wf → N is the continuous map that appears in the Stein factorization of f , [N ] ∈ Hn (N ; Z2 ) is the fundamental class of N , and Ff ∈ Nm−n is the unoriented cobordism class of a regular fiber of f . Note that when N is a closed manifold, the unoriented cobordism class Ff can be determined by the Stiefel-Whitney classes of M

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Fig. 4.

The singular fiber that determines the cobordism class

together with f ∗ (1∗ ) ∈ H n (M ; Z2 ), where 1∗ ∈ H n (N ; Z2 ) is the Poincar´e dual of the canonical generator 1 ∈ H0 (N ; Z2 ). A similar remark is also valid in the oriented case. For details, see [12, §3]. Remark 3.2. Even if every component of every regular fiber is nullcobordant, the source manifold may not be null-cobordant. For example, consider a C ∞ stable map f : CP 2 → R3 . Every component of every regular fiber is diffeomorphic to S 1 , which is null-cobordant. However, CP 2 is not null-cobordant. In fact, for a C ∞ stable map f : M 4 → R3 of a closed oriented 4dimensional manifold M 4 , the cobordism class of M 4 is determined by singular fibers as depicted in Fig. 4 (see [13,14]). Remark 3.3. Let M be a closed connected m-dimensional manifold. For a given Morse function f  : M → R, we can modify it by homotopy so that we get an ordered Morse function f : M → R. Here, a Morse function f is ordered if for every pair of critical points p and q of f with index(p) < index(q), we have f (p) < f (q). As is observed in [4], if f is an ordered Morse function with m ≥ 3, then every fiber of f is connected, and hence Wf is homeomorphic to a line segment. In this case, we have γf = 0, although the source manifold M may not be null-cobordant. 4. Further results Let f  : M → S 1 be a continuous map of a smooth closed connected manifold M of dimension m ≥ 3 into the circle. We assume that the induced homomorphism f∗ : π1 (M ) → π1 (S 1 ) is surjective. Then, it is known that f  is homotopic to a Morse map f : M → S 1 whose fibers are all non-empty and connected, where a Morse map is a smooth map whose critical points are all non-degenerate (for example, see [4, Theorem 1.3]). In this case, the quotient space Wf is canonically homeomorphic to S 1 through f¯. In the

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following, a map is said to be fiber-connected if all of its fibers are nonempty and connected. For such a map, the quotient space is canonically homeomorphic to the target manifold. Definition 4.1. Let fi : Mi → N be smooth maps of closed m-dimensional manifolds Mi into a manifold N , i = 0, 1. We say that f0 and f1 are bordant if there exists a cobordism Q between M0 and M1 (i.e. Q is a compact (m + 1)-dimensional manifold with ∂Q being identified with M0 ∪ M1 ), and a smooth map F : Q → N × [0, 1] such that fi = F |Mi : Mi → N × {i}, i = 0, 1. Such a map F is often called a bordism between f0 and f1 . If everything is oriented, then we say that f0 and f1 are oriented bordant (for details, see [2], for example). For example, if two smooth maps are homotopic, then they are bordant. Proposition 4.2. Let fi : Mi → N be smooth maps of m-dimensional closed connected manifolds Mi into a connected n-dimensional manifold N with m ≥ n ≥ 1, i = 0, 1. We suppose that fi are topologically stable and are fiber-connected, i = 0, 1. If f0 and f1 are bordant, then we have γf0 = γf1 ∈ Hn (N ; Nm−n ). If M0 , M1 and N are oriented, and f0 and f1 are oriented bordant, then we have γf0 = γf1 ∈ Hn (N ; Ωm−n ). Proof. Let F : Q → N × [0, 1] be the map as in Definition 4.1. By the Sard theorem, we can choose a point y ∈ N which is a common regular value of f0 and f1 . Then, by slightly perturbing F on the interior of Q, we may assume that F is transverse to the line segment {y} × [0, 1]. Then, we see that Q = F −1 ({y} × [0, 1]) gives a cobordism between regular fibers of f0 and f1 . Since, for each i = 0, 1, the regular fibers of fi are all cobordant, we get the required result. The above proposition implies that if two topologically stable fiberconnected maps fi : Mi → N , i = 0, 1, satisfy γf0 = γf1 , then they are not bordant. In particular, when M0 = M1 , f0 and f1 cannot be homotopic. Remark 4.3. Suppose that fi : Mi → N , i = 0, 1, are bordant as in Definition 4.1. We further assume that fi are topologically stable, i = 0, 1, and that γf0 does not vanish. Under these assumptions, we have a sufficient condition for the non-vanishing of γf1 as follows. Let F : Q → N × [0, 1] be the smooth map that gives a bordism between f0 and f1 . Let y be a regular value of f0 such that f0−1(y) contains a component which is not null-cobordant. Let α : [0, 1] → N × [0, 1] be a smooth

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embedding with α(0) = y × {0} such that α(1) ∈ N × {1} corresponds to a regular value of f1 and that α is transverse to F . Let R be the component of F −1 (α([0, 1])) which contains the component of f0−1(y) which is not nullcobordant. By choosing α and F generic enough, we may further assume that the map h = α−1 ◦ F : R → [0, 1] is a Morse function. If this Morse function has no critical points of index 1, then we can show that γf1 does not vanish. Proposition 4.4. Let m be an integer with m ≥ 3. For an arbitrary c ∈ Nm and for an arbitrary finite number of elements cj ∈ Nm−1 , j = 1, 2, . . . , k, there exist a smooth closed connected m-dimensional manifold M and fiber-connected Morse maps fj : M → S 1 such that [M ] = c and γfj ∈ H1 (S 1 ; Nm−1 ) ∼ = Nm−1 corresponds to cj , j = 1, 2, . . . , k. We also have a corresponding proposition for maps between oriented manifolds. Proof of Proposition 4.4. Take a closed connected (m − 1)-dimensional manifold Fj in the cobordism class cj for each j, and a closed connected m-dimensional manifold M  with [M  ] = c. Let us consider the closed connected m-dimensional manifold M given by & ' M = M   kj=1 (S 1 × Fj ) . Since S 1 × Fj bounds D 2 × Fj , it is null-cobordant, and hence we have [M ] = [M  ] = c. Note that for each j, M naturally decomposes as ((S 1 × Fj )  Int Dm ) ∪ Mj , where (S 1 × Fj )  Int Dm and Mj are attached along their sphere boundaries. Let us construct a continuous map fj : M → S 1 as follows. On (S 1 × Fj )  Int Dm , it is homotopic to the restriction of the projection S 1 × Fj → S 1 to the first factor and is a constant map on the boundary of (S 1 × Fj )  Int Dm . We define fj on Mj to be the constant map to the same value in S 1 . We may assume that fj is smooth on (S 1 × Fj )  Int D m and it has {∗} × Fj as a regular fiber. Then by [4, Theorem 1.3], fj is homotopic to a fiber-connected Morse map fj : M → S 1 . By construction, the regular fibers of fj are all cobordant to Fj . Hence, fj satisfy the desired properties.

Proposition 4.4 shows that at least for fiber-connected maps f , γf carries no information on the cobordism class of the source manifold. We end this paper by posing some problems.

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Problem 4.5. (1) How about the case of maps of manifolds with non-empty boundaries? (2) By associating an “invariant” of a (regular or singular) fiber component corresponding to certain dimensional simplices of Wf , can we define a homology class of Wf ? (3) Study such kind of homology classes and their relations to the geometry and topology of the manifolds and the map. For example, can we define Vassiliev type invariants for maps using such homology classes?

Acknowledgements The authors would like to thank the anonymous referee for stimulating questions and comments. The propositions in §4 were added after the referee’s question. The first author has been supported in part by CAPES, Brazil. The second author has been supported in part by JSPS KAKENHI Grant Number 23244008, 23654028.

References 1. O. Burlet and G. de Rham, Sur certaines applications g´ en´eriques d’une vari´et´e close ` a 3 dimensions dans le plan, Enseignement Math. (2) 20 (1974), 275–292. 2. P.E. Conner and E.E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1964. 3. S.K. Donaldson and P.B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. 4. D.T. Gay and R. Kirby, Indefinite Morse 2-functions; broken fibrations and generalizations, arXiv:1102.0750v2 [math.GT]. 5. J.T. Hiratuka and O. Saeki, Triangulating Stein factorizations of generic maps and Euler characteristic formulas, RIMS Kˆ okyˆ uroku Bessatsu, B38 (2013), 61–89. 6. J.F.P. Hudson, Piecewise linear topology, W.A. Benjamin, Inc., New York, 1969. 7. M. Kobayashi and O. Saeki, Simplifying stable mappings into the plane from a global viewpoint, Trans. Amer. Math. Soc. 348 (1996), 2607–2636. 8. L. Kushner, H. Levine and P. Porto, Mapping three-manifolds into the plane I, Boletin de la Sociedad Matem´ atica Mexicana 29 (1984), 11–31. 9. H. Levine, Classifying immersions into R4 over stable maps of 3-manifolds into R2 , Lecture Notes in Math., Vol. 1157, Springer-Verlag, Berlin, New York, 1985.

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10. W. Motta, P. Porto Jr. and O. Saeki, Stable maps of 3-manifolds into the plane and their quotient spaces, Proc. London Math. Soc. (3) 71 (1995), 158–174. 11. P. Porto Jr and Y.K.S. Furuya, On special generic maps from a closed manifold into the plane, Topology Appl. 35 (1990), 41–52. 12. O. Saeki, Studying the topology of Morin singularities from a global viewpoint, Math. Proc. Camb. Phil. Soc. 117 (1995) 223–235. 13. O. Saeki, Singular fibers and 4-dimensional cobordism group, Pacific J. Math. 248 (2010), 233–256. 14. O. Saeki and T. Yamamoto, Singular fibers of stable maps and signatures of 4-manifolds, Geom. Topol. 10 (2006), 359–399. 15. K. Sakuma, On the topology of simple fold maps, Tokyo J. Math. 17 (1994), 21–31. 16. R. Thom, Quelques propri´et´es globales des vari´et´es diff´erentiables, Comment. Math. Helv. 28 (1954), 17–86. 17. C.T.C. Wall, Determination of the cobordism ring, Ann. of Math. 72 (1960), 292–311.

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The reconstruction and recognition problems for homogeneous hypersurface singularities A. V. Isaev Department of Mathematics, The Australian National University, Canberra, ACT 0200, Australia [email protected] By the well-known Mather-Yau theorem, a complex hypersurface germ V with isolated singularity is fully determined by its moduli algebra A(V). The proof of this theorem does not provide an explicit procedure for recovering V from A(V), and finding such a procedure is a long-standing open question, called the reconstruction problem. In the present paper we survey a recently proposed method for reconstructing V from A(V) up to biholomorphic equivalence under the assumption that the singularity of V is homogeneous, in which case A(V) coincides with the Milnor algebra of V. As part of our discussion of the method, we give a characterization of the algebras arising from finite polynomial maps with homogeneous components of equal degrees. For gradient maps, the question of describing such algebras is a special case of the so-called recognition problem for the moduli algebras of general isolated hypersurface singularities. Keywords: isolated hypersurface singularities, the Mather-Yau theorem, Milnor algebras AMS classification numbers: 32S25, 13H10

1. Introduction Let On be the local complex algebra of holomorphic function germs at the origin in Cn with n ≥ 2. For a hypersurface germ V at the origin (considered with its reduced complex structure) denote by I(V) the ideal of elements of On that vanish on V. Fix a generator f of I(V) and let A(V) be the quotient of On by the ideal generated by f and all its first-order partial derivatives. The algebra A(V), called the moduli algebra or Tjurina algebra of V, is in fact independent of the choice of f as well as the coordinate system near the origin. Furthermore, the moduli algebras of biholomorphically equivalent hypersurface germs are isomorphic. It is clear that A(V) is non-zero if and only if V is singular. We assume that 0 < dimC A(V) < ∞, which occurs if and only if the singularity of V is isolated (see, e.g. Chapter 1 in [GLS]).

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By the well-known Mather-Yau theorem (see [MY]), two hypersurface germs V1 , V2 in Cn with isolated singularities are biholomorphically equivalent if their moduli algebras A(V1 ), A(V2 ) are isomorphic. Thus, given the dimension n, the moduli algebra A(V) determines V up to biholomorphism. For example, if dimC A(V) = 1, then V is biholomorphic to the germ of the hypersurface {z12 + · · · + zn2 = 0}, and if dimC A(V) = 2, then V is biholo2 morphic to the germ of the hypersurface {z12 + · · · + zn−1 + zn3 = 0}. The proof of the Mather-Yau theorem does not provide an explicit procedure for recovering the germ V from the algebra A(V) in general, and finding a way for reconstructing V (or at least some invariants of V) from A(V) is an interesting open problem, called the reconstruction problem (cf. [Y1], [Y2], [S]). Recently, we have proposed a method for restoring V from A(V) up to biholomorphic equivalence under the assumption that the singularity of V is homogeneous (see [IK]). In the present paper we discuss this method and related questions. Let V be a hypersurface germ having an isolated singularity. The singularity of V is said to be homogeneous if for some (hence for every) generator f of I(V) there is a coordinate system near the origin in which f becomes the germ of a homogeneous polynomial. In this case f lies in the Jacobian ideal J (f ) in On , which is the ideal generated by all first-order partial derivatives of f . Hence, for a homogeneous singularity, A(V) coincides with the Milnor algebra On /J (f ) for any generator f of I(V). Now, we let Q(z), with z := (z1 , . . . , zn ), be a holomorphic (m + 1)form on Cn , i.e. a homogeneous polynomial of degree m + 1 in the variables z1 , . . . , zn , where m ≥ 2. Consider the germ VQ of the hypersurface {Q(z) = 0} and assume that: (i) the singularity of VQ is isolated, and (ii) the germ of Q generates I(VQ ). These two conditions are equivalent to the nonvanishing of the discriminant Δ(Q) of Q (see Chapter 13 in [GKZ]). The method proposed in [IK] recovers the form Q (hence the germ VQ ) from the algebra A(VQ ) up to linear equivalence, where two forms Q1 , Q2 on Cn are called linearly equivalent if there exists a non-degenerate linear transformation L of Cn such that Q2 = Q1 ◦ L. We review the method of [IK] in Section 2. Define Q to be the gradient map Q : Cn → Cn , z → grad Q(z). The condition Δ(Q) = 0 means that the fiber Q−1 (0) consists of 0 alone, i.e. Q is finite at the origin. The main content of this method is recovery of the map Q from A(VQ ) up to linear equivalence, where we say that two maps Φ1 , Φ2 : Cn → Cn are linearly equivalent if there exist non-degenerate linear transformations L1 , L2 of Cn such that Φ2 = L1 ◦ Φ1 ◦ L2 . In fact, in [IK] we consider a more general situation.

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Let p1 , . . . , pn , be holomorphic m-forms on Cn and I the ideal in On generated by the germs of these forms at the origin. Define P to be the map of Cn given by z → (p1 (z), . . . , pn (z)) and set AP := On /I. We assume that dimC AP < ∞, which occurs if and only if P is finite at the origin, i.e. P−1 (0) = {0} (see Chapter 1 in [GLS]). The latter condition is equivalent to the non-vanishing of the resultant of p1 , . . . , pn (see Chapter 13 in [GKZ]). In [IK] we proposed a procedure (which requires only linearalgebraic manipulations) for explicitly recovering the map P from AP up to linear equivalence. Applying this procedure to the algebra A(VQ ) arising from Q, one obtains a map Q linearly equivalent to Q. It is then not hard to derive from Q an (m + 1)-form Q linearly equivalent to Q. A priori, one can attempt to apply the procedure of [IK] to any complex commutative associative finite-dimensional algebra. Therefore, a natural problem is to characterize the algebras isomorphic to AP for some P as above. In the case of gradient maps, this characterization problem is a special case of the well-known recognition problem for the moduli algebras of general isolated hypersurface singularities and the corresponding Lie algebras of derivations (see, e.g. [Y1], [Y2], [S]). In Section 2 we give an elementary solution of the recognition problem for the algebras AP prior to our review of the reconstruction method of [IK]. We conclude the paper by giving an example of application of the method of [IK] to the Milnor algebras of simple elliptic singularities of type /7 in Section 3. An analogous example for the Milnor algebras of simple E /6 was discussed in [IK]. elliptic singularities of type E Acknowledgement. This work is supported by the Australian Research Council. 2. The reconstruction method For fixed m, n ≥ 2 let P : Cn → Cn be a map that is finite at the origin and whose components p1 , . . . , pn are holomorphic m-forms on Cn . Define I to be the ideal in On generated by the germs of these forms at the origin and set AP := On /I. Since m ≥ 2, one has dimC AP ≥ n + 1 > 1. Before reviewing the method of [IK] for recovering the map P from AP , we will give a simple characterization of the algebras that arise from finite homogeneous polynomial maps as above among all complex commutative associative finite-dimensional algebras. Let A be a local unital complex commutative associative algebra with 1 < dimC A < ∞. We say that A has Property (∗) if it satisfies conditions (∗)1 –(∗)3 stated below.

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(∗)1 : A is a standard graded algebra, i.e. one has 0 A= Li , i≥0

where Li are linear subspaces of A with L0  C, Li Lj ⊂ Li+j for all i, j, and Ll = Ll1 for all l ≥ 1. In this case, if m is the (unique) maximal ideal of A, then 0 m= Li , i≥1

and Li is a complement to mi+1 in mi for all i ≥ 0, where m0 := A. Hence A is a standard graded algebra if and only if A is isomorphic to its associated graded algebra Gr(A) :=

ν 0

mi /mi+1 ,

i=0

where ν is the nil-index of m, i.e. the largest integer μ with mμ =  0 (note that m is a nilpotent algebra by Nakayama’s lemma). (∗)2 : A is a complete intersection. (∗)3 : for some M ≥ 2 the following holds: dimC mi /mi+1 = dimC PiN

for i = 1, . . . , M − 1,

N dimC mM /mM+1 = dimC PM − N,

where N := dimC m/m2 is the embedding dimension of A and for every i ≥ 1 we denote by PiN the vector space of all i-forms on CN (observe that N > 0). Note that the numbers dimC PiN are well-known:   i+N −1 N . dimC Pi = i Remark 2.1. Verification of conditions (∗)2 and (∗)3 for a given algebra A is not hard. Indeed, (∗)2 means that dimC H1 (Kf1 ,...,fN ) = N,

(2.1)

where Kf1 ,...,fN is the Koszul complex constructed from a basis f1 , . . . , fN in a complement to m2 in m (see, e.g. pp. 169–172 in [M]); computing the left-hand side of (2.1) is straightforward from the definitions. Further, computation of all the dimensions dimC mi /mi+1 required for verifying (∗)3

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is easily done as well. In contrast, verification of condition (∗)1 may be quite difficult, especially if one is interested in constructing a standard grading on A explicitly. For example, to deal with condition (∗)1 , one can first check whether Gr(A) is Gorenstein and, if this is the case, use the criterion for isomorphism of Gorenstein algebras obtained in [FIKK]. We note that a necessary condition for the existence of a standard grading is given in Proposition 8.1 in [FK]. We are now ready to state our result. Theorem 2.2. Let A be a local unital complex commutative associative algebra with 1 < dimC A < ∞. Then A is isomorphic to the algebra AP for some P if and only if A has Property (∗), in which case N = n and M = m = ν/n + 1. Proof. First, we let A = AP for some map P as above and show that A has Property (∗). Indeed, p1 , . . . , pn is a regular sequence in On (see Theorem 2.1.2 in [BH]), hence A satisfies (∗)2 . Next, we clearly have N = n, and for any i ≥ 0 define Li to be the linear subspace of A that consists of all elements represented by germs of forms in Pin . These subspaces form a standard grading on A, hence (∗)1 is satisfied. Further, condition (∗)3 obviously holds with M = m. Finally, as we noted in [IK], the embedding dimension n divides ν and one has m = ν/n + 1. Conversely, let A be a local unital complex commutative associative algebra with 1 < dimC A < ∞ having Property (∗). Choose a basis f1 , . . . , fN in a complement to m2 in m. The elements f1 , . . . , fN generate A (as an algebra), hence A is isomorphic to C[z1 , . . . , zN ]/R, where R is the ideal of all relations, i.e. polynomials P ∈ C[z1 , . . . , zN ] with P (f1 , . . . , fN ) = 0. Observe that R contains all homogeneous polynomials of degree greater than ν. It then follows that A is isomorphic to ON /R0 , where R0 is the ideal in ON generated by the germs of all relations. We will now use condition (∗)1 . Namely, choosing f1 , . . . , fN to be a basis in L1 , we see that R is generated by finitely many homogeneous relations, which yields R0 ⊂ m2N , where mN is the maximal ideal of ON . Next, since A is isomorphic to the quotient of the regular local ring ON by the ideal R0 ⊂ m2N , condition (∗)2 means that the minimal number of generators of R0 is equal to N (see, e.g. pp. 170–171 in [M]). Further, condition (∗)3 means that the ideal R contains no relations of degree less than M and that the linear subspace H ⊂ R of all homogeneous relations of degree M has dimension N .

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Choose a set of generators of R0 consisting of N elements, let g1 , . . . , gN be holomorphic functions representing the generators, and fix a basis p1 , . . . , pN of H. Then in a neighborhood of the origin we have

pi =

N 

aij gj ,

i = 1, . . . , N

(2.2)

j=1

for some holomorphic functions aij . On the other hand, if we complete the basis p1 , . . . , pN to a set of generators of R by adding homogeneous elements q1 , . . . , qL ∈ R of degree greater than M , each function gj can be written as gj =

N 

bjk pk +

k=1

L 

cj q ,

j = 1, . . . , N,

(2.3)

=1

where bjk , cj are holomorphic functions. Plugging (2.3) into (2.2) and collecting terms of degree M in both sides of the resulting identity, we see that the matrix (aij (0)) is non-degenerate. Letting (djk ) be the inverse of (aij ) near the origin (with djk (0) = bjk (0)), from equation (2.2) we then obtain gj =

N 

djk pk ,

j = 1, . . . , N,

k=1

which implies that the germs of p1 , . . . , pN generate R0 . Thus, we have shown that the germs of the elements of every basis of H generate R0 . Let p1 , . . . , pN be any basis and P := (p1 , . . . , pN ). By the condition dimC A < ∞, the map P is finite at the origin. We have thus proved that A is isomorphic to AP . The proof of Theorem 2.2 provides a method for producing a map P from any algebra A having Property (∗), but this method requires that a standard grading on A be explicitly defined. As we noted in Remark 2.1, finding such a grading may be hard. On the other hand, in article [IK] we proposed an algorithm for recovering P up to linear equivalence from the algebra AP , where one does not need to know a standard grading explicitly. Assuming that AP is given as an abstract algebra (i.e. by a multiplication table with respect to some basis), we summarize the main steps of the

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algorithm as follows (see [IK] for details): 1. Find m and its nil-index ν. 2. Determine n from the formula n = dimC m/m2 . 3. Determine m from the formula m = ν/n + 1. 4. Choose a complement to m2 in m and an arbitrary basis f1 , . . . , fn in this complement. 5. Calculate q1 (f ), . . . , qK (f ), where f := (f1 , . . . , fn ), q1 (z), . . . , qK (z) n . are all monomials of degree m in z := (z1 , . . . , zn ), and K := dimC Pm 6. Choose a complement S to mm+1 in mm . 7. Compute π(q1 (f )), . . . , π(qK (f )), where π : mm → S is the projection onto S with kernel mm+1 . 8. Find n linearly independent linear relations among the vectors π(q1 (f )), . . . , π(qK (f )): K 

γσρ π(qρ (f )) = 0,

σ = 1, . . . , n,

γσρ ∈ C.

ρ=1

9. The following formula then gives a map linearly equivalent to P: Φ : Cn → Cn ,

z → Γq(z),

where Γ := (γσρ )σ=1,...,n, ρ=1,...,K, and q := (q1 , . . . , qK ). Remark 2.3. The algorithm of [IK] shows, in particular, that if two algebras AP and AP are isomorphic, then the maps P and P are linearly equivalent. Hence, the sufficiency implication of Theorem 2.2 can be strengthened as follows: for every algebra A having Property (∗) there exist exactly one, up to linear equivalence, map P such that A is isomorphic to AP . One can apply the above algorithm to P = Q := grad Q for a holomorphic (m + 1)-form Q on Cn with Δ(Q) = 0. Let Φ be a map linearly equivalent to Q derived from the algebra A(VQ ), where VQ is the germ of the hypersurface {Q = 0} at the origin. For any n × n-matrix D we now n D n introduce the holomorphic differential 1-form ω Φ := i=1 ΦD i dzi on C , D D D where Φ1 , . . . , Φn are the components of the map Φ := D Φ. Consider the equation D

dω Φ = 0

(2.4)

as a linear system with respect to the entries of the matrix D. It is explained in [IK] that in order to recover Q from A(VQ ) up to linear equivalence one

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needs to complement the above algorithm with the following two steps: 10. Find a matrix D ∈ GL(n, C) satisfying system (2.4). 11. Integrate ΦD to obtain an (m + 1)-form linearly equivalent to Q.

3. An example of application of the reconstruction method We will now illustrate our reconstruction method, as well as Property (∗), by an example of a 1-parameter family of algebras. Namely, for t ∈ C, t = ±2, let At be the complex commutative 9-dimensional algebra given with respect to a certain basis e1 , . . . , e9 by the following multiplication table: e1 ej = ej f orj = 1, . . . , 9, e22 = e6 − e2 e3 =

2 1 e7 + e8 − e9 , 3 3

1 1 1 1 2 1 e7 + e8 − e9 , e2 e4 = e5 − e7 + e9 , e2 e5 = e7 + 3e8 + e9 , 3 3 3 3 3 3

t t t e2 e6 = − e7 + e9 , e2 e7 = e8 , e2 e8 = 0, e2 e9 = −2e8 , e23 = − e8 , 6 6 2 t t e3 e4 = − e7 − e9 , e3 ej = 0, j = 5, 7, 8, 9, e3 e6 = e8 , 3 6 e24 = e3 −

(3.1)

1 1 1 1 2 1 e7 + 2e8 + e9 , e4 e5 = e7 − e9 , e4 e6 = e7 + 3e8 + e9 , 3 3 3 3 3 3

t e4 e7 = e8 , e4 e8 = 0, e4 e9 = e8 , e25 = e8 , e5 ej = 0, j = 6, 7, 8, 9, e26 = − e8 , 2 e6 ej = 0, j = 7, 8, 9, e7 ej = 0, j = 7, 8, 9, e8 ej = 0, j = 8, 9, e29 = 0.

As we will see later, every algebra At has Property (∗). It is clear from (3.1) that e1 = 1 (the identity element) and At is a local algebra with maximal ideal mt = e2 , . . . , e9 , where · denotes linear span. We then have m2t = e3 , e5 , e6 , e7 , e8 , e9 , m3t = e7 , e8 , e9 , m4t = e8 , m5t = 0, hence ν = 4. Further, by the formula at Step 2 of the algorithm given in Section 2, we obtain n = 2, which together with formula at Step 3 yields m = 3. We now list all monomials of degree 3 in z := (z1 , z2 ) as follows: q1 (z) := z13 ,

q2 (z) := z12 z2 ,

q3 (z) := z1 z22 ,

q4 (z) := z23

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(here K = 4). Next, we let f1 := e2 , f2 := e4 , which for f := (f1 , f2 ) yields t t 2 1 q1 (f ) = − e7 + e9 , q2 (f ) = e7 + 2e8 + e9 , 6 6 3 3 1 1 t t e7 − e9 , q4 (f ) = − e7 − e9 . 3 3 3 6 Further, define S := e7 , e9 . Clearly, S is a complement to m4t in m3t . Then for the projection π : m3t → S with kernel m4t one has t t 2 1 π(q1 (f )) = − e7 + e9 , π(q2 (f )) = e7 + e9 , 6 6 3 3 q3 (f ) =

1 1 t t e7 − e9 , π(q4 (f )) = − e7 − e9 . 3 3 3 6 The vectors π(q1 (f )), π(q2 (f )), π(q3 (f )), π(q4 (f )) satisfy the following two linearly independent linear relations: π(q3 f )) =

2π(q1 (f )) + tπ(q3 (f )) = 0, tπ(q2 (f )) + 2π(q4 (f )) = 0. Hence we have

 Γ=

20t0 0t02

 ,

which for q(z) := (q1 (z), q2 (z), q3 (z), q4 (z)) yields ⎞ ⎛ 2z13 + tz1 z22 ⎠. Φ(z) = Γq(z) = ⎝ 2 3 tz1 z2 + 2z2 Further, for

 D=

d11 d12 d21 d22



system (2.4) is equivalent to the following system of equations: t(d11 − d22 ) = 0, td12 − 6d21 = 0,

(3.2)

6d12 − td21 = 0. If t = 0, ±6, the only non-degenerate solutions of (3.2) are non-zero scalar matrices. Integrating ΦD for such a matrix D we obtain a form proportional to Qt := z14 + tz12 z22 + z24 .

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If t = 0, any non-degenerate solution of (3.2) is a diagonal matrix with non-zero d11 , d22 . Integrating ΦD for such a matrix D we obtain the form ' 1& d11 z14 + d22 z24 , 2 which is linearly equivalent to Q0 := z14 + z24 by suitable dilations of the variables. The remaining case t = ±6 is more interesting. In this situation D is a solution of (3.2) if and only if d11 = d22 ,

d12 = ±d21 .

Such a matrix D is non-degenerate if and only if d211 ∓ d212 = 0. One obvious solution is given by d11 = 1, d12 = 0, in which case integrating ΦD one obtains a form proportional to Q±6 := z14 ± 6z12 z22 + z24 (observe that Q6 and Q−6 are in fact linearly equivalent). If Q is a form arising from any other solution, then the algebras A(VQ ) and A(VQ±6 ) are isomorphic, which by the Mather-Yau theorem implies that VQ and VQ±6 are biholomorphically equivalent hence Q and Q±6 are linearly equivalent. For example, letting d11 = 0, d12 = 1 one obtains ⎞ ⎛ ±6z12 z2 + 2z23 ⎠, ΦD = ⎝ ±2z13 + 6z1 z22 and integration of ΦD leads to the form Q±6 := ±2z13 z2 + 2z1 z23 . As explained above, by the Mather-Yau theorem each of Q±6 is linearly equivalent to each of Q±6 . This last fact can also be understood without referring to the MatherYau theorem as follows. It is well-known that all non-equivalent binary quartics with non-vanishing discriminant are distinguished by the invariant J :=

I32 , Δ

where for any quartic Q = a4 z14 + 4a3 z13 z2 + 6a2 z12 z22 + 4a1 z1 z23 + a0 z24

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one has

⎛

⎞ a4 a3 a2 I3 (Q) = det ⎝ a3 a2 a1 ⎠ a2 a1 a0

(see, e.g. pp. 28–29 in [O]). We then have I(Q±6 ) = I(Q±6 ) = 0, and therefore each of Q±6 is linearly equivalent to each of Q±6 as stated above. Thus, assuming that the algebra At has Property (∗), we have shown that it is isomorphic to A(VQt ) for all t = ±2. Note that A(VQt ) is a well-known family of algebras. Indeed, A(VQt ) is isomorphic to the moduli algebra A(Vt ) of the germ Vt of the following hypersurface in C3 : 1 2 (z1 , z2 , z3 ) ∈ C3 : z14 + tz12 z22 + z24 + z32 = 0 , t = ±2. These hypersurface singularities are called simple elliptic singularities of /7 . We stress that, unlike VQ , these singularities are not homogetype E t neous. Observe that one basis in which the algebra A(Vt ) is given by multiplication table (3.1) is as follows: e1 = 1, e2 = Z1 + Z1 Z2 , e3 = Z22 + Z1 Z22 , e4 = Z2 + Z12 Z2 , e5 = Z1 Z2 + 2Z1 Z22 , e6 = Z12 + 3Z12 Z2 , e7 = Z12 Z2 + Z1 Z22 , e8 = Z12 Z22 , e9 = Z12 Z2 − 2Z1 Z22 , where Zj is the element represented by the germ of the coordinate function zj , j = 1, 2. We will now independently check that At has Property (∗) for every t = ±2. First of all, the subspaces 3 4 2 2 2 1 L0 := e1 , L1 := e2 − e5 + e7 − e9 , e4 − e7 − e9 , 3 3 3 3 4 3 1 1 2 2 L2 := e3 − e7 + e9 , e5 − e7 + e9 , e6 − 2e7 − e9 , 3 3 3 3 L3 := e7 , e9 ,

L4 := e8 

form a standard grading on At , thus At satisfies condition (∗)1 . Further, we clearly have N = 2 and dimC Pi2 = i + 1, which immediately yields that condition (∗)3 is satisfied with M = 3. It remains to verify condition (∗)2 . This condition means that dimC H1 (Kf1 ,f2 ) = 2,

(3.3)

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where Kf1 ,f2 is the Koszul complex constructed from any basis f1 , f2 in a complement to m2t in mt . As before, we choose f1 = e2 , f2 = e4 , in which case (3.3) is equivalent to dimC

{(u, v) ∈ At × At : ue2 + ve4 = 0} = 2. {(u, v) ∈ At × At : u = −we4 , v = we2 for some w ∈ At }

The above identity easily follows by expanding u, v, w with respect to the basis e1 , . . . , e9 and utilizing multiplication table (3.1). Thus, the algebra At has Property (∗) as stated. References BH.

FIKK.

FK. GKZ.

GLS.

IK.

MY. M. O. S. Y1. Y2.

Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993. Fels, G., Isaev, A., Kaup, W. and Kruzhilin, N., Isolated hypersurface singularities and special polynomial realizations of affine quadrics, J. Geom. Analysis 21 (2011), 767–782. Fels, G. and Kaup, W., Nilpotent algebras and affinely homogeneous surfaces, Math. Ann. 353 (2012), 1315–1350. Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, Resultants and Multidimensional Determinants, Modern Birkh¨ auser Classics, Birkh¨ auser Boston, Inc., Boston, MA, 2008. Greuel, G.-M., Lossen, C. and Shustin, E., Introduction to Singularities and Deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. Isaev, A. V. and Kruzhilin, N. G., Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras, to appear in Proc. Amer. Math. Soc. Mather, J. and Yau, S. S.-T., Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982), 243–251. Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986. Olver, P., Classical Invariant Theory, London Mathematical Society Student Texts 44, Cambridge University Press, Cambridge, 1999. Schulze, M., A solvability criterion for the Lie algebra of derivations of a fat point, J. Algebra 323 (2010), 2916–2921. Yau, S. S.-T., Solvable Lie algebras and generalized Cartan matrices arising from isolated singularities, Math. Z. 191 (1986), 489–506. Yau, S. S.-T., Solvability of Lie algebras arising from isolated singularities and nonisolatedness of singularities defined by sl(2, C) invariant polynomials, Amer. J. Math. 113 (1991), 773–778.

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Openings of differentiable map-germs and unfoldings

Dedicated to Professor Satoshi Koike for his 60th birthday

Goo Ishikawa Department of Mathematics, Hokkaido University, Japan [email protected] The algebraic notion of openings of a map-germ is introduced in this paper. An opening separates the self-intersections of the original map-germ, preserving its singularities. The notion of openings is different from the notion of unfoldings. Openings do not unfold the singularities. For example, the swallowtail is an opening of the Whitney cusp map-germ from plane to plane and the open swallowtail is a versal opening of them. Openings of map-germs appear as typical singularities in several problems of geometry and its applications. The notion of openings has close relations to isotropic map-germs in a symplectic space and integral map-germs in a contact space. We describe the openings of Morin singularities, namely, stable unfoldings of map-germs of corank one. The relation of unfoldings and openings are discussed. Moreover we provide a method to construct versal openings of map-germs and give versal openings of stable map-germs (R4 , 0) → (R4 , 0). Lastly the relation of lowerable vector fields and openings is discussed. Keywords: versal opening, ramification module, lowerable vector field AMS classification numbers: 58C27, 14E40, 32S45

1. Introduction There is a sequence of well-known singularities of map-germs: The Whitney cusp f : (R2 , 0) → (R2 , 0), f (x, u) = (x3 + ux, u), the swallowtail F : (R2 , 0) → (R3 , 0), F (x, u) = (f (x, u), x4 + 23 ux2 ), and the open swallowtail F/ : (R2 , 0) → (R4 , 0), F/(x, u) = (F (x, u), x5 + 49 ux3 ). They have the same singular locus and the same kernel field of the differential along the singular locus, while the self-intersections are resolved. What is the algebraic structure behind them? One of answers to the above question is presented in this paper. In fact we observe, for the swal-

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lowtail F (x, u) = (x3 + ux, u, x4 + 23 ux2 ), we see that 5 6 2 4 4 d(x4 + ux2 ) = x d(x3 + ux) − x2 du ∈ d(x3 + ux), du E2 . 3 3 9

a)

)

I

For the open swallowtail F/ (x, u) = (x3 + ux, u, x4 + 23 ux2 , x5 + 49 ux3 ), we have that 5 6 4 5 10 d(x5 + ux3 ) = x2 d(x3 + ux) − x3 du ∈ d(x3 + ux), du E2 . 9 3 9 Here d means the exterior differential and E2 denotes the R-algebra of C ∞ function-germs on (R2 , 0). As the key construction, we introduce the notion of openings of multigerms of mappings. To do this, first we summarise the auxiliary notions in this paper. Let f : (Rn , A) → (Rm , b) be a multi-germ of a C ∞ map with n ≤ m. Here A is a finite subset of Rn , b ∈ Rm and f (A) = {b}. We define the Jacobi module Jf of f by Jf = {

m  j=1

pj dfj | pj ∈ ERn ,A (1 ≤ j ≤ m) } ⊂ Ω1Rn ,A

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in the space Ω1Rn ,A of 1-form-germs on (Rn , A). Note that Jf is just the first order component of the graded differential ideal Jf• in Ω•Rn ,A generated by df1 , . . . , dfm . Then the singular locus, the non-immersive locus, of f is given by Σf = {x ∈ (Rn , A) | rank Jf (x) < n}. Also we consider the kernel field Ker(f∗ : T Rn → T Rm ) of the differential of f , along Σf .  For another map-germ f  : (Rn , A) → (Rm , b ), n ≤ m , if Jf  = Jf , then Σf  = Σf and Ker(f∗ ) = Ker(f∗ ). Then define the ramification module Rf of f by Rf = {h ∈ ERn ,A | dh ∈ Jf }, (cf. [12] [15]).  For f : (Rn , A) → (Rm , b), f  : (Rn , A) → (Rm , b ), easily we see that Jf  = Jf if and only if Rf  = Rf (Lemma 2.1). Definition 1.1. Given h1 , . . . , hr ∈ Rf , the map-germ F : (Rn , A) → Rm × Rr = Rm+r defined by F = (f1 , . . . , fm , h1 , . . . , hr ) is called an opening of f , while f is called a closing of F . Then, for any opening F of f , we have RF = Rf , JF = Jf , ΣF = Σf and Ker(F∗ ) = Ker(f∗ ). For example, the swallowtail is an opening of the Whitney cusp. The open swallowtail is an opening of the swallowtail and of the Whitney cusp. Note that an opening of an opening of f is an opening of f . Definition 1.2. An opening F = (f, h1 , . . . , hr ) of f is called a versal opening (resp. a mini-versal opening) of f : (Rn , A) → (Rm , b), if 1, h1 , . . . , hr form a (minimal) system of generators of Rf as an ERm ,b -module via f ∗ : ERm ,b → ERn ,A . Note that a versal opening of an opening of f is a versal opening of f . An opening of a versal opening of f is a versal opening of f . A mini-versal opening F : (Rn , A) → Rm+r of f is unique up to leftequivalence and a versal opening G : (Rn , A) → Rm+s of f is left-equivalent to a mini-versal opening composed with an immersion (Rn , A) → Rm+r → Rm+s (Proposition 2.14).

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Openings of map-germs appear as typical singularities in several problems of geometry and its applications. The openings naturally appear in the classification problem of “tangential singularities”([5] [16] [17]). Open swallowtails, open folded umbrellas, etc. appear as tangent varieties ([1]). We have applied opening constructions to the solution of the “stable”classification problem of tangent varieties to generic submanifolds in [17]. Moreover openings are related to singularities of isotropic mappings in symplectic spaces. Let T ∗ Rn = R2m be the 2m-dimensional symplectic space with the  n symplectic form ω = m → R2m a multi-germ i=1 dpi ∧ dxi , and f : (R , A) m ∗ of isotropic mapping. Since f ω = 0, we have that i=1 (pi ◦ f )d(xi ◦ f ) is closed, so it is exact and there exists e ∈ ERn ,A such that de =

m  (pi ◦ f )d(xi ◦ f ). i=1

Define g : (R , A) → R by g(x) = (x1 ◦f (x), . . . , xm ◦f (x)). Then e ∈ Rg . m Conversely, given e ∈ Rg , we have de = i=1 ai dgi for some functions a1 , . . . , am , and we obtain an isotropic multi-germ f : (Rn , A) → R2m by pi ◦ f = ai , xi ◦ f = gi , (1 ≤ i ≤ m). The opening (g, e) : (Rn , A) → Rm+1 is the frontal germ associated to f . For example the frontal of a open WhitneyMorin umbrella is the folded umbrella (see Proposition 4.1). The open folded umbrella appears also as a “frontal-symplectic singularity” [18]. Several geometric applications of the theory of openings are given in [16] [17]. n

m

Remark 1.3 (Relations with known notions). Pellikaan (his thesis and [25], p.358), de Jong and van Straten ([19], p.185) introduced the notion of primitive ideal  ∂h ∈ I, 1 ≤ i ≤ n} I = {h ∈ On | h ∈ I, ∂xi of an ideal I of the ring On of holomorphic function-germs on (Cn , 0). It is motivated for the deformation theory of non-isolated singularities. We can introduce the analogous notion to it in C ∞ case and also the notion of primitive ring by   ∂h I = {h ∈ En | ∈ I, 1 ≤ i ≤ n} ∂xi It is related to the notion of ramification module as follows: Let f : (Rn , 0) → (Rm , 0), n ≤ m be a map-germ. We take as I the ideal Jf generated by n-minors of the Jacobi matrix of f , which is associated with the singular locus of f . If n = 1 (the case of curves), then we have that

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7

Jf = Rf . However, they 7 7  are different in general in the case n ≥ 2. In ∗ fact f Em ⊆ Jf and Jf does not have the natural Em -module structure via f ∗ in general, while R7f does. For example, let f (x1 , x2 ) = (x1 , x22 ),  Jf = R + x22 E2 , and Rf = E1 + x22 E2 = f : (R2 , 0) → (R2 , 0). Then ∗ 3 ∗ f E2 + x2 f E2 where E1 means the ring of functions of x1 . Mond [24] introduced a notion R0 (f ), which is more closely related to Rf , also from the motivation of deformation theory on map-germs in complex analytic category. Its C ∞ analogue is given by R0 (f ) = {h ∈ En | dfi1 ∧ · · · ∧ dfn−1 ∧ dh ∈ Jf dx1 ∧ · · · ∧ dxn , 1 ≤ i1 < · · · < in−1 ≤ m}. Then clearly we have Rf ⊆ R0 (f ). We have that, for two map-germ7f, f  ,  Jf . Rf = Rf  implies R0 (f ) = R0 (f  ). If n = 1, then Rf = R0 (f ) = Moreover we have that, if n = m and the singular locus of f is nowhere dense, then Rf = R0 (f ), using Cramer’s rule in linear algebra. We conjecture that the equality Rf = R0 (f ) holds also in the case n < m under a rather mild condition. Note that, in Propositions 4.1 and 4.3 of [24], they were already given the related results to the results in the present paper (Lemma 2.2, Corollary 2.10, Proposition 2.16 and Proposition 5.2). In §2, we give a detailed exposition on ramification modules and openings of multi-germs. In §3, the relation of unfoldings and openings are discussed. We treat the problem to find a versal opening of an unfolding of a given map-germ. Then the notion of extendability of an unfolding is introduced. If the given mapgerm is of corank one, then any unfolding is extendable and, in particular its versal opening is obtained from that of its stable unfolding. Then, in §4, we give the explicit presentation on versal openings of stable unfoldings of map-germs of corank 1, namely, versal openings of Morin maps. In §5, we remark the existence of versal openings in finite analytic case. In §6, we give a direct method to find versal openings for several examples and show the existence of the versal opening for any stable map-germ (R4 , A) → (R4 , b), explicitly. In §7, the relation of lowerable vector fields of map-germs and openings is discussed. In this paper we often abbreviate E(Rn ,A) by EA , the R-algebra of C ∞ function-germs on (Rn , A). If A = {a1 , . . . , as }, then we denote by mi the maximal ideal consisting of h ∈ EA with h(ai ) = 0. We set mA = ∩si=1 mi . If A consists of the origin, then we use En , mn instead of EA , mA respectively.

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All manifolds and mappings which we treat in this paper are assumed to be of class C ∞ , unless otherwise stated. 2. Ramification modules and openings In Introduction, we have introduced the notion of openings based on that of Jacobi modules and ramification modules. Lemma 2.1. For map-germs f : (Rn , A) → (Rm , b), f  : (Rn , A) →  (Rm , b ), we have that Jf  = Jf if and only if Rf  = Rf . Proof. It is clear that Jf  = Jf implies Rf  = Rf . Conversely suppose Rf  = Rf . Then any component fj of f  belongs to Rf  = Rf , hence dfj ∈ Jf . Therefore Jf  ⊆ Jf . By the symmetry we have Jf  = Jf . Lemma 2.2. Let f : (Rn , A) → (Rm , b) be a map-germ. Then we have (1) f ∗ Eb ⊆ Rf ⊆ EA . (2) Rf is an Eb -submodule via f ∗ : Eb → EA of EA . (3) Rf is C ∞ -subring of EA . (4) If τ : (Rm , b) → (Rm , b ) is a diffeomorphism-germ, then Rτ ◦f = Rf . If σ : (Rn , A ) → (Rn , A) is a diffeomorphism-germ, then Rf ◦σ = σ ∗ (Rf ). Proof. Assertions (1) and (2) follow from the fact that, if h ∈ Rf and  dh = m j=1 pj dfj , then d{(k ◦ f )h} =

m 

{(k ◦ f )pj + h (∂k/∂yj )} dfj .

j=1

Assertion (3) follows from the fact that, if h1 , . . . , hr ∈ Rf and if τ : Rr → R is a C ∞ function, then r  ∂τ d{τ (h1 , . . . , hr )} = (h1 , . . . , hr ) dhi ∈ Jf . ∂y i i=1

Assertion (4) follows from the fact that Jτ ◦f = Jf and Jf ◦σ = σ ∗ (Jf ). Lemma 2.3. Let f : (Rn , A) → (Rm , b) be a map-germ with A = {a1 , . . . , as }. We denote by fi : (Rn , ai ) → (Rm , b) the restriction of f 8s to (Rn , ai ). Then Rf ∼ = i=1 Rfi as Eb -module. Proof. We have the isomorphism ϕ : Rf → (h|(Rn ,ai ) )si=1 .

8s i=1

Rfi defined by ϕ(h) =

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Remark 2.4. Any multi-germ f : (Rn , A) → (Rm , b), #(A) = s, is right9 9 equivalent to a map-germ of form s fi : s (Rn , 0) → (Rm , b) from the disjoint union of s-copies of (Rn , 0). A map-germ f : (Rn , A) → (Rm , b) is called finite if EA is a finite Eb module. The condition is equivalent to that dimR EA /(f ∗ mb )EA < ∞ by the preparation theorem (see for example [4]). Moreover f is finite if and only if K-finite and n ≤ m ([26]). Proposition 2.5. If f : (Rn , A) → (Rm , b) is finite and of corank at most one. Then we have (1) Rf is a finite Eb -module. Therefore there exists a versal opening of f . (2) 1, h1 , . . . , hr ∈ Rf generate Rf as Eb -module if and only if they generate the vector space Rf /(f ∗ mb )Rf over R. Proof. In the case A consists of a point, the assertions are proved in Theorem 1.3 of [13] and Corollary 2.4 of [15]. For a general finite set A, the assertions are reduced to the case that A consists of a point by Lemma 2.3.

Remark 2.6. In §5, we define the analytic counterpart Rω f of the notion of ramification modules for an analytic map-germ f : (Rn , A) → (Rm , 0). Then it is essentially obvious that, if f is a finite map-germ, then Rω f is a finite module over the ring O(Rn ,A) of analytic function-germs. In fact O(Rn ,A) is a Noetherian ring and it is a finite O(Rm ,0) -module via f ∗ . Moreover Rω f is an O(Rm ,0) -submodule of O(Rn ,A) . Since every submodule of a finite module over a Noetherian ring is finite, we have that Rω f is a finite n module over the ring O(R ,A) . Furthermore, in §5, we show that Rf is a finite Eb -module if f is finite and analytic (Proposition 5.2). For a map-germ f : (Rn , A) → (Rm , b), n ≤ m, we have defined in Introduction the notions of openings and versal openings of f . 2.7. (1) Let h : (R, 0) → (R, 0), h(x) = x2 . Then Rh = 6 5Example 3 1, x f ∗ (E1 ) . The map-germ H : (R, 0) → (R2 , 0), H(x) := (x2 , x3 ), the simple cusp map, is the mini-versal opening of h. 5 6 (2) Let g : (R, 0) → (R, 0), g(x) = x3 . Then Rg = 1, x4 , x5 f ∗ (E1 ) . The map-germ G : (R, 0) → (R3 , 0), G(x) := (x3 , x4 , x5 ) is the mini-versal opening of g.

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94 3 (3) Let f5 : (R2 , 0) → (R2 , 0), f (x, u) 6 = (x +ux, u), an unfolding of 2g. Then 2 4 4 2 5 3 Rf = 1, x + 23 ux , x + 9 ux f ∗ (E2 ) . The map-germ F : (R , 0) →

2 ux2 , x5 + 49 ux3 ), the (R4 , 0) defined by F (x, u) := (x3 + ux, u, x4 + 23 open swallowtail is the mini-versal opening of Whitney cusp f . (4) Let consider the multi-germ k : (R, A) → (R2 , 0), A = {0, 1} defined by k(t) = (t, 0) near t = 0 and k(t) = (0, t − 1) near t = 1. Then Rk is generated by 1 ∈ EA and s ∈ EA defined by s(t) = 1 near t = 0 and s(t) = 0 near t = 1, over k ∗ E2 . Then K = (k, s) : (R, A) → R3 is the versal opening of k. In fact K resolves the self-intersection by the over and underpasses.

Example 2.8. Here we add several additional illustrative examples. Let us consider the following five map-germs: f : (R2 , 0) → (R2 , 0), g : (R2 , 0) → (R3 , 0), h : (R2 , 0) → (R3 , 0), k : (R2 , 0) → (R2 , 0),  : (R2 , 0) → (R2 , 0) defined by f (x, t) = (x, t2 ),

g(x, t) = (x, xt, t2 ),

k(x, t) = (x2 , t2 ),

h(x, t) = (x2 , xt, t2 ),

(x, t) = (x2 − t2 , xt).

Then we have Rk  Rh  Rg ,

R  Rh ,

Rf  Rg .

In fact Jf = dx, tdtE2 ,

Jg = dx, xdt, tdtE2 ,

Jk = xdx, tdxE2 ,

Jh = xdx, xdt + tdx, tdtE2 ,

J = xdx − tdt, tdx + xdtE2 .

Then we see that Rf is minimally generated by 1, t3 over f ∗ E2 , Rg is minimally generated by 1, t3 over g ∗ E3 , and Rh is minimally generated by 1, x3 , x2 t, xt2 , t3 over h∗ E3 . Moreover we have that Rk is minimally generated by 1, x3 , t3 , x3 t3 over k ∗ E2 and that R is minimally generated by 1, x3 − 3xt2 , 3x2 t − t3 , x2 (x2 + t2 )2 over ∗ E2 . 9 9 Remark 2.9 (continued with Remark 2.4). Let s fi : s (Rn , 0) → (Rm , b) be a multi-germ of map. Suppose Fi = (fi ; hi1 , . . . , hiri ) : (Rn , 0) → (Rm+ri , b × 0) be a versal opening of fi . Then, setting r = max1≤i≤s ri 9 9 and hij = 0 if j > ri , then F = s (fi , hi1 , . . . , hiri , . . . ) : s (Rn , 0) → 9 (Rm+r , b × 0) is a versal opening of s fi . By Proposition 2.5, we have Corollary 2.10. Let f : (Rn , A) → (Rm , b) be finite and of corank at most one. Then there exists a versal opening of f .

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Remark 2.11. The existence of a versal opening for a finite C ∞ map-germ of arbitrary corank is still open. However, using Proposition 5.2 together with the analytic result, we can show that Corollary 2.10 for finitely Adetermined map-germs (Rn , A) → (Rm , b) with n ≤ m without the corank condition. Moreover we have the following: Corollary 2.12. Let f : (Rn , A) → (Rm , b) be finite and of corank at most one. Then an opening F = (f, h1 , . . . , hr ) of f is a mini-versal opening of f , namely, 1, h1 , . . . , hr ∈ Rf form a minimal system of generators of Rf as Eb -module if and only if they form a basis of R-vector space Rf /(f ∗ mb )Rf . The following is useful for the classification problem of map-germs in a geometric context ([16,17]). Proposition 2.13. Let f : (Rn , A) → (Rm , b), n ≤ m be a C ∞ mapgerm. (1) For any versal opening F : (Rn , A) → (Rm+r , F (A)) of f and for any opening G : (Rn , A) → (Rm+s , G(A)), there exists an affine bundle map Ψ : (Rm+r , F (A)) → (Rm+s , G(A)) over (Rm , b) such that G = Ψ ◦ F . (2) For any mini-versal openings F : (Rn , A) → (Rm+r , F (A)) and F  : (Rn , A) → (Rm+r , F  (A)) of f , there exists an affine bundle isomorphism Φ : (Rm+r , F (A)) → (Rm+r , F  (A)) over (Rm , b) such that F  = Ψ ◦ F . In particular, the diffeomorphism class of mini-versal opening of f is unique. (3) Any versal openings F  : (Rn , A) → (Rm+s , F  (A)) of f is diffeomorphic to (F, 0) for a mini-versal opening F of f . Proof. (1) Let F = (f, h1 , . . . , hr ) and G = (f, k1 , . . . , ks ). Since kj ∈ Rf , there exist cj0 , cj1 , . . . , cjr ∈ Eb such that kj = cj0 ◦ f + (cj1 ◦ f )h1 + · · · + (cjr ◦ f )hr . Then it suffices to set Ψ(y, z) = (y, (cj0 (y) + cj1 (y)z1 + · · · + cjr (y)zr )1≤j≤s ). (2) By (1) there exists an affine bundle map Ψ with F  = Ψ ◦ F . From the minimality, we have that the matrix (cji (b)) is regular. (See Corollary 2.12.) Therefore Ψ is a diffeomorphism-germ. (3) Let F = Ψ ◦ F  for some affine bundle map Ψ. Then the matrix (cji (b)) is of rank r. Therefore F  is diffeomorphic to (F, k1 , . . . , ks−r ) for some kj ∈ Rf . Write each kj = Kj ◦ F for some Kj ∈ EF (a) . Then we set Ξ(y, z, w) = (y, z, w − K ◦ F ). Then Ξ is a local diffeomorphism on Rm+r+(s−r) and Ξ ◦ (F, k1 , . . . , ks−r ) = (F, 0).

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Two map-germs F : (Rn , A) → (Rp , B) and G : (Rn , A) → (Rq , C) is called L-equivalent, or, left-equivalent, if there exists a diffeomorphismgerm Ψ : (Rp , B) → (Rq , C) such that G = Ψ ◦ F . Then, by Proposition 2.13, we have: Corollary 2.14. Let f : (Rn , A) → (Rm , b) be a C ∞ map-germ (n ≤ m). Then a mini-versal opening of f is unique up to L-equivalence. A versal opening of f is L-equivalent to a mini-versal opening composed with an immersion. We introduce further the following notion: Definition 2.15. An opening F = (f, h1 , . . . , hs ) of a map-germ f = (f1 , . . . , fm ) : (Rn , A) → (Rm , b) is called L-minimal if f1 , . . . , fm , h1 , . . . , hr minimally generate the C ∞ -ring Rf over R, in other words, if Rf = (f, h)∗ (Em+r ) and (f, h) in minimal with this property. Then we have a similar uniqueness result for L-equivalence of L-minimal openings. In Example 2.8, we have seen that 1, x3 , t3 , x3 t3 minimally generate Rk as k ∗ E2 -module. However 1, x3 , t3 already minimally generate Rk as k ∗ E2 C ∞ -ring. Therefore K = (x2 , y 2 , x3 , t3 , x3 t3 ) is a mini-versal opening of k = (x2 , y 2 ) : (R2 , 0) → (R2 , 0) and K  = (x2 , y 2 , x3 , t3 ) is a L-minimal opening of k. Lastly we show injectivity of versal openings: Proposition 2.16. Let f : (Rn , A) → (Rm , b) be a finite map-germ. Suppose F : (Rn , A) → (Rm+r , F (A)) is a versal opening of f . Then F has an injective representative. Remark 2.17. The corresponding result to Proposition 2.16 in the analytic case is already proved in an earlier paper (Corollary 1.2 of [13]). By Proposition 5.2 proved in §5 together with the analytic result, we can show that Proposition 2.16 for finitely A-determined map-germs (Rn , A) → (Rm , b) with n ≤ m. Note that the proof of Corollary 1.2 in [13] heavily depends, via the finite coherence theorem in analytic geometry, on that the map-germ is finite and analytic. It seems very difficult to find a unified proof of Proposition 2.16 for C ∞ case with a similar vein to the analytic

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case, because of the lack of the notion of “coherence” in the general C ∞ case. Here we provide an alternative proof applicable to any finite C ∞ mapgerms, which can be applied to finite real analytic map-germs, of course. The proof is similar to the proof for a similar result which was given in [9]. Proof of Proposition 2.16. Let A = {a1 , . . . as }. If each restriction F |(Rn ,ai ) for i = 1, 2, . . . , s is injective, then F is injective. In fact, for each i and for each v ∈ Rm , we can take a map-germ F i,v : (Rn , A) → (Rm , Fi (A)) which coincides with F near aj , (j = i) and F +v near ai . Since any F i,v is an opening of f , and F is a versal opening, different branches of F can not intersect each other. (See Proposition 2.13 (1)). Now suppose that f is a mono-germ f : (Rn , a) → (Rm , b). Assume the versal opening map-germ F = (f, h) : (Rn , a) → (Rm+r , F (a)) of f has no injective representative. Then there must be a sequence of points bi in Rm+r which tends to F (a) when i → ∞, and a1i , a2i , a1i = a2i in Rn which tend to a respectively when i → ∞, such that F (a1i ) = F (a2i ) = bi . We may suppose bi = bj if i = j. Take a C ∞ function h on Rm such that the support is a disjoint union of small balls centred at bi , i = 1, 2, . . . . Note that such function must be infinitely flat at b, that is, j ∞ h(b) = 0. Take the C ∞ function k = F ∗ h on Rn . Since F is finite, the support of k is a disjoint union of closed neighbourhoods of a1i and those of a2i , after shrinking the neighbourhood of {bi} on which h is non-zero if necessary. Take the function k on Rn which coincides with k except on the closed neighbourhoods of a2i , and is identically zero there. Then we see that k  is C ∞ , k  belongs to RF = Rf and k  (a1i ) = k (a2i ) for any i = 1, 2, . . . . In fact, k is C ∞ on Rn  {0} and it extends to an infinitely flat m ∗ function on Rn at a. Moreover we have dk = i=1 F (∂h/∂yi )dfi + r ∗ j=1 F (∂h/∂zj )dhj , where y1 , . . . , ym , z1 , . . . , zr are the coordinates of m+r . Take the function ai (resp. bj ) on Rn which coincides with R ∗ F (∂h/∂yi) (resp. F ∗ (∂h/∂zj )) except on the closed neighbourhoods of a2i , and is identically zero there. Then ai and bj are C ∞ and, then we have  r  dk  = m i=1 ai dfi + j=1 bj dhj ∈ JF = Jf . Therefore we have k ∈ Rf .  n m+1 , (b, 0)) of f . Since F Consider the opening (f, k ) : (R , a) → (R is a versal opening of f , we must have k  = τ ◦ F for a function-germ τ : (Rn , a) → R. See Proposition 2.13 (1), or, the definition of versal openings (Definition 1.2). Then we must have k (a1i ) = τ (F (a1i )) = τ (F (a2i )) = k  (a2i ) for a sufficiently large i. This leads a contradiction. Therefore we have that F is injective.

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Remark 2.18. By Proposition 2.16, the versal opening F = (f, h1 , . . . , hr ) of a finite map-germ f is injective. However F is not necessarily finitely Adetermined. For example, the cuspidal edge is a versal opening of the fold map and it is not finitely A-determined (Example 2.8 and Remark 2.18). In fact a higher order perturbation of F destroys the singular locus of F . On the other hand, whether an opening F of f is versal or not depends only on finite jets of h1 , . . . , hr . This follows from the preparation theorem (cf. Proposition 2.13). Remark 2.19. Related to the above Proposition 2.16, T. Gaffney suggested a relation of the notion of openings and that of weak normalisations [6], which is left to be our open problem. 3. Unfoldings and openings We recall the notion of unfolding of map-germs ([21]). Let f : (Rn , A) → (Rm , b) be a map-germ. An unfolding of f is a mapgerm F : (Rn+ , A × 0) → (Rm+ , (b, 0)) of form F (x, u) = (F1 (x, u), u) and F1 (x, 0) = f (x), for (x, u) ∈ (Rn+ , A × 0). For another unfolding G : (Rn+ , A × 0) → (Rm+ , (b, 0)), F and G are called isomorphic if there exist an unfolding Σ : (Rn+ , A×0) → (Rn+ , A× 0) of the identity map on (Rn , A) and an unfolding T : (Rm+ , (b, 0)) → (Rm+ , (b, 0)) of the identity map on (Rm , b) such that G ◦ Σ = T ◦ F . Proposition 3.1 (Unfoldings and openings). Let f : (Rn , A) → (Rm , b) be a C ∞ map-germ and F : (Rn+ , A × 0) → (Rm+ , (b, 0)) be an unfolding of f . Let i : (Rn , A) → (Rn+ , A × 0) be the inclusion, i(x) = (x, 0). Then we have: (1) i∗ RF ⊂ Rf . (2) If f is of corank ≤ 1 with n ≤ m, then i∗ RF = Rf . If 1, H1 , . . . , Hr generate RF via F ∗ , then 1, i∗ H1 , . . . , i∗ Hr generate Rf via f ∗ . Proof. For the mono-germ case the assertions are proved in Proposition 1.6 of [13], Lemma 2.4 of [14]. Here we present the proof for the general case: (1) is clear. (2) Let H ∈ RF . Then dH ∈ JF . Hence d(i∗ H) = i∗ (dH) ∈ i∗ JF ⊂ Jf . Therefore i∗ H ∈ Rf . Let f be of corank at most one. Suppose h ∈ Rf . m Then dh = j=1 aj dfj for some aj ∈ Ea . There exist Aj , Bk ∈ E(a,0) such m  that i∗ Aj = aj and the 1-form j=1 Aj d(F1 )j + k=1 Bk dλk is closed (cf. Lemma 2.5 of [15]). Then there exists an H ∈ E(a,0) such that dH =  m ∗ ∗ j=1 Aj d(F1 )j + k=1 Bk dλk ∈ JF and d(i H) = i (dH) = dh. Then

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there exists c ∈ R such that h = i∗ H + c = i∗ (H + c), and H + c ∈ RF . Therefore h ∈ i∗ RF . Since i∗ is a homomorphism over j ∗ : E(b,0) → Eb , where j : (Rm , 0) → (Rm+ , 0) is the inclusion j(y) = (y, 0), we have the consequence. An unfolding F : (Rn+ , A × 0) → (Rm+ , (b, 0)) of a map-germ f : (Rn , A) → (Rm , b) is called extendable if i∗ RF = Rf for the inclusion i : (Rn , A) → (Rn+ , A × 0). By Proposition 3.1, we have: Corollary 3.2. If corank of f is at most one, then any unfolding of f is extendable. In §6, we will see that there exist non-extendable unfoldings for mapgerms of corank ≥ 2. Therefore the opening constructions do not behave well under unfoldings in general. 4. Openings of stable maps of corank one We will give the explicit versal opening in the case of corank one. As is seen in Remark 2.9, it is sufficient to treat the case of mono-germs, namely, germs f : (Rn , 0) → (Rm , 0) of corank one. Moreover, by Corollary 3.2, it is sufficient to treat the case that f is stable, namely, f is a Morin map. Let k ≥ 0, m ≥ 0. To present the normal forms of Morin maps, consider variables t, λ = (λ1 , . . . , λk−1 ), μ = (μij )1≤i≤m,1≤j≤k and polynomials F (t, λ) = tk+1 +

k−1 

λj tj ,

i=1

Let f : (R

k+km

, 0) → (R

m+k+km

Gi (t, μ) =

k 

μij tj , (1 ≤ i ≤ m).

j=1

, 0) be a Morin map defined by

f (t, λ, μ) := (F (t, λ), G(t, μ), λ, μ), for the above polynomials F and G. For  ≥ 0, we denote by F() , Gi() the polynomials  t  t  F() (t, λ) = s F (s, λ)ds, Gi () (t, μ) = s Gi (s, μ)ds. 0

0

Then we have: Proposition 4.1 (Theorem 3 of [12]). The ramification module Rf of the Morin map f is minimally generated over f ∗ Em+k+km by the 1 + k + (k − 1)m elements 1, F(1) , . . . , F(k) , G1 (1) , . . . , G1 (k−1) , . . . , Gm (1) , . . . , Gm (k−1) .

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The map-germ F : (Rk+mk , 0) → (Rm+k+km × Rk+(k−1)m , 0) = (R2(k+km) , 0) defined by & ' F = f, F(1) , . . . , F(k) , G1 (1) , . . . , G1 (k−1) , . . . , Gm (1) , . . . , Gm (k−1) is a mini-versal opening of f . Proof. The first half is proved in [12]. The second half follows from the definition. Remark 4.2. It is shown in [12] moreover that F is an isotropic map for a symplectic structure on R2(k+km) . Note that the fact that the open swallowtail is a Lagrangian variety with respect to a certain symplectic structure was found first by Arnol’d and Givental’ (see [8]). In particular we have: Lemma 4.3. Let  be a positive integer and F = (F1 (t, u), u) : (Rn , 0) → (Rn , 0) an unfolding of f : (R, 0) → (R, 0), f (t) = F1 (t, 0) = t . Suppose H1 , . . . , Hr ∈ RF ∩ mn . Then 1, H1 , . . . , Hr generate RF via F ∗ if and only 2 i∗ H1 , . . . , i∗ Hr generate m+1 1 /m1 . In particular F1(1) , . . . , F1(−1) form a system of generators of RF via F ∗ over En . Proof. It is easy to show that Rf = R + m1 . By Proposition 2.5 (2), 1, H1 , . . . , Hr generate RF as En -module via F ∗ if and only if they generate RF /F ∗ (mn )RF over R. Since 2 RF /F ∗ (mn )RF ∼ = (R + m1 )/(f ∗ m1 )(R + m1 ) ∼ = m+1 1 /m1

we have the consequence. 5. Versal openings of analytic map-germs In this section we discuss the case f is analytic. First we recall the complex analytic case briefly from [13]. Let (X, A) be a germ of complex analytic space at a finite set A with the structure sheaf OX,A , and f = (f1 , . . . , fm ) : (X, A) → (Cm , b) a finite analytic map-germ. In the graded differential OX,A -algebra (de Rham algebra) ΩX,A on (X, A), consider the graded differential ideal If generated by df1 , . . . , dfm . Then the differential d on ΩX,A induces the f ∗ OCm homomorphism d : ΩX,A /If → ΩX,A /If and then OCm ,b -homomorphism

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d : f∗ (ΩX,A /If ) → f∗ (ΩX,A /If ). Then we consider the i-th cohomology Hi (f∗ (ΩX,A /If ); d) for the complex (f∗ (ΩX,A /If ), d). It is evident that Hi (f∗ (ΩX,A /If ); d) is a finite OCm ,b -module (cf. Remark 2.6). In fact, we have moreover: Proposition 5.1 (Proposition 1.1 of [13]). Hi (f∗ (ΩX,A /If ); d) is a coherent OCm ,b -module (i = 0, 1, 2, . . . ). We remark that the stalk of H0 (f∗ (ΩX,A /If ); d) at A is the complex analytic counterpart of Rf in the real C ∞ case. We write it Rhol f to distin∞ guish with the real C case. By Proposition 5.1, in particular, that Rhol f is a finite OCm -module. Now let f : (Rn , A) → (Rm , b) be a finite real analytic map-germ. We denote by ORn ,A (resp. ORm ,b ) the germ of sheaf of analytic functions on (Rn , A) (resp. (Rm , b)). Then, besides with Rf , we consider the sheaf Rω f := {h ∈ ORn ,A | dh ∈ df1 , . . . , dfm ORn ,A } and the direct image f∗ (Rω f ) as ORm ,b -module. Then we see that, in parω ticular, f∗ (Rf ) is a finite ORm ,b -module by Proposition 5.1 in the case X = (Cn , A). Thus we have that f∗ (Rω f ) is generated over ORm ,b by . Moreover it turns out that F = (f, h1 , . . . , hr ) : some 1, h1 , . . . , hr ∈ Rω f (Rn , A) → (Rm+r , b × h(A)) is injective (See [13]). Then we show the following: Proposition 5.2. Let f : (Rn , 0) → (Rm , 0) be a finite analytic map-germ. ∗ Suppose 1, h1 , . . . , hr generate Rω f over ORm ,0 via f . Then 1, h1 , . . . , hr generate Rf over ERm ,0 via f ∗ . Proof. First we may suppose hi (0) = 0, (1 ≤ i ≤ r). The opening F = (f1 , . . . , fm , h1 , . . . , hr ) : (Rn , 0) → (Rm+r , 0) of f is injective by [13]. Let Fp stand for the R-algebra of formal functions on (Rn , p). (Fp = / ERn ,p /m∞ Rn ,p , and it is the completion of ORn ,p as well.) Define a sheaf FRn 8 on Rn by F/Rn (U ) = p∈U Fp , for any open subset U of Rn . Denote by F/n the stalk of F/Rn at 0. For the definition see also Ch. III §4, pp. 45–46 of [20]. Then F/n is faithfully flat over On (Ch. III §4, p. 47, Corollary 4.13 of [20]). Define the formal counterpart / f = {( R gp )p∈(Rn ,0) ∈ F/n | d gp ∈ df1,p , . . . , dfn,p Fp , p ∈ (Rn , 0)}

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of Rf . Here  gp means the formal function at a point p near 0 defined by g ∈ ERn ,p . Then, for our F = (f, h1 , . . . , hr ), we have that 1, h1 , . . . , hr / f over F/m . generate R For a map-germ F : (Rn , 0) → (Rm+r , 0), we say that a function-germ g ∈ En formally belongs to F ∗ Em+r if, for any q ∈ (Rm+r , 0), there exists a formal function  k ∈ Fq such that, for any p ∈ F −1 (q),  gp = F ∗  k. Then, it is known the following Glaeser’s type theorem on characterisation of composite differentiable functions: If g ∈ En formally belongs to F ∗ Em+r , then g belongs to F ∗ (Em+r ). In fact the theorem follows, for instance, from Theorem D and Theorem C (3) of [2]. (We apply Theorem D of [2] to the case that φ = F is a finite map-germ, and X = (Rn , 0), Y = (Rm+r , 0), s = 1, p = q = r = 1, A = 1, B = 0 for the notations in [2].) See also Theorem 11.8 of [3]. Since F is injective in our case, we have that any element of Rf formally belongs to F ∗ Em+r , therefore we have Rf ⊆ F ∗ Em+r . Since F is an opening of f , we have that Rf = F ∗ Em+r . Let π : (Rm+r , 0) → (Rm , 0) be the projection. Then π ∗ : Em → Em+r is the inclusion. Regard F ∗ Em+r as an Em+r -module via F ∗ . By the preparation theorem, 1, h1 , . . . , hr generate F ∗ Em+r = Rf as Em -module via F ∗ ◦ π ∗ = f ∗ if 1, h1 , . . . , hr generate F ∗ Em+r /(f ∗ mm )F ∗ Em+r over R. We will show that ∗ ∗ ∗ m∞ n ∩ F Em+r ⊆ (f mm )F Em+r .

m

Define h = i=1 fi2 : (Rn , 0) → (R, 0). Since f is finite, h−1 (0) = {0}, and moreover, the norms of 1/h and its partial derivatives up to order say  are bounded above by 1/xα for some α = α() > 0. Then 1/h is a multiplier for the ideal m∞ n in the sense of Malgrange (Ch. IV §1, p.54, Proposition 1.4 ∞ of [20]). Hence, for any k ∈ m∞ function on (Rn , 0) and it is n , k/h is a C ∞ ∗ an element of mn , if we set (k/h)(0) = 0. Moreover let k ∈ m∞ n ∩ F Em+r . Then k/h formally belongs to F ∗ Em+r . In fact, the Taylor series of k/h at 0 ∈ Rn is 0, and, outside of 0, k/h is a composite function of F . Therefore, again by the Glaeser’s type theorem as above, we have k/h ∈ F ∗ Em+r . Then m m   fi2 )(k/h) = fi (fi k/h) ∈ (f ∗ mm )F ∗ Em+r . k=( i=1

i=1

∗ ∗ ∗ Thus we have m∞ n ∩ F Em+r ⊆ (f mm )F Em+r . ∗ Now let H ∈ Rf = F Em+r . Then

H ≡ a0 ◦ f + a1 ◦ f · h1 + · · · + ar ◦ f · hr ≡ a0 (0) + a1 (0)h1 + · · · + ar (0)hr ,

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for some functions a0 , a1 , . . . , ar , modulo (f ∗ mm )F ∗ Em+r + m∞ n ∩ F ∗ Em+r ⊆ (f ∗ mm )F ∗ Em+r . Thus we see 1, h1 , . . . , hr generate F ∗ Em+r / (f ∗ mm )F ∗ Em+r over R, and therefore they generate Rf over ERm ,0 via f ∗ . 6. The cases of corank ≥ 2 If corank(f ) ≥ 2, then the restriction of a versal opening of an unfolding of f is not necessarily a versal opening of f . That phenomenon was observed already in [15]. We utilise Proposition 5.2 if necessary to treat the following examples. Example 6.1 (cf. Example 2.8). Let f : (R2 , 0) → (R2 , 0), h(x, y) = ( 12 x2 , 12 y 2 ) = (z, w). Then Rf is minimally generated by 1, x3 , y 3 , x3 y 3 over f ∗ ER2 ,0 . Therefore if we set F : (R2 , 0) → (R5 , 0) by 1 1 F (x, y) = ( x2 , y 2 , x3 , y 3 , x3 y 3 ), 2 2 then F is the mini-versal opening of f . Here we give a concrete method to find the minimal generators as above. Let h ∈ ER2 ,0 = E2 . Then by the preparation theorem we have h ≡ (a ◦ f )x + (b ◦ f )y + (c ◦ f )xy, (mod. f ∗ E2 ). The condition that h ∈ Rf is equivalent to that dh belongs to Jacobi module Jf . We calculate dh ≡ (a ◦ f )dx + (b ◦ f )dy + (c ◦ f )(ydx + xdy), (mod. Jf ), and set (a ◦ f )dx + (b ◦ f )dy + (c ◦ f )(ydx + xdy) = Axdx + Bydy, for some function A, B ∈ E2 . Again by the preparation theorem, we put A = (a1 ◦ f ) + (a2 ◦ f )x + (a3 ◦ f )y + (a4 ◦ f )xy, B = (b1 ◦ f ) + (b2 ◦ f )x + (b3 ◦ f )y + (b4 ◦ f )xy. Then Ax = (a1 ◦ f )x + (a2 ◦ f )x2 + (a3 ◦ f )xy + (a4 ◦ f )x2 y = 2(za2 ) + a1 x + 2(za4 )y + a3 xy, By = (b1 ◦ f )y + (b2 ◦ f )xy + (b3 ◦ f )y 2 + (b4 ◦ f )xy 2 = 2(wb3 ) + 2(wb4 ) + b1 y + b2 xy.

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omitting “◦f ”, where z = 12 x2 and w = 12 y 2 . Then we have a + cy = 2za2 + a1 x + (2za4 )y + a3 xy, b + cx = 2wb3 + 2(wb4 ) + b1 y + b2 xy and therefore (a − 2za2 ) + (−a1 )x + (c − 2za4 )y + (−a3 )xy = 0, (b − 2wb3 ) + (c − 2wb4 )x + (−b1 )y + (−b2 )xy = 0. Since E2 is free over f ∗ E2 in this example, we have a = 2za2 , a1 = 0, c = 2za4 , a3 = 0, b = 2wb3 , c = 2wb4 , b1 = 0, b2 = 0. Then we have (a4 , b4 ) = k(w, z) for a function k ∈ E2 and h ≡ (2za2 ◦ f )x + (2wb3 ◦ f )y + (2zwk ◦ f )xy, (mod. f ∗ E2 ). Thus we find a minimal system of generators 1, 2zx, 2wy, 2zwxy, namely 1, x3 , y 3 , x3 y 3 of Rf over f ∗ E2 . In each case of the following three examples, we have the mini-versal openings using Proposition 5.2. Example 6.2. Let g : (R3 , 0) → (R3 , 0) be a map-germ defined by 1 1 g(x, y, u) = (z, w, u) = ( x2 + uy, y 2 + ux, u), 2 2 which is an unfolding of f in Example 6.1. Then Rg is minimally generated over g ∗ ER3 ,0 by 1 and ψ3 ψ55,0 ψ50,5 ψ63,3

= = = =

x3 + y 3 + 3xyu, x5 + 5x3 yu − 12x2 u3 + 9yu4 , y 5 + 5xy 3 u − 12y 2 u3 + 9xu4 , x3 y 3 − 12x2 y 2 u2 − 11x3 u3 − 11y 3 u3 − 12xyu4 .

Therefore i∗ Rg  Rf , where i : (R2 , 0) → (R3 , 0), i(x, y) = (x, y, 0), and we see that g is not an extendable unfolding of f . The versal opening of g is given by G : (R3 , 0) → (R7 , 0) = (R3 ×R4 , 0), G(x, y, u) = (g(x, y, u), x3 + y 3 + 3xyu, x5 + 5x3 yu − 12x2 u3 + 9yu4 , y 5 + 5xy 3 u − 12y 2u3 + 9xu4 , x3 y 3 − 12x2 y 2 u2 − 11x3 u3 − 11y 3u3 − 12xyu4 ). Then 1 1 G(x, y, 0) = ( x2 , y 2 , x3 + y 3 , x5 , y 5 , x3 y 3 ) 2 2

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is not a versal opening of f = ( 12 x2 , 12 y 2 ). Note that the element ψ3 gives a Lagrange immersion of type D4+ , which is a Lagrange stable lifting of g. Other elements are obtained by operating lowerable vector fields of g to ψ3 . See §7. Example 6.3 (Hyperbolic case). Let h : (R4 , 0) → (R4 , 0) be the stable map-germ 1 1 h(x, y, λ, μ) = (z, w, λ, μ) = ( x2 + yλ, y 2 + xμ, λ, μ) 2 2 of K-class I2,2 ([22]). Then Rf is minimally generated over h∗ ER4 ,0 by 1 and ϕ4 = x3 μ + y 3 λ + 3xyλμ 3 2 2 2 2 ϕ3,2 5 = x y − 2x yλμ + xλ μ 2 3 2 2 2 ϕ2,3 5 = x y − 2xy λμ + yλ μ 5 3 3 ϕ5,0 5 = x + 5x yλ + 15yλ μ 5 3 3 ϕ0,5 5 = y + 5xy μ + 15xλμ

ϕ6 = x3 y 3 − 3xyλ2 μ2 . We have the mini-versal opening H : (R4 , 0) → (R4 × R6 , 0) = (R10 , 0) of h by 2,3 5,0 0,5 H = (f, ϕ4 , ϕ3,2 5 , ϕ5 , ϕ5 , ϕ5 , ϕ6 ).

Moreover we see that j ∗ Rh  Rg ( ER3 ,0 ),

(j ◦ i)∗ Rh  i∗ Rg  Rf ( ER3 ,0 ),

where j : (R3 , 0) → (R4 , 0), j(x, y, u) = (x, y, u, u). Thus the unfolding h of f is not extendable, which is also not extendable regarded as an unfolding of g as well. Now we show the concrete way of calculations for Example 6.3 to make sure ourselves: Let k ∈ ER4 ,0 = E4 . By the preparation theorem, we set, k = (a0 ◦ h) + (a1 ◦ h)x + (a2 ◦ h)y + (a3 ◦ h)xy. Then dk ≡ ((a1 ◦ h) + (a3 ◦ h)y)dx + ((a2 ◦ h) + (a3 ◦ h)x)dy (mod. Jh ).

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We suppose dk is equal to the form Adz + Bdw + Cdλ + Ddμ = (Ax + Bμ)dx + (Aλ + By)dy + (Ay + C)dλ + (Bx + D)dμ. Then we have (a1 ◦ h) + (a3 ◦ h)y = Ax + Bμ, (a2 ◦ h) + (a3 ◦ h)x = Aλ + By, Ay + C = 0, Bx + D = 0. Then C = −Ay, D = −Bx. Now set A = (A0 ◦ h) + (A1 ◦ h)x + (A2 ◦ h)y + (A3 ◦ h)xy, B = (B0 ◦ h) + (B1 ◦ h)x + (B2 ◦ h)y + (B3 ◦ h)xy. Then we have Ax = (2zA1 − 4wA3 ) + (A0 + 4λμA3 )x + (−2λA1 + 2zA3 )y + A2 xy, omitting “◦h”. Similarly we have By = (2wB2 − 4zμB3 ) + (−2μB2 + 2wB3 )x + (B0 + 4λμB3 )y + B1 xy, Aλ = (λA0 ) + (λA1 )x + (λA2 )y + (λA3 )xy. Bμ = (μB0 ) + (μB1 )x + (μB2 )y + (μB3 )xy. Then we set, to find analytic or formal generators, a1 = 2zA1 − 4wλA3 + μB0 , a3 = −2λA1 + 2zA3 + μB2 , 0 = A0 + 4λμA3 + μB1 , 0 = A2 + μB3 , and a2 = 2wB2 − 4zμB3 + λA0 , a3 = −2μB2 + 2wB3 + λA1 , 0 = B0 + 4λμB3 + λA2 , 0 = B1 + λA3 . Then we are led to the relation: 3λA1 − 3μB2 − 2zA3 + 2wB3 = 0 · · · · · · · · · (∗). If A1 , A2 , B2 , B3 satisfy the relation (*), then A0 , A2 , B0 , B1 are determined from them and so a1 ◦ h, a2 ◦ h, a3 ◦ h: A0 = −3λμA3 , A2 = −μB3 , B0 = −3λμB3 , B1 = −λA3 , and a1 = 2zA1 − 4wλA3 − 3λμ2 B3 , a2 = −3λ2 μA3 + 2wB2 − 4zμB3 , a3 = −2λA1 + 2zA3 + μB2 (= λA1 − 2μB2 + 2wB3 ).

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Since (3λ, −3μ, −2x, 2w) are regular sequence in E4 , the first Koszul cohomology for them vanishes (see for instance, [23]). Then, by setting (A1 , B2 , A3 , B3 ) as (μ, λ, 0, 0), (2z, 0, 3λ, 0), (2w, 0, 0, −3λ), (0, 2z, −3μ, 0), (0, 2w, 0, 3μ), (0, 0, w, z), 2,3 5,0 0,5 respectively, we have elements ϕ4 , ϕ3,2 5 , ϕ5 , ϕ5 , ϕ5 , ϕ6 . such that 2,3 5,0 0,5 1, ϕ4 , ϕ3,2 5 , ϕ5 , ϕ5 , ϕ5 , ϕ6 ∗ generate Rω h over h O4 . Then, by Proposition 5.2, they generate Rh over ∗ h E4 .

Similarly we have the following. Example 6.4 (Elliptic case). Let k : (R4 , 0) → (R4 , 0) be the stable map-germ given by 1 k(x, y, λ, μ) = ( (x2 − y 2 ) + λx + μy, xy + μx − λy, λ, μ), 2 of K-class II2,2 . Then Rk is minimally generated over k ∗ E4 by 1 and 2,3 5,0 0,5 ρ4 , ρ3,2 5 , ρ5 , ρ5 , ρ5 , ρ6 , where ρ = a1 x + a2 y + 12 a3 (x2 + y 2 ), a1 = 2zA1 + 2wA2 + (− 32 λ3 − 32 λμ2 − 3zλ − 3wμ)A3 +(− 32 λ2 μ − 32 μ3 − 3zμ + 3wλ)B3 , a2 = −2wA1 + 2zA2 + ( 32 λ2 μ − 32 μ3 + zμ − wλ)A3 , +( 32 λ3 + 32 λμ2 − zλ − 3wμ)B3 , a3 = −λA1 − μA2 + (z − 12 λ2 + 12 μ2 )A3 + (w − λμ)B3 , 2,3 5,0 0,5 and ρ = ρ4 , ρ3,2 5 , ρ5 , ρ5 , ρ5 , ρ6 respectively for

(A1 , A2 , A3 , B3 ) = (λ, μ, 0, 0), (0, z − 32 λ2 + 32 μ2 , 0, 3λ), (0, w − 3λμ, −3λ, 0), (z − 32 λ2 + 32 μ2 , 0, 0, −3μ), (w − 3λμ, 0, 3μ, 0), (0, 0, z − 32 λ2 + 32 μ2 , w − 3λμ)). Note that, in the process of calculations, we see that A1 , A2 , A3 , B3 obey the relation 3 3 (3μ)A1 + (−3λ)A2 + (−w + 3λμ)A3 + (z − λ2 + μ2 )B3 = 0. 2 2

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and that 3μ, −3λ, −w + 3λμ, z − 32 λ2 + 32 μ2 form a regular sequence in E4 . Theorem 6.5. Any stable mono-germ (R4 , a) → (R4 , b), and therefore any stable multi-germ (R4 , A) → (R4 , b) has a versal opening. Proof. Let f : (R4 , a) → (R4 , b) be a stable map-germ. Then f is of corank ≤ 1 and is diffeomorphic to a Morin map or f is of corank 2 and is diffeomorphic to the germ h of Example 6.3 or k of Example 6.4 (see for instance [7]). In the case of corank one, we have constructed the versal opening in Proposition 4.1. In the case of corank two, we have constructed the versal opening, using a normal form of f in Examples 6.3 and Example 6.4. If a map-germ  has a versal opening and f is diffeomorphic to , then f has a versal opening (cf. Lemma 2.2 (3)). Therefore f has a versal opening. Let f : (R4 , A) → (R4 , b) be a stable multi-germ. Then, for each ai ∈ A, the restriction fi : (R4 , ai ) → (R4 , b) of f to (Rn , ai ) is a stable germ. Since each fi has a versal opening, we see that f itself has a versal opening by Remark 2.9. 7. Openings and lowerable vector fields Let f : (Rn , A) → (Rm , b) a map-germ. A germ of vector field ξ over (Rn , A) is called lowerable for f , or f -lowerable, if there exists a germ of vector field η over (Rm , b) such that f∗ ξ = η ◦ f as a germ of vector field along f . The lowerable vector fields form an f ∗ ERm ,b -module, which is denoted by Xf . Lemma 7.1 (cf. [14]). Let f : (Rn , A) → (Rm , b) be a map-germ and ξ a lowerable vector field for f . Then ξ(f ∗ ERm ,b ) ⊆ f ∗ ERm ,b and ξ(Rf ) ⊆ Rf . Therefore Xf (f ∗ ERm ,b ) ⊆ f ∗ ERm ,b and Xf (Rf ) ⊆ Rf . Proof. Suppose f∗ ξ = η ◦ f as in the above definition. Let a ∈ f ∗ ERm,b . Then ξ(f ∗ a)(x) = d(f ∗ a)(x), ξ(x) = f ∗ (da)(x), ξ(x) = (da)(f (x)), f∗ (ξ(x)) = (da)(f (x)), η(f (x)) = (ηa)(f (x)) = (f ∗ (ηa))(x). Therefore ξ(f ∗ a) = f ∗ (ηa). m Let b ∈ Rf . Then db = i=1 pi dfi for some pi ∈ ERn ,A . Then we have m m m    d(ξb) = Lξ ( pi dfi ) = (ξpi )dfi + pi d(ξfi ). i=1

i=1

i=1

Since each ξfi ∈ f ∗ ERm ,b ⊆ Rf , we see d(ξb) ∈ Jf . Therefore ξb ∈ Rf .

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The differential operator D : ERn ,A → ERn ,A is called lowerable for f : (Rn , A) → (Rm , b), if D is a finite sum of operators of form ξ1 ξ2 · · · ξs ∗

with coefficients in f ERn ,b , where ξ1 , ξ2 , . . . , ξs are lowerable vector field (regarded as first order differential operators) for f . The lowerable differential operators form a (non-commutative) f ∗ ERn ,b -algebra, which is denoted by Lf . By Lemma 7.1, we have: Corollary 7.2. Let f : (Rn , A) → (Rm , b) be a map-germ and D : ERn ,A → ERn ,A be a lowerable differential operator for f . Then D(f ∗ ERm ,b ) ⊆ f ∗ ERm ,b and D(Rf ) ⊆ Rf . Therefore Lf (f ∗ ERm ,b ) ⊆ f ∗ ERm ,b and Lf (Rf ) ⊆ Rf . We conclude the present paper by examining how lowerable vector fields act on the ramification modules and the structure of mini-versal openings for Example 6.1, Example 6.2 and Example 6.3. Example 7.3. The module of lowerable vector fields for f in Example 6.1 is generated by ξ1 = x

∂ , ∂x

ξ2 = y

∂ ∂y

over f ∗ E2 . Then we have ξ1 (x3 ) = 3x3 , ξ2 (x3 ) = 0, ξ1 (y 3 ) = 0, ξ2 (y 3 ) = 3y 3 , 3 3 ξ1 (x y ) = 3x3 y 3 , ξ2 (x3 y 3 ) = 3x3 y 3 . Define the module S = 1, x3 , y 3 f ∗ Em  Rf over f ∗ Em . Then we see Xf (S) = S,

Lf (S) = S.

The module of lowerable vector fields for g in Example 6.2 is generated by

⎧ ∂ ∂ ∂ ⎪ ⎪ ξ0 = x +y +u , ⎪ ⎪ ∂x ∂y ∂u ⎪ ⎪ ⎪ ∂ ∂ ∂ ⎪ ⎪ ⎨ ξ1 = (− 32 u3 + 13 zx) + (− 21 u2 x) + (− 13 zu) , ∂x ∂y ∂u ∂ ∂ ∂ ⎪ ⎪ ξ2 = ( 13 wx + 12 u2 y) + ( 32 u3 − 12 uxy) + 23 wu , ⎪ ⎪ ∂x ∂y ∂u ⎪ ⎪ ⎪ ∂ ∂ ∂ ⎪ ⎪ ⎩ ξ3 = (− 32 wu2 − 12 zuy + 12 wxy) + ( 32 u3 + 12 zxy) + (−zw) , ∂x ∂y ∂u

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over g ∗ E2 , where z = 12 x2 + uy, w = 12 y 2 + ux. Then we have that 1, ξ0 ψ3 , ξ1 ψ3 , ξ2 ψ3 , ξ3 ψ3 generate Rg over g ∗ E3 . This fact is a consequence of Lagrange stability of induced Lagrangian immersion from (g, ψ3 ), g/ : (R3 , 0) → T ∗ R3 = R6 defined by g/ = (z, w, u, p1 , p2 , p3 ) = (z =

1 2 1 x + uy, w = y 2 + ux, u, 3x, 3y, −3xy). 2 2

See [11]. Therefore, if we consider the g ∗ E3 -module T generated by 1, ψ3 , then we have that Xg (T ) = Rg ,

Lg (T ) = Rg .

The lowerable vector fields for h in Example 6.3 is generated by ⎧ ∂ ∂ ∂ ∂ ⎪ ⎪ +y +λ +μ , ξ0 = x ⎪ ⎪ ∂x ∂y ∂λ ∂μ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ∂ ∂ ⎪ ⎪ ⎪ ξ1 = x −y + (3λ) + (−3μ) , ⎨ ∂x ∂y ∂λ ∂μ ⎪ ∂ ∂ ∂ ⎪ ⎪ ξ2 = (−3λμ + xy) + (−μx) + (−x2 − 2λy) , ⎪ ⎪ ∂x ∂y ∂λ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ∂ ⎪ ⎪ + (−3λμ + xy) + (−y 2 − 2μx) . ⎩ ξ3 = (−λy) ∂x ∂y ∂μ Then we have ξ0 ϕ4 = 4ϕ4 , ξ1 ϕ4 = 0 and 2,3 2 2 ∗ ξ2 ϕ4 = −ϕ2,3 5 − 8λ(w + zμ ) ≡ −ϕ5 (mod. h E4 ), 3,2 2 2 ∗ ξ3 ϕ4 = −ϕ3,2 5 − 8μ(z + wλ ) ≡ −ϕ5 (mod. h E4 ).

Moreover 3,2 2,3 2,3 5,0 5,0 0,5 0,5 ξ0 ϕ3,2 5 = 5ϕ5 , ξ0 ϕ5 = 5ϕ5 , ξ0 ϕ5 = 5ϕ5 , ξ0 ϕ5 = 5ϕ5 , 3,2 2,3 2,3 5,0 5,0 0,5 0,5 ξ1 ϕ3,2 5 = ϕ5 , ξ1 ϕ5 = −ϕ5 , ξ1 ϕ5 = 5ϕ5 , ξ1 ϕ5 = −5ϕ5 ,

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and 3,3 3 3 ξ2 ϕ3,2 5 = 3ϕ6 + 18λμϕ4 − 36zwλμ − 3λ μ ∗ ≡ 3ϕ3,3 6 + 18λμϕ4 (mod. h E4 ), 0,5 2 2 2 2 ξ2 ϕ2,3 5 = −4λϕ5 − 32z μ + 68wλ μ ∗ ≡ −4λϕ0,5 5 (mod. h E4 ), 3,2 2 2 ξ2 ϕ5,0 5 = 5λϕ5 − 16z μ − 4wλ μ ∗ ≡ 5λϕ3,2 5 (mod. h E4 ), 2 2 2 4 ξ2 ϕ0,5 5 = 15μ ϕ4 − 60zwμ − 45λ μ

≡ 15μ2 ϕ4 (mod. h∗ E4 ), 5,0 2 2 3 2 ξ3 ϕ3,2 5 = −4μϕ5 − 32w λ + 68zλ μ ∗ ≡ −4μϕ5,0 5 (mod. h E4 ), 3,3 3 2 ξ3 ϕ2,3 5 = 3ϕ6 + 18λμϕ4 − 36zwλμ − 3λ μ ∗ = 3ϕ3,3 6 + 18λμϕ4 (mod. h E4 ), 2 2 4 2 ξ3 ϕ5,0 5 = 15λ ϕ4 − 60zwλ − 45λ μ

≡ 15λ2 ϕ4 (mod. h∗ E4 ), 2,3 2 2 ξ3 ϕ0,5 5 = 5μϕ5 − 16w λ − 4zλμ ∗ ≡ 5μϕ2,3 5 (mod. h E4 ).

We consider the h∗ E4 -module U generated by 1, ϕ4 and V generated by 2,3 5,0 0,5 1, ϕ4 , ϕ3,2 5 , ϕ5 , ϕ5 , ϕ5 . Then we have Xh (U ) = V,

Lh (U ) = Rh .

It would be an interesting open problem, for the geometry of openings, to find a submodule U , as small as possible, of Rf which satisfies Xf (U ) = Rf or Lf (U ) = Rf , for any stable map-germ f : (Rn , A) → (Rm , b), n ≥ 5. In this paper we have solved the problem just for the case n = 4, m = 4. To understand the structure of openings in general case, it seems to be necessary to study more higher dimensional cases n ≥ 5. References 1. V.I. Arnol’d, Catastrophe theory, 3rd edition, Springer-Verlag, (1992). 2. E. Bierstone, P.D. Milman, Relations among analytic functions I, Ann. Inst. Fourier, Grenoble, 37–1 (1987), 187–239.

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3. E. Bierstone, P.D. Milman, Relations among analytic functions II, Ann. Inst. Fourier, Grenoble, 37–2 (1987), 49–77. 4. T. Br¨ ocker, Differentiable Germs and Catastrophes, London Math. Soc. Lecture Note Series 17, Cambridge Univ. Press (1975). 5. J.W. Bruce, P.J. Giblin, Curves and singularities, A geometrical introduction to singularity theory, 2nd ed., Cambridge Univ. Press, (1992). 6. T. Gaffney, M.A. Vitulli, Weak subintegral closure of ideals, Adv. in Math., 226 (2011), 2089–2117. 7. M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Graduate Texts in Math., Springer-Verlag (1974). 8. A.B. Givental’, Varieties of polynomials having a root of fixed co-multiplicity and generalized Newton equation, Funct. Anal. Appl. 16–1 (1982), 13–18. 9. G. Ishikawa, Families of functions dominated by singularities of mappings, Master thesis, Kyoto University (1982), in Japanese. (A part of [9] is published as [10]). 10. G. Ishikawa, Families of functions dominated by distributions of C-classes of mappings, Ann. Inst. Fourier, 33-2 (1983), 199–217. 11. G. Ishikawa, Lagrangian stability of a Lagrangian map-germ in a restricted class — open Whitney umbrellas and open swallowtails, Proceedings of the Symposium “Singularity Theory and its Applications”, 1989, Hokkaido Univ., ed. by G. Ishikawa, S. Izumiya and T. Suwa, Hokkaido Univ. Technical Report Series in Math., 12 (1989), 267–273. 12. G. Ishikawa, Parametrization of a singular Lagrangian variety, Trans. Amer. Math. Soc., 331–2 (1992), 787–798. 13. G. Ishikawa, Parametrized Legendre and Lagrange varieties, Kodai Math. J., 17–3 (1994), 442–451. 14. G. Ishikawa, Developable of a curve and determinacy relative to osculationtype, Quart. J. Math. Oxford, 46 (1995), 437–451. 15. G. Ishikawa, Symplectic and Lagrange stabilities of open Whitney umbrellas, Invent. math., 126 (1996), 215–234. 16. G. Ishikawa, Singularities of tangent varieties to curves and surfaces, Journal of Singularities, 6 (2012), 54–83. 17. G. Ishikawa, Tangent varieties and openings of map-germs, to appear in RIMS K¯ oky¯ uroku Bessatsu. 18. G. Ishikawa, S. Janeczko, Symplectic bifurcations of plane curves and isotropic liftings, Quarterly J. Math. Oxford, 54 (2003), 73–102. 19. T. de Jong, D. van Straten, A deformation theory for nonisolated singularities, Anh. Math. Sem. Univ. Hamburg, 60 (1990), 177–208. 20. B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press (1966). 21. J. Martinet, Singularities of Smooth Functions and Maps, London Math. Soc. Lecture Note Series 58, Cambridge Univ. Press (1982). 22. J.N. Mather, Stability of C ∞ mappings IV: The nice dimensions, Lecture Notes in Math. 192, Springer (1971), pp.192–253. 23. H. Matsumura, Commutative Algebra, Benjamin (1970). 24. D. Mond, Deformations which preserve the non-immersive locus of a mapgerm, Math. Scand., 66 (1990), 21–32.

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25. R. Pellikaan, Finite determinacy of functions with non-isolated singularities, Proc. London Math . Soc., 57 (1988), 357–382. 26. C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), 481–539.

Acknowledgement: This paper has the origin in my master thesis [9] [10]. From the time when I was an undergraduate student of Kyoto University under the supervision of Professor Masahisa Adachi, Professor Satoshi Koike kept giving me kind encouragements and friendship throughout. I would like to thank him in the occasion of his 60th birthday. I am grateful to the referee for helpful comments to correct and improve the original manuscript.

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Non concentration of curvature near singular points of two variable analytic functions Satoshi Koike, Tzee-Char Kuo and Laurentiu Paunescu Department of Mathematics, Hyogo University of Teacher’s Education, Hyogo, Japan [email protected] School of Mathematics, University of Sydney, Sydney, NSW, 2006, Australia [email protected] School of Mathematics, University of Sydney, Sydney, NSW, 2006, Australia [email protected] In this paper we study the phenomenon of non concentration of curvature of level curves of two variable complex analytic function germs, and we characterise it in terms of tree models and topological types. In addition, we also discuss the relationship in the real case, between the phenomenon of non concentration of curvature and real tree models (or blow-analytic types). In particular, we give an example to demonstrate that the corresponding characterisation does not hold in the real case. Keywords: concentration of curvature, tree model, blow-analyticity, curvature tableland. AMS classification numbers: 14B05, 14H50, 32S15, 58K20

1. Introduction Let f : (K2 , 0) → (K, 0) be an analytic function germ not identically zero, where K = R or C. The singular point set of f is contained in the zero-set f −1 (0). What kind of relationship can we find between the singularity type of f : K2 → K and its level curves f = c, 0 < |c| < ,  small? In this respect N. A’Campo made a profound observation. R. Langevin explored the A’Campo phenomenon and found (in [16]) an interesting relationship between the integration of the total Gaussian curvatures of the level curves of a complex analytic function and its Milnor number. In addition, E. Garcia Barroso and B. Teissier analysed the concentration of curvature of the level curves of a complex analytic function in [1], and J.-J. Risler investigated the

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curvature problem for the real Milnor fibre in [18]. In a couple of previous papers ([9], [10]), using the language of infinitesimals as introduced in [14], [15], we studied the A’Campo’s curvature bumps for both real and complex analytic functions. The curvature formula in the real case is KfR (x, y) := ±

Δf (x, y) 3

(fx (x, y)2 + fy (x, y)2 ) 2

,

and that in the complex case is KfC (z, w) := −

2|Δf (z, w)|2 , (|fz (z, w)|2 + |fw (z, w)|2 )3

where Δf := 2fx fy fxy − fx2 fyy − fy2 fxx . These are known formulae for computing the Gaussian curvature of level curves f = c, 0 < |c| <  (cf. John A. Thorpe [19]). Let us consider the polynomial functions f1 , f2 : (R2 , 0) → (R, 0) defined by f1 (x, y) = x2 − y 3 ,

f2 (x, y) = x4 − y 5 .

Using the real formula above, we can easily see that if |c| is sufficiently small, the level curves of f1 and f2 are very close to be vertical in a (wide) hornneighbourhood of the x-axis. Similarly the level curves of f2 are also nearly horizontal in a horn-neighbourhood of the y-axis. Therefore the union of the level curves f2 = ±c, c = 0, looks like a rectangle outside some thin hornlike region tangent to the y-axis. Intuitively the concentration of curvature of f2 happens in this thin region. On the other hand, we can see that the concentration of curvature of f1 happens in a thin horn-neighbourhood of the y-axis. These phenomena are illustrated in the pictures below. (See also [9].) In [6] the first named author and A. Parusi´ nski gave a complete blowanalytic classification of two variable real analytic function germs in terms of their real tree model. See §3 for the definitions of blow-analytic equivalence and real tree model. The real tree models of the above f1 and f2 are drawn as follows:

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1.0

0.5

0.5

0.0

0.0

0.5

0.5

1.0

1.0 1.0

0.5

0.0

0.5

1.0

1.0

Fig. 1.

−

+ 2 (0, −1)

3 2

0.5

0.0

0.5

1.0

f1 = c , f2 = c

+

+

3 2

2 (0, 1) 4

−

5 4

4

+ 5 4

4

(0, −1)

8

(0, 1)

RT (f2 )

RT (f1 )

By an easy computation of the curvature using the above formula, we can observe that the concentration of curvature of f1 and f2 happens on the bars of height 32 and 54 , respectively. Note that, in both cases, the concentration of curvature does not happen directly on the ground bar. Remark 1.1. In [9] we made a more detailed analysis of the curvature of the level curves of the above f1 and f2 . We let the trunks supporting the bars of height 32 in RT (f1 ) and of height 54 in RT (f2 ) grow upward and put provisional bars of height 2 and 43 on the grown trunks in RT (f1 ) and RT (f2 ), respectively. In each case the concentration of curvature happens on the provisional bar. We do not elaborate on this observation in this paper. In the real case the zero set f −1 (0) can be just {(0, 0)} as a set germ at the origin. Let us consider such polynomial functions f3 , f4 : (R2 , 0) → (R, 0) defined by f3 (x, y) = x4 + y 4 ,

f4 (x, y) = x4 + y 6 .

The real tree models of f3 and f4 are drawn below.

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3 2

+

4

8

(0, −1)

RT (f3 )

+

3 2

4 8

(0, 1)

RT (f4 )

In the case of f3 we can see that |Kf3 (x, y)| on each level curve takes a maximum along some curves near y = ±x, but non concentration of curvature happens. Note that there is no bar of height bigger than 1 in RT (f3 ). On the other hand, a level curve f4 = c, c > 0, looks like a rectangle outside some thin horn-like region tangent to the y-axis. The concentration of curvature of f4 happens in this thin region. We can see that the concentration of curvature of f4 happens on the bars of height 32 . The above observations give rise to natural questions. Question 1.2. For a two variable real analytic function germ f : (R2 , 0) → (R, 0) is there any relationship between its real tree model and the concentration of curvature? Question 1.3. For a two variable complex analytic function germ f : (C2 , 0) → (C, 0) is there any relationship between its tree model and the concentration of curvature? In this paper we give an affirmative answer to Question 1.3. Namely, we show that non concentration of curvature happens if and only if no bar of height bigger than 1 appears in the tree model T (f ) (Theorem 5.4). This condition is also equivalent to the homogeneous-likeness. (See §3 for the definition of homogeneous-like.) From this result, we can see that the appearance of concentration of curvature is a topological invariant in the complex case (Corollary 5.5). On the other hand, we have a negative answer to Question 1.2. More precisely, the corresponding condition on the real tree model, which is equivalent to the homogeneous-likeness, implies non concentration of curvature, but the converse is not valid. Indeed, we give an example to demonstrate that non concentration of curvature does not always imply homogeneouslikeness (Proposition 5.7). In order to introduce our notions of concentration of curvature and non concentration of curvature, we need to also consider the curvature of the level curve f = 0 in a punctured neighbourhood of 0 ∈ K2 , even if the zero-

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set is a singular locus of f . The sign of the curvature is not essential for these notions. In the next section we define the non-directed curvature for the level curves f = c, |c| < , in a punctured neighbourhood of 0 ∈ K2 , and using this definition, we introduce the notions of concentration of curvature and non concentration of curvature in §5. In §3 we recall the notions of tree model and real tree model, mention some related results, and give a characterisation of the homogeneous-likeness in terms of tree models. In §4 we review several results on A’Campo curvature bumps proved in [9,10], which are necessary for the proofs of our main result in the complex case mentioned in §5. Throughout our paper we will use the following convention and notations. We call an analytic function germs f : (K2 , 0) → (K, 0) mini-regular in x, if f (x, y) := Hm (x, y) + Hm+1 (x, y) + · · · ,

Hm (1, 0) = 0,

where m = O(f ) is the order of f and Hk (x, y) are homogeneous forms of degree k ≥ m. For two non-negative functions f, g : [0, δ) → [0, ∞), δ > 0, (i) we write f ≈ g if there exist 0 < K1 ≤ K2 and 0 < δ1 ≤ δ such that K1 g() ≤ f () ≤ K2 g() for 0 ≤  ≤ δ1 , and (ii) we write f g if g() > 0 for 0 <  < δ1 with δ1 ≤ δ and the quotient f () tends to 0 as  → 0. g() 2. Non-directed curvature In this section we introduce the non-directed curvature for level curves of an analytic function germ, including the zero locus. We first consider the real case. Let g : (R2 , 0) → (R, 0) be an irreducible analytic function germ, and let f = g m for m ≥ 1. Then we have Δf = m3 g 3(m−1) Δg

3

3

and (fx2 + fy2 ) 2 = ±m3 g 3(m−1) (gx2 + gy2 ) 2 .

Therefore, after cancellation of |g 3(m−1) |, we have |Kf | =

(fx2

|Δf | |Δg | 3 = 3 = |Kg | 2 2 + fy ) 2 (gx + gy2 ) 2

in a punctured neighbourhood of 0 ∈ R2 . Note that in case m is odd, Kf = Kg after cancellation of g 3(m−1) .

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Let f (x, y) = x2 y and g(x, y) = xy, then |Kf | does not coincide with |Kg | in a punctured neighbourhood of 0 ∈ R2 . Nevertheless we still have some reasonable results on comparing their curvature. We first prepare some lemmas. Lemma 2.1. Let g : (R2 , 0) → (R, 0) be an irreducible analytic function such that g −1 (0) = {0} as germs at 0 ∈ R2 . Then gx2 + gy2 is not divisible by g. Proof. Since g is irreducible, g has an isolated singularity at 0 ∈ R2 (cf. [11]). Suppose that gx2 + gy2 is divisible by g. Then gx = 0 and gy = 0 along g −1 (0). This contradicts the isolated singularity of g. Remark 2.2. We cannot drop the assumption that g −1 (0) = {0} at 0 ∈ R2 . Let g(x, y) = x2 + y 2 , g is irreducible and g −1 (0) = {0}. On the other hand, gx2 + gy2 = 4(x2 + y 2 ) is divisible by g. Let f , g, h : (R2 , 0) → (R, 0) be analytic function germs such that f = g m h for m ≥ 2. Suppose that g is an irreducible analytic function such that g −1 (0) = {0} as germs at 0 ∈ R2 and that h is not divisible by g. Then we have the following lemmas. 3

Lemma 2.3. |Δf | and (fx2 + fy2 ) 2 are divisible by |g 3(m−1) |. Proof. By an easy computation, we have fx = (mgx h + ghx )g m−1 , fy = (mgy h + ghy )g m−1 , fxx = (mgxx gh + m(m − 1)gx2 h + 2mgxghx + g 2 hxx )g m−2 , fxy = (mgxy gh + m(m − 1)gx gy h + mgx ghy + mgy ghx + g 2 hxy )g m−2 , fyy = (mgyy gh + m(m − 1)gy2 h + 2mgy ghy + g 2 hyy )g m−2 . Then we have Δf = {m3 h3 (2gx gy gxy − gx2 gyy − gy2 gxx ) + Hg}g 3(m−1) , fx2 +fy2 = {m2 h2 (gx2 +gy2 )+2mh(gx hx +gy hy )g +(h2x +h2y )g 2 }g 2(m−1) , where H : R2 → R is an analytic function germ at 0 ∈ R2 . Therefore |Δf | 3 and (fx2 + fy2 ) 2 are divisible by |g 3(m−1) |. / f := After this, we put Δ

|Δf |

|g 3(m−1) |

3

and f∇ :=

(fx2 +fy2 ) 2 |g3(m−1) |

.

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Lemma 2.4. (1) f∇ = 0 over g −1 (0)  {0}. f / f = |Kg | over g −1 (0)  {0}. / f := Δ . Then K (2) Let K f∇ Proof. (1) By Lemmas 2.1 and 2.3, m2 h2 (gx2 + gy2 ) = 0 over g −1 (0)  {0}. Therefore f∇ = 0 over g −1 (0)  {0}. (2) Over g −1 (0)  {0}, /f = K =

|m3 h3 (2gx gy gxy − gx2 gyy − gy2 gxx )| 3

(m2 h2 (gx2 + gy2 )) 2 |2gx gy gxy − gx2 gyy − gy2 gxx | 3

(gx2 + gy2 ) 2

= |Kg |.

Remark 2.5. Lemma 2.4 holds also for m = 1. In particular, (2) becomes the following: Kf = ±Kg over g −1 (0)  {0}, where the sign ± depends on the sign of h at (x, y) ∈ g −1 (0)  {0}. Let f : (R2 , 0) → (R, 0) be an analytic function germ. Then f has a decomposition of the following form: f = f1m1 · · · fkmk h, mi ≥ 1 (1 ≤ i ≤ k), k ∈ N∪{0}, fi = fj (i = j), (2.1) where each fi : (R2 , 0) → (R, 0), 1 ≤ i ≤ k, is an irreducible analytic component of f such that fi−1 (0) = {0} as germs at 0 ∈ R2 , and h : (R2 , 0) → (R, 0) is an analytic function germ such that h−1 (0) ⊆ {0} as germs at 0 ∈ R2 . Note that k ∈ N in case h is a unit. Let us define the non-directed curvature of level curves of f as follows:  |KfR (x, y)| if (x, y) ∈ R2  f −1 (0) R K f (x, y) := |Kfj (x, y)| if (x, y) ∈ fj−1 (0)  {0} (1 ≤ j ≤ k). The next proposition follows from Lemmas 2.3 and 2.4. R

Proposition 2.6. K f is continuous in a punctured neighbourhood of 0 ∈ R2 . As mentioned in the introduction, using the concept of non-directed curvature, we shall define the notions of concentration of curvature and non concentration of curvature for two variable real analytic function germs in §5. Concerning the reduction of the problem of concentration of curvature, it may be natural to ask the following question.

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Question 2.7. Let f : (R2 , 0) → (R, 0) be an analytic function germ with a decomposition of form (2.1), and let g : (R2 , 0) → (R, 0) be the function germ defined by g = f1 · · · fk h. One can ask whether there are positive numbers 0 < C1 < C2 such that C1 |Kf | ≤ |Kg | ≤ C2 |Kf | . If the answer would be affirmative, then it would be enough to consider only reduced analytic function germs. Unfortunately this does not hold in general. In fact, we have Example 2.1. Let f, g : (R2 , 0) → (R, 0) be real analytic function germs defined by f (x, y) = x2 (x − y 2 ), g(x, y) = x(x − y 2 ). By simple computations, we can see |Kf | =

|Δf | (fx2

+

3 fy2 ) 2

≈

1 |Δg | , |Kg | = 3 ≈ 1 3 2 |y| (gx + gy2 ) 2

on the curve {x = 23 y 2 }. Therefore there does not exist C1 > 0 such that C1 |Kf | ≤ |Kg |. We next consider the complex case. Lemma 2.8. Let g : (C2 , 0) → (C, 0) be an irreducible analytic function germ. Then g has an isolated singularity at 0 ∈ C2 . Let f , g, h : (C2 , 0) → (C, 0) be analytic function germs such that f = g m h for m ≥ 2. Suppose that g is irreducible and h is not divisible by g. Then we have the following. Lemma 2.9. |Δf |2 and (|fz |2 + |fw |2 )3 are divisible by |g|6(m−1) . Proof. Similarly to Lemma 2.3, we have Δf = {m3 h3 (2gx gy gxy − gx2 gyy − gy2 gxx ) + Hg}g 3(m−1) , |fz |2 + |fw |2 = (|mgz h + ghz |2 + |mgw h + ghw |2 )|g|2(m−1) , where H : C2 → C is an analytic function germ at 0 ∈ C2 . It follows that |Δf |2 and (|fz |2 + |fw |2 )3 are divisible by |g|6(m−1) . 2|Δ |2

2

2 3

f z | +|fw | ) C / C := − 6(m−1) After this, we put Δ and f∇ := (|f|g| . Using the 6(m−1) f |g| same arguments as in Lemma 2.4, we can show a similar lemma.

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123 C Lemma 2.10. (1) f∇ = 0 over g −1 (0)  {0}. C  / C := ΔCf . Then K / C = K C over g −1 (0)  {0}. (2) Let K f

f

f∇

g

Let f : (C2 , 0) → (C, 0) be an analytic function germ. Then f has a decomposition of the following form: f = f1m1 · · · fkmk ,

mi ≥ 1 (1 ≤ i ≤ k), k ∈ N, fi = fj (i = j),

where each fi : (C2 , 0) → (C, 0), 1 ≤ i ≤ k, is an irreducible analytic component of f . Let us define the curvature of level curves of f as follows:  C 2 −1 C  f (z, w) := Kf (z, w) if (z, w) ∈ C  f (0) K −1 KfCj (z, w) if (z, w) ∈ fj (0)  {0} (1 ≤ j ≤ k). The next proposition follows from Lemmas 2.9 and 2.10.  C is continuous in a punctured neighbourhood of 0 ∈ Proposition 2.11. K f C2 . When we discuss the phenomenon of concentration of curvature and non concentration of curvature for a complex analytic function germ f : (C2 , 0) → (C, 0), the minus sign is not essential. Accordingly we set C  fC (z, w)| K f (z, w) := |K

over a punctured neighbourhood of 0 ∈ C2 . In §5 we shall use this definiC tion of K f to define the notions of concentration or non concentration of curvatures in the complex case. 3. Tree model and real tree model In this section we briefly review the definitions of tree model and real tree model introduced in [12] and [6,7], respectively. We first recall the notion of tree model. This is a kind of geometric interpretation of the classical Zariski theorem on the topology of complex plane curves ([20]). Let f (x, y) be a complex analytic function germ of multiplicity m and mini-regular in x. Let x = λi (y), i = 1, . . . , m, be the complex NewtonPuiseux roots of f . We denote the contact order at zero of λi and λj by O(λi , λj ) := ord0 (λi − λj )(y). Let h ∈ Q. We say that λi , λj are congruent modulo h+ if O(λi , λj ) > h.

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The tree model T (f ) of f is defined as follows. Draw a vertical line segment as the main trunk of the tree. Mark m = mult0 f (x, y) alongside the trunk. Let h0 := min{O(λi , λj ) | 1 ≤ i, j ≤ m}. Then draw a bar, B0 , on top of the main trunk. Call h(B0 ) := h0 the height of B0 . The roots are divided into equivalence classes modulo h+ 0 . We then represent each equivalence class by a vertical line segment drawn on top of B0 , and call it trunk. If a trunk consists of s roots we say it has multiplicity s, and mark s alongside. The same construction is repeated recursively. The construction terminates at the stage where all trunks contain only single, possibly multiple, roots of f . Example 3.1. Let f : (C2 , 0) → (C, 0) be a polynomial function defined by f (x, y) = x(x + y)2 (x2 − y 3 )(x3 + y 7 )3 . The tree model T (f ) is drawn as follows:

3

3 2

3

10

7 3 3 2

12 1 14 T (f )

We do not take care on the position of trunks on bars in the complex case. We call the sets of roots corresponding to trunks bunches. Each bunch A is a set of roots growing through a unique bar B(A). Fix a bunch A with finite height h(A). Take a root λi (y) ∈ A. Let λA (y) denote λi (y) with all terms y e , e ≥ h(A), omitted. Then we can write λi (y) ∈ A as λi (y) = λA (y) + ci y h(A) + · · · ,

ci ∈ C.

(3.2)

We next recall the notion of real tree model. Let f (x, y) be a real analytic function germ. Consider the Newton-Puiseux roots as arcs x = λi (y) defined for y ∈ R, y ≥ 0. The complex conjugation acts on the Newton-Puiseux

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roots, and hence on the tree model T (f ). A bunch A of T (f ) is called real if it is stable under complex conjugation. A bar or a trunk is real if and only if so are the corresponding bunches growing on it. We denote by T+ (f ) the conjugation invariant part of T (f ). Fix v a unit vector of R2 . Fix any local system of coordinates x, y such that f (x, y) is mini-regular in x and v is of the form (v1 , v2 ) with v2 > 0. Consider the Newton-Puiseux roots of f as arcs x = λi (y) defined for y ∈ R, y ≥ 0. We define the real tree model of f relative to v, denoted by RTv (f ), as the part of T+ (f ) consisting only of those roots tangent to v with the following additional information. Let A be a real bunch such that B = B(A) is a bar of RTv (f ). Then: - draw the trunks on B realising the sub-bunches of A keeping the clockwise order of the roots (i.e. the order of the coefficients ci in (3.2)), - whenever B gives a new Puiseux pair of some roots of A we mark 0 on B and draw from it the unique sub-bunch of A with ci = 0, i.e. consisting of the roots that do not have the new Puiseux pair at B. Hence we are also able to determine from the tree the sub-bunches with positive and negative ci . Graphically, we identify 0 ∈ B with the point of B that belongs to the trunk supporting B. The real tree model RT (f ) of f is defined as follows: • Draw a bar B0 (identified with S 1 ). We define h(B0 ) = 1 and call B0 the ground bar. We mark m(B0 ) := 2 mult0 f (x, y) below the ground bar. We call m(B0 ) the multiplicity of the ground bar. • Draw on B0 the non-trivial RTv (f ) for v ∈ S 1 , keeping the clockwise order. • Let v1 , v2 be any two subsequent unit vectors for which RTv (f ) is nontrivial. Mark on B0 of RT (f ) the sign of f in the sector between v1 and v2 . Definition 3.1. We call two real tree models RT (f ), RT (g) isomorphic, if there is a homeomorphism ϕ between the trees which maps the ground bar to the ground bar preserving the multiplicity, bars to bars preserving the heights, trunks to trunks preserving the multiplicities, and the signs of the characteristic coefficients, after we move trunks, if necessary, on the bars whose heights are not giving Puiseux pairs. Blow-analytic equivalence is a notion defined by the second author in [13] as a natural equisingularity condition for real analytic function germs : (Rn , 0) → (R, 0). We say that two real analytic function germs f : (R2 , 0) →

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(R, 0) and g : (R2 , 0) → (R, 0) are blow-analytically equivalent, if there exist compositions of finite point blowings-up μ : (M, μ−1 (0)) → (R2 , 0), μ : (M  , μ−1 (0)) → (R2 , 0) and an analytic isomorphism Φ : (M, μ−1 (0)) → (M  , μ−1 (0)) which induces a homeomorphism φ : (R2 , 0) → (R2 , 0) such that f = g ◦ φ. Remark 3.2. In general blow-analytic equivalence is defined using real modifications ([13]). In the two variable case a real modification is attained by a composition of finite point blowings-up ([6]). For properties on blowanalyticity, see the surveys [3], [5]. Blow-analytic equivalence of two variable real analytic function germs is completely determined by the real tree models as follows: Theorem 3.3 ([6]). Let f : (R2 , 0) → (R, 0) and g : (R2 , 0) → (R, 0) be real analytic function germs. Then f and g are blow-analytically equivalent if and only if the real tree models of f and g are isomorphic. Example 3.2. Let f, g : (R2 , 0) → (R, 0) be two polynomial functions defined by f (x, y) = x3 − y 4 , g(x, y) = x3 + y 4 . The real tree models RT (f ) and RT (g) are drawn as follows:

+ 3

4 3

(0, −1)

−

+ 3 6

RT (f )

(0, 1)

−

+ 4 3

+ 4 3

4 3

3 (0, −1)

3 6

(0, 1)

RT (g)

Since RT (f ) and RT (g) are not isomorphic, we can see by Theorem 3.3 that f and g are not blow-analytically equivalent. Definition 3.4. We call a real analytic function germ f : (R2 , 0) → (R, 0) homogeneous-like, if f is blow-analytically equivalent to a homogeneous polynomial function germ. Concerning the homogeneous-likeness of two variable real analytic function germs, we give a characterisation in terms of real tree models.

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Lemma 3.5. For a real analytic function germ f : (R2 , 0) → (R, 0), the following are equivalent. (1) f is homogeneous-like. (2) Up to isomorphisms of real tree models, no bar except the ground bar appears in the real tree model RT (f ). Proof. Suppose that f is homogeneous-like, namely f is blow-analytically equivalent to a two variable real homogeneous polynomial function germ, and by Theorem 3.3, their real tree models are isomorphic. Since any two variable real homogeneous polynomial function germ clearly has a real tree model consisting of only the ground bar with finite trunks (including also the empty case) on it, (2) follows immediately. On the other hand, consider a real analytic function germ f : (R2 , 0) → (R, 0) having a real tree model RT (f ) which consists of only the ground bar with finite trunks on it. Taking into account also the multiplicities of the ground bar and the trunks and signs, it is easy to construct a two variable real homogeneous polynomial function having the real tree model isomorphic to RT (f ). Therefore (1) follows immediately from Theorem 3.3.

We say that two complex analytic function germs f : (C2 , 0) → (C, 0) and g : (C2 , 0) → (C, 0) are topologically equivalent, if there exists a homeomorphism φ : (C2 , 0) → (C2 , 0) such that f = g ◦ φ. Now we recall the improved version of the Zariski theorem on the topology of two variable complex analytic function germs. Theorem 3.6 (O. Zariski [20], Kuo - Lu [12], A. Parusi´ nski [17]). Let f , g : (C2 , 0) → (C, 0) be analytic function germs. Then f and g are topologically equivalent if and only if the tree models of f and g coincide. Definition 3.7. We call a complex analytic function germ f : (C2 , 0) → (C, 0) homogeneous-like, if f is topologically equivalent to a homogeneous polynomial function germ. Using Theorem 3.6 and a similar argument to Lemma 3.5, we can also characterise the homogeneous-likeness of a two variable complex analytic function germ as follows: Lemma 3.8. For a complex analytic function germ f : (C2 , 0) → (C, 0), the following are equivalent. (1) f is homogeneous-like.

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(2) No bar of height bigger than 1 appears in the tree model T (f ). Remark 3.9. Any real tree model has at least the ground bar. Some tree models in the complex case, however, can consist of only one trunk without any bar. This case also satisfies condition (2) in the above lemma. 4. Computation of A’Campo bumps In [9,10] we introduced and computed A’Campo bumps of two variable analytic function germs using infinitesimals. Here we review some notions and results which will be used to show our main results. Take an analytic function germ α : (C, 0) −→ (C2 , 0),

α(t) ≡ 0.

Let α∗ := Im(α) be the image set germ. Being an irreducible curve germ in C2 , it has a unique tangent T (α∗ ) at 0, T (α∗ ) is a point of the Riemann Sphere CP 1 . We call α∗ an infinitesimal at T (α∗ ). The Enriched Riemann Sphere is CP∗1 := {α∗ }. The absolute value of the curvature computed along α∗ , if not zero, can be written as |KfC (α(t))| = asL + · · · ,

a > 0,

L ∈ Q,

where s = s(t) is the arc length. This is dominated by the leading term asL as s → 0. If |KfC | ≡ 0 along α∗ , we write (a, L) := (0, ∞). Hence we introduce the notations (a, L) := aδ L ,

0V := 0δ ∞ ,

V(R) := {aδ L | a = 0} ∪ {0V },

where δ is a symbol. A lexicographic ordering on V(R) is defined: 0ν is the smallest element, and 

aδ L > a δ L if and only if either L < L , or else L = L , a > a . The curvature function K∗ on CP∗1 , and the component L∗ are defined as follows: K∗ : CP∗1 → V(R), α∗ → aδ L ;

L∗ : CP∗1 → Q, α∗ → L.

Recall the field F of convergent fractional power series in an indeterminate y is algebraically closed. A non-zero element of F is a convergent series α(y) = a0 y n0 /N + · · · + ai y ni /N + · · · , n0 < n1 < · · · ; ni ∈ Z,

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where 0 = ai ∈ C, N ∈ Z+ , GCD(N, n0 , n1 , ...) = 1. The conjugates of α are √  2π −1 (k) ai θkni y ni /N , 0 ≤ k ≤ N − 1, θ := e N . αconj (y) := The order is Oy (α) := n0 /N , Oy (0) := +∞. The Puiseux multiplicity is mpuis (α) := N . The following are integral domains: D1 := {α ∈ F | Oy (α) ≥ 1},

D1+ := {α | Oy (α) > 1},

having quotient field F. As in Projective Geometry, CP∗1 is a union of two charts: CP∗1 = C∗ ∪C∗ , C∗ := {β∗ ∈ CP∗1 | T (β∗ ) = [1 : 0]},

C∗ := {β∗ ∈ CP∗1 | T (β∗ ) = [0 : 1]}.

Next we define the contact order Cord (α∗ , β∗ ). We can assume α∗ , β∗ ∈ C∗ . Then  ∞ if α∗ = β∗ , Cord (α∗ , β∗ ) := (i) (j) maxi,j {Oy (αconj (y) − βconj (y))} if α∗ = β∗ . The horn subspaces of CP∗1 centered at α∗ of order e, e+ are He (α∗ ) := {β∗ | Cord (α∗ , β∗ ) ≥ e},

He+ (α∗ ) := {β∗ | Cord (α∗ , β∗ ) > e},

respectively. In particular, Cord (α∗ , β∗ ) = 1 if T (α∗ ) = T (β∗ ); and H1 (α∗ ) = CP∗1 for all α∗ . When there is no need to specify α∗ , we write He := He (α∗ ). Let He (α∗ ) be given. If η(y) = α(y) + [cy e + · · · ], c a generic number, then L∗ (η∗ ) is a constant. We write this constant as L∗ (Hegrc (α∗ )). A horn interval of radius r, r > 0, is, by definition, He (α∗ , r) := {β∗ | β(y) = α(y) + (cy e + · · · ), |c| ≤ r}. Definition 4.1. A horn subspace He (α∗ ) is a curvature tableland if (1) β∗ ∈ He (α∗ ) =⇒ L∗ (β∗ ) ≥ L∗ (Hegrc (α∗ )); and (2) in the case e > 1, there exists e , 1 ≤ e < e, such that ν∗ ∈ He (α∗ )  He (α∗ ) =⇒ L∗ (ν∗ ) > L∗ (Hegrc (α∗ )). Example 4.1. Let f (x, y) = x2 − y 5 . Then we have fx = 2x, fy = −5y 4 , fxx = 2, fxy = 0, fyy = −20y 3. Therefore we have KfC (x, y) =

200|y|6|8x3 − 5y 5 |2 . (4|x|2 + 25|y|8 )3

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We consider the horn subspace H4 (0∗ ). Then we can see L∗ (H4grc (0∗ )) = −8 = L∗ (β∗ ) for all β∗ ∈ H4 (0∗ ). Let 0 <  < 32 . If ν∗ ∈ H4− (0∗ )  H4 (0∗ ), then −8 < L∗ (ν∗ ) ≤ −8 + 6. It follows that L∗ (ν∗ ) > L∗ (H4grc (0∗ )). Thus H4 (0∗ ) is a curvature tableland. Definition 4.2. Let He be a curvature tableland. Take β∗ ∈ He . We say K∗ has an A’Campo bump on He+ (β∗ ), or simply say He+ (β∗ ) is an A’Campo bump, if there exists  > 0 such that μ∗ ∈ He (β∗ , ) =⇒ K∗ (β∗ ) ≥ K∗ (μ∗ ). Let us apply a generic unitary transformation so that f is mini-regular in z where m = O(f ). Then the initial form Hm is written as follows: Hm (z, w) = c(z−c1 w)m1 · · · (z−cr w)mr ; mi ≥ 1 ; ci = cj ; if ; i = j, (4.3)  where 1 ≤ r ≤ m, mi = m, c = 0. Thus Hm (z, w) is degenerate if and only if r < m. Let ζi denote the Newton-Puiseux roots of f (z, w), and γj those of fz : f (z, w) = unit ·

m (z − ζi (w)),

fz (z, w) = unit ·

i=1

m−1

(z − γj (w)),

j=1

where Ow (ζi ), Ow (γj ) ≥ 1. Each γj is called a polar, and so is γj∗ . Definition 4.3. Given a polar γ, let dgr (γ) denote the smallest number e such that Ow (Grad f (γ(w), w)) = Ow (Grad f (γ(w) + uwe , w)), where u ∈ C is a generic number. We call dgr (γ) the gradient order of γ. The D-gradient canyon of γ, and the ∗-gradient canyon of γ∗ are G(γ) := {α ∈ D1 | Oy (α − γ) ≥ d},

G∗ (γ∗ ) := Hd (γ∗ ),

d := dgr (γ),

respectively. When there is no confusion, we call them “canyons”; we also write d := dgr (γ),

G := G(γ),

G∗ := G∗ (γ∗ ).

The degree and multiplicity of G, G∗ are, respectively, dgr (G)v := dgr (G∗ ) := d,

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m(G) := {k | G(γk ) = G}, m(G∗ ) := {k | G∗ (γk∗ ) = G∗ }. Finally, we say G(γ) and G∗ (γ∗ ) are minimal if G(γj ) ⊆ G(γ) =⇒ G(γj ) = G(γ). Let γ be a polar with d < ∞. We now define Lγ ∈ Q, and a rational function Rγ (u), u ∈ C. We can assume γ ∈ D1+ so that T (γ∗ ) = [0 : 1]. If d > 1, define Lγ , Rγ (u) by |KfC (γ(y) + uy d , y)| := 2Rγ (u)y 2Lγ + · · · ,

Rγ (u) ≡ 0,

where y can be considered as the arc length of (γ(y)+uy d )∗ , since lim y/s = 1. In the case d = 1, define Lγ and Rγ (u) by |KfC (γ(y) + "

uy y ," )| := 2Rγ (u)y 2Lγ + · · · , 2 1 + |u| 1 + |u|2

Rγ (u) ≡ 0.

Theorem 4.4 ([9]). A minimal ∗-canyon with d < ∞ is a curvature tableland, and vice versa. Take a minimal G(γ), d < ∞. Take a local maximum Rγ (c) of Rγ (u). Then Hd+ (γ∗+c ) is an A’Campo bump, where γ +c (y) := γ(y) + cy d . All A’Campo bumps can be found in this way. Addendum 4.5. ([9]) Every G(γ) with d > 1 is minimal. In this case, m(G) = {k | γk ∈ G},

m(G∗ ) = {k | γk∗ ∈ G∗ }.

Let r be as in (4.3). There are exactly r − 1 polars of gradient degree 1; moreover, dgr (γ) = 1 =⇒ G(γ) = D1 , ; m(G(γ)) = r − 1. A minimal G(γ) with d = 1 exists if and only if Hm (z, w) is nondegenerate. In this case every polar has d = 1, CP∗1 is the only curvature tableland and the only ∗-canyon. Next we recall the Newton polygon relative to a polar. Let γ be a given polar, not a multiple root of f (z, w), i.e., f (γ(w), w) = 0. We can apply a unitary transformation, if necessary, so that T (γ∗ ) = [0 : 1], γ ∈ D1+ . Let us change coordinates: Z := z − γ(w),

W := w,

F (Z, W ) := f (Z + γ(W ), W ).

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Then Δf (z, w) = ΔF (Z, W ) + γ  (W )FZ3 ,

Gradz,w f  ≈ GradZ,W F .

Recall that a monomial term aZ i W q , a = 0, q ∈ Q, is represented by a “Newton dot” at (i, q). We shall simply say (i, q) is a dot. If i ≥ 1, then (i, q) is a dot of F (Z, W ) if and only if (i − 1, q) is one of FZ . Since γ is a polar, FZ has no dot of the form (0, q); F (Z, W ) has no dot of the form (1, q). As f (γ(w), w) = 0, we know F (0, W ) = 0. Hence F (0, W ) := aW h + · · · ,

a = 0,

h = OW (F (0, W )),

and then (0, h) is a vertex of the Newton polygon N P(F ), (0, h − 1) is one of the Newton polygon N P(FW ). Let Etop denote the top edge of the Newton polygon N P(F ), i.e., the edge with left vertex (0, h). Let (mtop , qtop ) denote the right vertex of Etop , and θtop the angle of Etop , tan θtop = co-slope of Etop , where the co-slope of a line passing through (x, 0) and (0, y) is, by definition, y/x.  ) = (0, h) be the dot of F on Etop which is closest to Let (mtop , qtop (0, h). Then, clearly, 2 ≤ mtop ≤ mtop ,

 h − qtop h − qtop = = tan θtop .  mtop mtop

Lemma 4.6 ([9]). Let L∗ denote the line joining (0, h − 1) (which is not a dot of FZ ) and a dot of FZ such that no dot of FZ lies below L∗ . Let σ ∗ denote the co-slope of L∗ . Then tan dgr (γ) = σ∗ ≥ tan θtop , where σ ∗ = tan θtop if and only if σ∗ = 1. In order to show the main result in the complex case (Theorem 5.4), we shall use the above Theorem 4.4, Addendum 4.5 and Lemma 4.6. 5. Characterisations of no concentration of curvature Let K = R or C. Consider the families of convergent Puiseux demi-arcs on R2 and the families of convergent Puiseux arcs on C2

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133 ni

n1

αa : x = a0 y + a1 y N + · · · + ai y N + · · · , or

ni

n1

α(0,b) : y = b1 x N + · · · + bi x N + · · · ,

where 1 < nN1 < nN2 · · · , ai , bi ∈ K, ni , N ∈ Z+ . Here the demi-arc means that they are defined for y ∈ R, y ≥ 0 or x ∈ R, x ≥ 0. Let us denote this family by A. Let f : (K2 , 0) → (K, 0) be an analytic function germ not identically zero, mini-regular in x. For γ ∈ A, we define the curvature exponent along γ by K

e(γ) := O(K f (γ)). Then we define the curvature exponent of f by e(f ) := inf {e(γ)}. γ∈A

Remark 5.1. We note that K

− e(f ) = min{s | |(x, y)|s |K f |  1}. Therefore e(f ) is attained by some arc in A. Now we define the notion of concentration of curvature. Let S := {v ∈ K | v = 1}. 2

Definition 5.2. (1) We say that f has concentration of curvature at 0 ∈ K2 , if the set {v ∈ S | ∃γ ∈ A s.t. lim

t→0

γ(t) = v & e(γ) = e(f )} γ(t)

is finite. (2) We say that f has no concentration of curvature at 0 ∈ K2 , if f does not have concentration of curvature at 0 ∈ K2 . Lemma 5.3. Let f : (K2 , 0) → (K, 0) be a homogeneous-like analytic function germ. Then f has no concentration of curvature. Proof. We show only the real case. The complex case follows similarly. Since f is homogeneous-like, it follows from Theorem 3.3 that there is a homogeneous polynomial H : (R2 , 0) → (R, 0) such that RT (f ) is isomorphic to RT (H). Therefore f has a decomposition of the following form: f = f1m1 · · · fkmk h, mi ≥ 0 (1 ≤ i ≤ k), k ∈ N ∪ {0},

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where fi (x, y) = x − ci y + φi (x, y) with j 1 φi (0, 0) = 0, and c1 < c2 < · · · < ck , and either In(h)−1 (0) = {0} as germs at 0 ∈ R2 or h is a unit. Here In(h) means the initial homogeneous form of h. / f after cancellation of / f be the numerator of K Let Δ 3(m1 −1)

f1

3(mk −1)

· · · fk

/ f ≡ 0, then non concentration of curvature happens. as in subsection 2. If Δ We next consider the directions (e, 1), e ∈ R, and (1, 0). Note that if / f is not identically 0, then there are only finitely many e’s such that Δ / In(Δ)(e, 1) = 0. 1 / f )(e, 1) = 0, K R • In the case where In(Δ f ≈ |y| along any convergent Puiseux arc αa with a0 = e. 1 / f )(e, 1) = 0, K R • In the case where In(Δ f |y| along any convergent Puiseux arc αa with a0 = e. Similarly for convergent Puiseux arcs with direction (1, 0), we can see the following: 1 / f )(1, 0) = 0, K R • In the case where In(Δ f ≈ |x| for any α(0,b) . 1 / f )(1, 0) = 0, K R • In the case where In(Δ for any α(0,b) . f

|x|

Therefore non concentration of curvature happens. Using the above lemmas, we give some characterisations of the non concentration of curvature in the complex case. Theorem 5.4. For an analytic function germ f : (C2 , 0) → (C, 0) not identically zero, the following are equivalent. (1) f is homogeneous-like. (2) No bar of height bigger than 1 appears in the tree model T (f ). (3) f has no concentration of curvature at 0 ∈ C2 . Proof. By Lemmas 3.8 and 5.3, it suffices to show (3) implies (1). Taking a unitary transformation if necessary, we may assume that f is mini-regular in z. Suppose that there exists a bar of height bigger than 1 in the tree model T (f ). Then it follows from the Kuo-Lu theorem ([12]) that there is a polar γ which leaves a trunk on this bar. Consider the Newton polygon relative to that polar. The top edge Etop has co-slope bigger than 1, because the co-slope is the height of the bar. Hence, by Lemma 4.6, the gradient order d of γ is bigger than 1. By Addendum 4.5 and Theorem 4.4, G(r) with d > 1 is minimal and G(γ) is a curvature tableland. If f has another polar γ  with

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gradient order d = 1, G(γ  ) with d = 1 is not minimal by Addendum 4.5. It follows from Theorem 4.4 that G(γ  ) is not a curvature tableland. Thus f has concentration of curvature at 0 ∈ C2 . The curvature of a complex analytic function itself can change under a non-unitary linear transformation. Nevertheless we can note the following. Corollary 5.5. For the two variable complex analytic function germs, the appearance of concentration of curvature is a topological invariant. By Lemmas 3.5 and 5.3 we have the following result in the real case. Theorem 5.6. For an analytic function germ f : (R2 , 0) → (R, 0) not identically zero, the following are equivalent. (1) f is homogeneous-like. (2) Up to isomorphisms of real tree models, no bar except the ground bar appears in the real tree model RT (f ). In addition, the above equivalent conditions imply (3) f has no concentration of curvature at 0 ∈ R2 . In the real case non concentration of curvature does not always imply the homogeneous-likeness. We give an example to demonstrate it. In order to see it, we prepare some notations. Let g, h : [0, δ) → R be convergent fractional power series functions, where h is not identically zero. Then we g(y) write g ∼ h if h(y) tends to 1 as y → +0. If g(y) − h(y) consists of terms with higher orders than the order of h(y), we write g(y) = h(y) + HOT . Note that this HOT is not the high order terms in the usual sense. Proposition 5.7. Let f : (R2 , 0) → (R, 0) be a polynomial function defined by f (x, y) = (x − y 2 )3 + x2 y 6 − xy 8 + y 12 . R

Then the non-directed curvature K f tends 2 along any convergent Puiseux demi-arc αa and α(0,b) on R2 as y → +0 and x → +0 respectively, namely non concentration of curvature happens with a constant coefficient. On the other hand, f is not homogeneous-like. Proof. Let us express αa and α(0,b) as follows: αa : x = a0 y s0 + a1 y s1 + a2 y s2 + · · · , 1 ≤ s0 < s1 < s2 < · · · , α(0,b) : y = b1 xs1 + b2 xs2 + · · · , 1 < s1 < s2 < · · · .

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We set G(x, y) := (fx (x, y)2 + fy (x, y)2 )3/2 . By an easy computation, we have fx = 3(x − y 2 )2 + 2xy 6 − y 8 , fy = −6y(x − y 2 )2 + 6x2 y 5 − 8xy 7 + 12y 11 , fxx = 6(x − y 2 ) + 2y 6 , fxy = −12y(x − y 2 ) + 12xy 5 − 8y 7 , fyy = −6(x − y 2 )2 + 24y 2 (x − y 2 ) + 30x2 y 4 − 56xy 6 + 132y 10. Then we have

Δf = −(−6y(x − y 2 )2 + 6x2 y 5 − 8xy 7 + 12y 11 )2 (6(x − y 2 ) + 2y 6 ) − (3(x − y2 )2 + 2xy6 − y8 )2 (−6(x − y 2 )2 + 24y 2 (x − y2 ) + 30x2 y 4 − 56xy 6 + 132y10 ) + 2(3(x − y 2 )2 + 2xy 6 − y 8 ) × (−6y(x − y 2 )2 + 6x2 y 5 − 8xy 7 + 12y 11 )(−12y(x − y 2 ) + 12xy 5 − 8y 7 ),

G = {(3(x−y 2 )2 +2xy 6 −y 8 )2 +(−6y(x−y 2 )2 +6x2 y 5 −8xy 7 +12y 11 )2 }3/2 . In order to see that non concentration of curvature happens, we have R to know the order of K f in y and x along αa and α(0,b) , respectively. We R

first compute the order of K f along αa . If there is no cancellation of the coefficients of the lowest order terms of −fy2 fxx , −fx2 fyy and 2fx fy fxy , it is enough to consider the orders and coefficients of the lowest order terms of fx , fy , fxx , fxy and fyy . But if the cancellation happens, we have to pay attention to more terms of them. We divide the situation into three cases. In the first two cases the cancellation of the coefficients of the lowest order terms does not happen, but such a cancellation happens in the third case. Case (I; αa ) : 1 ≤ s0 < 2, a0 = 0. Along αa we have fx ∼ 3a20 y 2s0 , fxx ∼ 6a0 y s0 ,

fy ∼ −6a20 y 2s0 +1 ,

fxy ∼ −12a0 y s0 +1 ,

fyy ∼ 6a0 y 2s0 .

Therefore we have −fx2 fyy ∼ 54a60 y 6s0 , −fy2 fxx ∼ −216a50y 5s0 +2 , 2fxfy fxy ∼ 432a50 y 5s0 +2 ,

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and Δf ∼ 54a60 y 6s0 , G ∼ 27a60 y 6s0 R

along αa . It follows that K f =

|Δf | G

tends to 2 along αa as y → +0.

Case (II; αa ) : s0 = 2, a0 = 1. Along αa we have fx ∼ 3(a0 − 1)2 y 4 ,

fy ∼ −6(a0 − 1)2 y 5 ,

fxy ∼ −12(a0 − 1)y 3 ,

fxx ∼ 6(a0 − 1)y 2 ,

fyy ∼ −6(a0 − 1)(a0 − 5)y 4 .

Therefore we have −fx2 fyy ∼54(a0 − 1)5 (a0 − 5)y 12 , −fy2 fxx ∼ − 216(a0 − 1)5 y 12 , 2fx fy fxy ∼432(a0 − 1)5 y 12 , and Δf ∼ 54(a0 − 1)6 y 12 , G ∼ 27(a0 − 1)6 y 12 R

along αa . It follows that K f tends to 2 along αa as y → +0. Case (III; αa ) : s0 = 2, a0 = 1. In this case the cancellation of the coefficients mentioned above happens. We set A := a1 y s1 +a2 y s2 +· · · . We further divide this case into three cases. (1) Case (III-1) : 2 < s1 < 4, a1 = 0. Along αa we have fx = 3A2 + y 8 + HOT, fy = − 6A2 y − 2y 9 + HOT = −2y(3A2 + y 8 ) + HOT, fxx = 6A + 2y 6 + HOT, fxy = − 12Ay + 4y 7 + HOT, fyy = − 6A2 + 24Ay 2 − 26y 8 + HOT. Therefore we have −fx2 fyy = (6A2 − 24Ay 2 + 26y 8 )(3A2 + y 8 )2 + HOT, −fy2 fxx = − (24Ay 2 + 8y 8 )(3A2 + y 8 )2 + HOT, 2fx fy fxy = (48Ay 2 − 16y 8 )(3A2 + y 8 )2 + HOT, and Δf ∼ (6A2 + 2y 8 )(3A2 + y 8 )2 ∼ 54a61 y 6s1 , G ∼ 27a61 y 6s1

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along αa . It follows that K f tends to 2 along αa as y → +0. (2) Case (III-2) : s1 = 4, a1 = 0. Along αa we have fx ∼ (3a21 + 1)y 8 ,

fy ∼ −2(3a21 + 1)y 9 ,

fxy = −12Ay + 4y 7 + HOT,

fxx = 6A + 2y 6 + HOT,

fyy = −6a21 y 8 + 24Ay 2 − 26y 8 + HOT.

Therefore we have −fx2 fyy = (6a21 y 6 − 24A + 26y 6 )(3a21 + 1)2 y 18 + HOT, −fy2 fxx = − (24A + 8y 6 )(3a21 + 1)2 y 18 + HOT, 2fx fy fxy = (48A − 16y 6 )(3a21 + 1)2 y 18 + HOT, and Δf ∼ 2(3a21 + 1)3 y 24 ,

G ∼ (3a21 + 1)3 y 24

R

along αa . It follows that K f tends to 2 along αa as y → +0. (3) Case (III-3) : s1 > 4. Along αa we have fx ∼ y 8 , fy ∼ −2y 9 ,

fxx = 6A + 2y 6 + HOT,

fxy = −12Ay + 4y 7 + HOT,

fyy = 24Ay 2 − 26y 8 + HOT.

Therefore we have −fx2 fyy =(−24A + 26y 6 )y 18 + HOT −fy2 fxx = − (24A + 8y 6 )y 18 + HOT, 2fx fy fxy =(48A − 16y 6 )y 18 + HOT, and Δf ∼ 2y 24 ,

G ∼ y 24

R

along αa . It follows that K f tends to 2 along αa as y → +0. R

We next compute the order of K f along α(0,b) . In the case where b1 = 0, namely y is identically zero, we have fx = 3x2 ,

fy = 0,

fxx = 6x,

fxy = 0,

fyy = −6x2

along α(0,b) . Therefore we have −fx2 fyy = 54x6 ,

−fy2 fxx = 0,

2fx fy fxy = 0,

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and Δf = 54x6 ,

G = 27x6

R

along α(0,b) . It follows that K f tends to 2 along α(0,b) as x → +0. In the case where b1 = 0, we have fx ∼ 3x2 , fy ∼ −6b1 xs1 +2 ,

fxx ∼ 6x,

fxy ∼ −12b1 xs1 +1 ,

fyy ∼ −6x2

along α(0,b) . Therefore we have −fx2 fyy = 54x6 ,

−fy2 fxx = −216b21x2s1 +5 ,

2fx fy fxy = 432b21 x2s1 +5 ,

and Δf ∼ 54x6 ,

G ∼ 27x6

R

along α(0,b) . It follows that K f tends to 2 along α(0,b) as x → +0. Lastly we show that f is not homogeneous-like. After taking an analytic transformation of R2 at 0 ∈ R2 , x = X 2 + Y, y = Y , f has the following form: f (X, Y ) = X 3 + X 2 Y 6 + XY 8 + Y 12 . Then we consider the family of polynomials ft : (R2 , 0) → (R, 0), t ∈ I = [0, 1], defined by ft (x, y) = x3 + x2 y 6 + txy 8 + y 12 . For any t ∈ I, the weighted initial form of ft with respect to the system of 1 weights ( 13 , 12 ) is gt (x, y) = x3 + txy 8 + y 12 , and gt has an isolated singularity at 0 ∈ R2 . Therefore, by the blow-analytic triviality theorem in [4] (or [2]), we can see that f is blow-analytically equivalent to h(x, y) = x3 + y 12 . Since h clearly has a bar of height 4, it follows from Theorem 3.3 that f is not homogeneous-like. References 1. E. Garc´ıa Barroso and B. Teissier, Concentration multi-´echelles de courbure dans des fibres de Milnor, Comment. Math. Helv. 74 (1999), 398–418. 2. T. Fukui and E. Yoshinaga, The modified analytic trivialization of family of real analytic functions, Invent. math. 82 (1985), 467–477. 3. T. Fukui, S. Koike and T.-C. Kuo, Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities (T. Fukuda, T. Fukui, S. Izumiya and S. Koike, ed), Pitman Research Notes in Mathematics Series, 381 (1998), pp. 8–29.

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4. T. Fukui and L. Paunescu, Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan 52 (2000), 433–446. 5. T. Fukui and L. Paunescu, On blow-analytic equivalence, Arc-Spaces and Additive Invariants in Real Algebraic Geometry, Panoramas et Syntheses 24 (2008), SMF, pp. 87–125. 6. S. Koike and A. Parusi´ nski, Blow-analytic equivalence of two variable real analytic function germs, Journal of Algebraic Geometry 19 (2010), 439–472. 7. S. Koike and A. Parusi´ nski, Equivalence relations of two variable real analytic function germs, Jour. Math. Soc. Japan 65 (2013), 237–276. 8. S. Koike, T.-C. Kuo and L.Paunescu, A study of curvature using infinitesimals, Proc. Japan Acad. Ser. A, Math. Sci. 88, No. 5, 70–74 (2012). 9. S. Koike, T.-C. Kuo and L.Paunescu, A’Campo curvature bumps and the Dirac phenomenon near a singular point, arXiv:1206.0525 10. S. Koike, T.-C. Kuo and L.Paunescu, A’Campo bumps near singular points of real analytic two variable function germs, in preparation. 11. T.-C. Kuo, The jet space J r (n, p), Proceedings of the Liverpool Singularities Symposium (C. T. C. Wall, ed.), Springer Lect. Notes in Math. 192 (1971), 169–177. 12. T.-C. Kuo and Y.C. Lu, On analytic function germs of complex variables, Topology 16 (1977), 299–310. 13. T.-C. Kuo, On classification of real singularities, Invent. math. 82 (1985), 257–262. 14. T.-C. Kuo and L. Paunescu, Equisingularity in R2 as Morse stability in infinitesimal calculus, (Communicated by Heisuke Hironaka) Proc. Japan Acad. Ser. A, Math. Sci. 81, No. 6, 115–120 (2005). 15. T.-C. Kuo and L. Paunescu, Enriched Riemann sphere, Morse stability and equisingularity in O2 , Jour. London Math. Soc. 85 (2012), 382–408. 16. R. Langevin, Courbure et singularit´es complexes, Comment. Math. Helv. 54 (1979), 6–16. 17. A. Parusi´ nski, A criterion for the topological equivalence of two variable complex analytic function germs, Proc. Japan Acad. Ser. A. Math. Sci. 84 No. 8 (2008), 147–150. 18. J.-J. Risler, On the curvature of the real Milnor fiber, Bull. London Math. Soc. 35 (2003), 445–454. 19. John A. Thorpe, Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics, 1979 Springer-Verlag. 20. O. Zariski, On the topology of algebroid singularities, Amer. Jour. Math. 54 (1932), 453–465.

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Saito free divisors in four dimensional affine space and reflection groups of rank four Jiro Sekiguchi∗ Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan [email protected] In this paper we collect examples of Saito free divisors in a four dimensional affine space. Some of them are already given in literatures but some are new. Typical examples are those defined as discriminant sets of real and complex reflection groups of rank four. Our interest is to study a relationship between such divisors and Saito free divisors obtained by restrictions of discriminants of polynomials to subspaces of the spaces of coefficients. In the last section, we show systems of uniformization equations with singularities along the discriminant sets of the reflection groups No. 28 and No. 31. Keywords: free divisor, complex reflection group AMS classification numbers: 20F65

1. Introduction In this paper, we report the progress on the construction of examples of Saito free divisors defined as zero sets of polynomials in four variables. The author studied in detail Saito free divisors in C3 and related topics in a series of papers [8], [9], [10], [11], [12], [13], [3]. Compared with three dimensional case, it is difficult to construct and classify Saito free divisors in C4 in a systematic way. (A systematic method of constructing Saito free divisors in the three dimensional case is explained in Remark 3.1 of the main context.) We explain an idea of finding polynomials which define Saito free divisors. Let Pn (t) = tn + x1 tn−1 + x2 tn−2 + · · · + xn−1 t + xn ∗ Partially

supported by Grand-in-Aid for Scientific Research (No. 20540066, No. 2354077), Japan Society of the Promotion of Science.

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be a polynomial of t. The discriminant Δn (x1 , x2 , . . . , xn ) of Pn (t) is regarded as a polynomial of coefficients of Pn (t). Let Y be an affine subspace of Cn and consider the restriction F (y1 , y2 , . . . , ym ) of Δn (x1 , x2 , . . . , xn ) to Y , where y = (y1 , y2 , . . . , ym ) is a coordinate on Y . Then it sometimes happens that the hypersurface F (y1 , y2 , . . . , ym ) = 0 turns out to be a Saito free divisor in Y . This is a basic idea of finding Saito free divisors employed in this paper. We need some modification of this idea when we apply this idea to individual cases. Some of the discriminants of real and complex reflection groups are obtained in this manner (cf. [3]). The main result of this paper is to collect Saito free divisors in C4 obtained by this idea and to compare such divisors with discriminants of real and complex reflection groups of rank four. The construction of systems of uniformization equations with singularities along Saito free divisors collected in this paper is an interesting subject to attack. We only report two examples of such systems in this paper and further studies on this subject will be postponed in further consideration. We now briefly explain the contents of this paper. In §2, we review Saito free divisors and complex reflection groups. In §3, we collect polynomials of four variables which define Saito free divisors. Some of them are discriminant sets of complex reflection groups and some are obtained as restrictions of discriminants of polynomials. In §4, we show systems of uniformization equations with singularities along the discriminant sets of the reflection groups No.28 and No.31 in the sense of [14]. In closing this introduction, we note that our computations in this paper are performed using the computer algebrac software Mathematica. Acknowledgement: The author thanks the referee for reading the contents carefully. Following his advice, the author improved the first draft.

2. Preliminaries 2.1. Saito free divisors The notion of Saito free divisors is introduced by K. Saito [6]. Let F (x) = F (x1 , x2 , . . . , xn ) be a reduced polynomial with the following conditions; n (A1) There is a vector field E = i=1 mi xi ∂xi such that EF = dF , where m1 , m2 , . . . , mn , d are positive integers with 0 < m1 ≤ m2 ≤ · · · ≤ mn . n (A2) There are vector fields V i = j=1 aij (x)∂xj (i = 1, 2, . . . , n) such that each aij (x) is a polynomial of x1 , x2 , . . . , xn , that the determinant of

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the n × n matrix (aij (x)) coincides with F (x), that V 1 = E, V i F (x) = ci (x)F (x) for polynomials ci (x), that [E, V i ] = ki V i for constants ki and  that [V i , V j ] ∈ nk=1 RV k for any i, j, where R = C[x1 , x2 , . . . , xn ]. Definition 2.1. Let F (x) be a reduced polynomial. Then D = {x ∈ Cn ; F (x) = 0} is a (weighted homogeneous) Saito free divisor if F (x) satisfies conditions (A1) and (A2). Remark 2.1. If a polynomial F (x) satisfies Condition (A1), we say that F (x) is weighted homogeneous with weight system (m1 , . . . , mn ; d). Remark 2.2. A Saito free divisor is also called a logarithmic free divisor (cf. [6]). n For simplicity, we put M = (aij (x)) and DerCn (log D) = k=1 RV k . In this paper V 1 , V 2 , . . . , V n are called basic vector fields for the polynomial F (x) and M is called the generating matrix defining basic vector fields. It follows from the definition that t

(V 1 , V 2 , · · · , V n ) = M t (∂x1 , ∂x2 , · · · , ∂xn ).

We may assume that each aij (x) is weighted homogeneous. 2.2. Irreducible real and complex reflection groups of rank four In this subsection, we collect elementary results on irreducible complex reflection groups of rank four. A basic reference is Shephard-Todd [14]. Reflection groups treated in this subsection are real reflection groups of types A4 , B4 , D4 , F4 , H4 and complex reflection groups of No.29, No.31, No.32 in the sense of [14]. Let G be one of the reflection groups mentioned above. Let V be a standard representation space of G over C. (In the case of real reflection groups, we consider their complexifications.) Let P1 , P2 , P3 , P4 be algebraically independent basic G-invariant polynomials and put kj = degξ (Pj ),where ξ = (ξ1 , ξ2 , ξ3 , ξ4 ) are coordinates of V . We may assume that k1 ≤ k2 ≤ k3 ≤ k4 . Let r be the greatest common divisor of k1 , k2 , k3 , k4 and put kj = kj /r (j = 1, 2, 3, 4). Since the discriminant of the group G is expressed as a polynomial of P1 , P2 , P3 , P4 , we write it by δG (P1 , P2 , P3 , P4 ). Putting  E = 4i=1 ki Pi ∂Pi , we find that EδG = (d/r)δG , where d is the degree of δG .

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TABLE I group A4 B4 D4 No.28 No.29 No.30 No.31 No.32

order k1 , k2 , k3 , k4 degree (k1 , k2 , k3 , k4 )

W (A4 ) 120 2, 3, 4, 5 20 W (B4 ) 384 2, 4, 6, 8 32 W (D4 ) 192 2, 4, 4, 6 24 W (F4 ) 1152 2, 6, 8, 12 48 G7680 7680 4, 8, 12, 20 80 W (H4 ) 14400 2, 12, 20, 30 120 G46080 46080 8, 12, 20, 24 120 G155520 155520 12, 18, 24, 30 120

(2, 3, 4, 5) (1, 2, 3, 4) (1, 2, 2, 3) (1, 3, 4, 6) (1, 2, 3, 5) (1, 6, 10, 15) (2, 3, 5, 6) (2, 3, 4, 5)

2.3. Discriminants of reflection groups We retain the notation of the previous subsection. It is underlined (cf. [6], [4]) that the hypersurface D : δG (P1 , P2 , P3 , P4 ) = 0 in C4 is a Saito free divisor. As a consequence, if M is the generating matrix for δG (P1 , P2 , P3 , P4 ), it follows that det M coincides with δG up to a non-zero constant factor. For this reason, the determination of M is important in the study of discriminants. A method how to construct M is shown in [6] for the case where G is a real reflection group and done in [4] for the case where G is a complex reflection group. Since the construction of M is one of central subjects in our consideration, we explain briefly the method of constructing M for the case of an arbitrary reflection group. In the real case, there exists a basic G-invariant P1 of degree two. The polynomial P1 plays a basic role in the construction of M . On the contrary to the real case, there is no basic G-invariant of degree two in the complex case, which is one of the reasons why the argument of showing the existence of M in the real case does not go well for the complex case. Orlik and Terao overcame this difficulty and established a method of the construction of M . For the details, see Chapter 6 and Appendix B of [4]. 3. Saito free divisors in C4 In this section, we introduce Saito free divisors in C4 some of which are already known and some of which are new. In Yano and Sekiguchi [15], we computed basic vector fields for discriminants of irreducible real reflection groups except those of types E7 , E8 . We collect the generating matrices which define basic vector fields for discriminants for those of types

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F4 , H4 , A4 in §3.1, §3.3, §3.5, respectively. We do not treat the cases B4 , D4 . For the readers who are interested in these cases, refer to [15]. We also collect the generating matrices in the cases of the complex reflection groups No.29, No.31 which are determined by the data in Appendix B of [4] and their explicit forms are shown in [1]. In §3.1, we treat polynomials with weight system (1,3,4,6;24). One is the discriminant of type F4 and the other is constructed by the method explained in Introduction. Both define Saito free divisors. In §3.2, we treat polynomial of weight system (1,2,3,5;20). One is the discriminant of the complex reflection group No.29. There is another polynomial which defines Saito free divisor. In §3.3, we show the generating matrix for the discriminant of the real reflection group of type H4 . Its generating matrix is the one given in [15]. The weight system of the discriminant is (1,6,10,15;60). In §3.4, we show the generating matrix in [1] for the discriminant of the complex reflection group No.31. In this case, the discriminant is regarded as the restriction of the discriminant of the polynomial of 6th degree to a one codimensional subspace. In §3.5, we show two matrices whose determinants define Saito free divisors. The first one corresponds to the discriminant of type A4 . Weight systems of both are (2,3,4,5;20). In the sequel, the variables will not called x1 , x2 , x3 , x4 but rather subscripts will indicate the degree of variable or weight degree of vector fields. Remark 3.1. In the subsequent subsections, we construct two examples of Saito free divisors in C4 by the method explained in Introduction (cf. §3.1, §3.2) and one example by direct computation (cf. §3.5) which are not defined by the zero sets of discriminants of reflection groups. In the three variables case, there is a systematic method of constructing Saito free divisors from isolated curve singularities (cf. [8]). We now explain this method briefly by taking a curve −27x43 + 256x34 = 0 which has E6 -singularity at the origin. Weights of x3 , x4 are given as 3,4, respectively. Then we consider a polynomial F = 256x34 − 27x43 − 128x22 x24 + (c1 x2 x23 + c2 x41 )x4 + c3 x32 x23 + c4 x62 , where c1 , c2 , c3 , c4 are constants and x2 is a parameter of weight 2. Then F is a weighted homogeneous polynomial of (x2 , x3 x4 ) and F = 0 is regarded as a 1-parameter family of curves in the (x3 , x4 )-space with the parameter x2 . Computing the condition for the constants c1 , c2 , c3 , c4 so that F = 0 is a Saito free divisor in C3 , we obtain two kinds of Saito free divisors. One is defined as the zero set of the discriminant of type A3 . The other is the one obtained by M. Sato in 1970’s. The author testify the method

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explained here to not only the cases of simple curve singularities (cf. [9]) but also some of fourteen families of exceptional singularities in the sense of Arnol’d and obtained a great number of examples of Saito free divisors including the discriminants of real and complex reflection groups of rank three (unpublished). Unfortunately the method of constructing Saito free divisors explained above is not generalized to the four variables case for the present. 3.1. The group No. 28 case (weight system (1, 3, 4, 6; 24)) The group No.28 is identified with the Weyl group W (F4 ) of type F4 . The discriminant of W (F4 ) is computed in [15] and its generating matrix M (F4 ) is given as follows: ⎛ ⎞ 3y6 4y8 6y12 y2 ⎜ ⎟ ⎜ ⎟ ⎜ 3y6 −15y2 y8 + 32 y25 −18y12 − 6y23 y6 4y82 − 4y24 y8 + 4y22 y62 ⎟ ⎜ ⎟ M (F4 ) = ⎜ ⎟ ⎜ 4y8 −18y12 − 6y 3 y6 −7y3 y8 + 7y2 y2 + 1 y7 9y2 y6 y8 − y3 − 3 y6 y6 ⎟ 2 6 2 6 2 2 2 2 2 ⎜ ⎟ ⎝ ⎠ 11 y3 y2 − 11 y y2 y 3 2 8 3 2 6 8 6y12 4y82 − 4y24 y8 + 4y22 y62 9y22 y6 y8 − y63 − 32 y26 y6 5 11 2 1 11 y y + y + 6

2 6

12 2

Here y2 , y6 , y8 , y12 are basic W (F4 )-invariants. In this case, we consider the polynomial of t defined by 27t8 + (4x31 + 54x3 )t6 + (54x6 + 27x23 + 12x21 x4 )t4 + (54x6 x3 + 12x1 x24 )t2 + 4x34 + 27x26 . By direct computation we find that its discriminant coincides with (x6 x21 − x1 x3 x4 + x24 )6 (27x26 + 4x34 ) ×(432x26 − x6 x61 − 36x6 x31 x3 − 216x6 x23 + x31 x33 + 27x43 +72x6 x21 x4 + x51 x3 x4 + 30x21 x23 x4 − x41 x24 − 96x1 x3 x24 + 64x34 )2 up to a constant factor. Then (27x26 + 4x34 ) ×(432x26 − x6 x61 − 36x6 x31 x3 − 216x6 x23 + x31 x33 + 27x43 +72x6 x21 x4 + x51 x3 x4 + 30x21 x23 x4 − x41 x24 − 96x1 x3 x24 + 64x34 ) coincides with the discriminant det M (F4 ) by the correspondence x1 x3 x4 x6

= y2 , 1 = 72 (−3y23 + 2y6 ), 1 = 576 (−y24 + 4y2 y6 + 4y8 ), 1 = 10368 (−6y12 − y23 y6 + y62 − 3y22 y8 ).

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Remark 3.2. The differences of the indices of x1 , x3 , x4 , x6 and y2 , y6 , y8 , y12 are confusing. This is caused by the notation of basic W (F4 )invariants employed in [15]. We are now going to find a polynomial of four variables with weight system (1,3,4,6;24) which defines a Saito free divisor not equal to that defined by the discriminant of W (F4 ). Let P6 (t) = t6 + x2 t4 + x3 t3 + x4 t2 + x5 t + x6 be a polynomial of t and let Δ6 (x2 , x3 , x4 , x5 , x6 ) be the discriminant of P6 (t). We treat a polynomial f (x1 , x3 , x4 , x6 ) obtained by restricting Δ6 (x2 , x3 , x4 , x5 , x6 ) to the subspace defined by x5 = a1 x4 x1 + a2 x3 x21 + a3 x41 , x2 = b1 x21 , where a1 , a2 , a3 , b1 are constants. We first consider the condition that f (x1 , x3 , x4 , x6 ) is divided by x6 + c1 x4 x21 + c2 x23 + c3 x3 x31 + c4 x61 . By direct computation, we find that if a2 = − 34 a21 , 1 3 a1 (−5a1 + 3a21 + 8b1 ) a3 = 16 and c1 c2 c3 c4

= = = =

− 41 a21 , 0, 1 3 a , 4 1 1 4 − 64 a1 (−8a1 + 5a21 + 12b1,

then f (x1 , x3 , x4 , x6 ) is reducible. As a consequence, if a2 , a3 , c1 , c2 , c3 , c4 satisfy the relations above, there is a polynomial f0 (x1 , x3 , x4 , x6 ) such that f (x1 , x3 , x4 , x6 ) = {64x6 −a41 (−8a1 +5a21 +12b1)x61 +16a31 x31 x3 −16a21x21 x4 } × f0 (x1 , x3 , x4 , x6 ). Next we consider the condition that f0 (x1 , x3 , x4 , x6 ) = 0 is a Saito free divisor. By direct computation, we conclude that if a1 =

1 (2 − p0 ), 6

b1 =

1 (10 − p20 ) 24

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for a non-zero constant p0 , then f0 (x1 , x3 , x4 , x6 ) = 0 is a Saito free divisor. It can be shown that the hypersurface f0 (x1 , x3 , x4 , x6 ) = 0 is isomorphic to each other for arbitrary p0 (= 0). In particular, in the case p0 = −2, f0 (x1 , x3 , x4 , x6 ) is equal to the polynomial fN o.28 (x1 , x3 , x4 , x6 ) defined as the determinant of the following matrix up to a non-zero constant factor: ⎛

x1

3x3

⎜ ⎜ ⎜ 54x3 x1 (5x1 x3 − 22x4 ) ⎜ ⎜ ⎜ ⎜ ⎜ 30x23 − 24x6 ⎜ 16x4 ⎜ ⎜ ⎜ ⎜ 2 2 ⎜ ⎝ 216x6 −19x1 x3 + 2x1 x3 x4 +48x24

4x4 7x31 x3 + 36x23 −20x21 x4 − 36x6 −7x1 x23 + 52x3 x4 −12x1 x6

22x31 x23 + 9x33 −116x21 x3 x4 + 132x1 x24

+72x3 x6

6x6

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 3 2 2 x1 x3 − 4x1 x3 x4 ⎟. ⎟ 2 +4x1 x4 + 48x3 x6 ⎟ ⎟ ⎟ −3x21 x33 + 8x1 x23 x4 ⎟ ⎟ −4x3 x24 − 84x21 x3 x6 ⎠ +144x1 x4 x6 −4x1 x3 x4 + 8x24 −12x21 x6

As a consequence, we have the following proposition: Proposition 3.1. There are at least two polynomials with weight system (1,3,4,6;24) of four variables which define Saito free divisors. One is the discriminant of W (F4 ) and the other is fN o.28 (x1 , x3 , x4 , x6 ). Remark 3.3. In the occasion of the workshop of Japan-Russia bilateral program held at Krasnoyarsk University in August, 2006, the author gave a talk on Saito free divisors and showed that fN o.28 (x1 , x3 , x4 , x6 ) = 0 is a Saito free divisor. 3.2. The group No. 29 case (weight system (1, 2, 3, 5; 20)) The generating matrix for the discriminant of the group No.29 is obtained in [1]. Its concrete form is given by ⎛ ⎞ 320x 640y 960z 1600t ⎜ 640y 2 1280(3200t + xy2 + 176yz) −640x(20t + yz) −640(txy + 80tz + 2yz2 ) ⎟ ⎟. M29 = ⎜ ⎝ 2xy + 960z ⎠ 4x2 y + 64xz 200t + 3x2 z − 5yz 8tx2 − 10ty − 4xz 2 1600t + 8yz −640tx + 16xyz + 1536z 2

10tx2 − 10ty − 8xz 2

72txz − 96z 3

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There is a relationship between the determinant of M29 which is equal to the discriminant of the group No.29 and the discriminant of a polynomial of fifth degree which we are going to explain. The discriminant of T 5 + x1 T 4 + x2 T 3 + x3 T 2 +

x22 T + x5 4

coincides with the determinant of the above matrix by the coordinate transformation x = 4x1 , y = −8x2 , z = 12 x1 x2 − 12 x3 , t = 18 x1 x22 − 14 x2 x3 + 12 x5 . There is another Saito free divisor which is the zero locus of the polynomial of x1 , x2 , x3 , x5 obtained by the restriction of the discriminant Δ5 (x1 , x2 , x3 , x4 , x5 ) of the polynomial T 5 + x1 T 4 + x2 T 3 + x3 T 2 + x4 T + x5 to a hypersurface of the (x1 , x2 , x3 , x4 , x5 )-space defined by x4 = a1 x3 x1 + a2 x22 + a3 x2 x21 + a4 x41 for some constants a1 , a2 , a3 , a4 . By direct computation, we find that if f (x1 , x2 , x3 , x5 ) is the polynomial of (x1 , x2 , x3 , x5 ) obtained by the substitution x4 = −

(2 + c) x1 {(−6 + c)(2 + c)2 x31 + 60(2 + c)x1 x2 − 400x3 } 2000

to Δ5 (x1 , x2 , x3 , x4 , x5 ), then f (x1 , x2 , x3 , x5 ) = 0 is a Saito free divisor, where c is a non-zero constant. To show its concrete form, it is better to change the coordinate. For this purpose, we introduce a coordinate transformation of the (x1 , x2 , x3 , x5 )-space to the (y1 , y2 , y3 , y5 )-space defined by 6 y1 , x1 = − 5c 1 2 2 2 2 x2 = − 500c 2 (−288y1 + 27c y1 − 500c y2 ),

x3 =

1 (−1728y13 12500c3

+ 486c2 y13 + 27c3 y13 − 9000c2y1 y2

−4500c3y1 y2 + 12500c3y3 ),

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+540000c2y13 y2 + 810000c3y13 y2 + 405000c4y13 y2 + 67500c5y13 y2 −2250000c3y12 y3 − 2250000c4y12 y3 − 562500c5y12 y3 − 39062500c5y5 ). Regarding f (x1 , x2 , x3 , x5 ) as a polynomial of (y1 , y2 , y3 , y5 ), its generating matrix is given by ⎛

y1

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 550000y22 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 2500y3 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜2812500y5 ⎜ ⎜ ⎝

2y2 ⎛

⎞ −3159y13y2 ⎜ −84600y1y22 ⎟ ⎜ ⎟ 2 ⎟ 2⎜ ⎜ +17145y1 y3 ⎟ ⎝ +202500y2y3 ⎠ −687500y5 ⎞ ⎛ 72y12 y2 2 ⎝ +800y22 ⎠ −285y1y3

⎞ 1440y12y22 ⎜ +16000y23 ⎟ ⎟ ⎜ 3 ⎟ 10 ⎜ ⎜ +108y1 y3 ⎟ ⎝ −40500y 2 ⎠ 3 +14375y1y5 ⎛

3y3

5y5

⎞

⎟ ⎟ ⎟  ⎟ ⎟ 3 ⎜ −15600y1y2 y3 ⎟ ⎟ 11907y1 ⎟ y5 ⎟ 3⎜ ⎝ −45000y 2 ⎠ −775000y3 ⎟ 3 ⎟ ⎟ −205000y1y5 ⎟ ⎟ ⎟ 2   ⎟ −27y1 y3 ⎟. −81y12 ⎟ y5 +2900y2y3 ⎟ +5500y2 ⎟ +2500y5 ⎟ ⎞ ⎛ ⎟ 4 108y1 y3 ⎟ ⎛ ⎞ ⎟ ⎜ +1500y 2y y ⎟ 243y14 ⎟ ⎜ 1 2 3⎟ ⎟ ⎜ ⎜ +3375y12y2 ⎟⎟ ⎜ +110000y22y3 ⎟ ⎟⎟ 3⎜ ⎟ 4y5 ⎜ ⎝ +167500y 2 ⎠⎟ ⎟ ⎜ −31500y1y32 ⎟ 2 ⎟ ⎟ ⎜ ⎝ +10775y12y5 ⎠ −68875y1y3 ⎠ ⎛

1323y13y3

⎞

+487500y2y5

In fact it is provable that the determinant fN o.29 (y1 , y2 , y3 , y5 ) of this matrix defines a Saito free divisor in the (y1 , y2 , y3 , y5 )-space. Note that fN o.29 (y1 , y2 , y3 , y5 ) has a factor y5 . As a consequence we obtain the following proposition. Proposition 3.2. There are at least two polynomials with weight system (1,2,3,5;20) of four variables which define Saito free divisors. One is the discriminant of the group No.29 and the other is fN o.29 (y1 , y2 , y3 , y5 ).

3.3. The group No. 30 case (weight system (1, 6, 10, 15; 60)) The group No.30 is identified with the real reflection group W (H4 ) of type H4 . The discriminant of W (H4 ) and its generating matrix is obtained in [15]. In particular the generating matrix is given as follows:

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⎛

2x2

12x12

20x20

⎜ ⎜ ⎜ −15x30 ⎜ 22x2 x20 ⎜ ⎜ 12x12 −12x52 x20 ⎜ −4x52 x12 ⎜ +4x32 x212 ⎜ ⎜ ⎜ ⎜ −15x30 10x42 x30 M (H4 ) = ⎜ ⎜ 20x 5 −28x32 x12 x20 20 −12x2 x20 ⎜ ⎜ 3 2 +4x2 x12 + 76 x x3 ⎜ 3 2 12 ⎜ ⎜ ⎜ −20x52 x30 −60x32 x12 x30 ⎜ 2 ⎜ +24x42 x220 ⎜ 30x30 −40x20 4 ⎝ −8x2 x12 x20 −56x22 x212 x20 − 83 x22 x312 r −16x412

30x30 −20x52 x30 −40x220 −8x42 x12 x20 − 38 x22 x312 −60x32 x12 x30 +24x42 x220 −56x22 x212 x20 −16x412 80x42 x20 x30 +120x22 x212 x30 +48x32 x12 x220 3 + 464 3 x2 x12 x20

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

We do not check whether the discriminant of W (H4 ) is identified with a polynomial which is obtained as a restriction of the discriminant of a polynomial of degree 15 or not because of the difficulty of the computation. 3.4. The group No. 31 case (weight system (2, 3, 5, 6; 24)) The defining matrix of the discriminant of the group No.31 is obtained in [1]. Its concrete form is shown as follows: ⎞ ⎛ 2160x 3240y 5400z 6480t ⎟ ⎜ 4860tx2 ⎜ 3240ty 2 2 −11340txz ⎟ −9720t + 16200xz ⎟ ⎜ −16200yz 2 ⎟ −26244000z 2 ⎜ M31 = ⎜ ⎟. ⎜ 2xy + 5400z −9720t + 3x3 −tx − 5yz −5ty − 2x2 z ⎟ ⎟ ⎜ ⎝ −3x2 y 5tx2 + 2xyz ⎠ 2 2 ty + 5x z 6480t − 2y −11340xz +5400z 2 There is a relationship between the determinant of M31 and the discriminant of a polynomial of sixth degree which we are going to explain. Let P6 (T ) = T 6 + x2 T 4 + x3 T 3 + x4 T 2 + x5 T + x6 be a polynomial of t and let Δ6 (x2 , x3 , x4 , x5 , x6 ) be the discriminant of P6 (T ). Consider the restriction of Δ6 (x2 , x3 , x4 , x5 , x6 ) to the subspace defined by x4 = px22 for a constant p and put f (x2 , x3 , x5 , x6 ; p) = Δ6 (x2 , x3 , px22 , x5 , x6 ).

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Then it is possible to show that the hypersurface f (x2 , x3 , x5 , x6 ; p) = 0 in the (x2 , x3 , x5 , x6 )-space is a Saito free divisor if and only if p = 14 . Moreover f (x2 , x3 , x5 , x6 ; 14 ) coincides with the determinant of M31 up to a constant factor by the coordinate change: x = −6x2 , y = −54x3 , z = − 14 x2 x3 + 12 x5 , t = 34 x23 − 3x6 . In the next section, we will introduce systems of uniformization equations with singularities along the hypersurface f (x2 , x3 , x5 , x6 ; 14 ) = 0 and discuss some of its properties. 3.5. The group No. 32 case (weight system (2, 3, 4, 5; 20)) The basic vector fields for the discriminant of the group No.32 are identified with those of the Weyl group W (A4 ) of type A4 by choosing the basic invariants appropriately. The generating matrix for the case W (A4 ) is, for example, given in [15]. Its concrete form is shown as follows: ⎞ ⎛ 2x2 3x3 4x4 5x5 ⎟ ⎜ ⎟ ⎜ 3x3 − 2 (3x22 − 10x4 ) 1 (−4x2 x3 + 25x5 ) − 52 x2 x4 5 5 ⎟ ⎜ ⎟. ⎜ ⎜ 4x4 1 (−4x2 x3 + 25x5 ) − 2 (3x2 − 5x2 x4 ) − 3 (x3 x4 − 5x2 x5 ) ⎟ 3 5 5 5 ⎠ ⎝ 5x5

− 25 x2 x4

− 35 (x3 x4 − 5x2 x5 ) − 52 (2x24 − 5x3 x5 )

The weight system of the discriminant in this case is (2,3,4,5;20). There is another polynomial of weight system (2,3,4,5;20) which defines a Saito free divisor. Its generating matrix is shown as follows (cf. [9]): ⎞ ⎛ 4x4 5x5 2x2 3x3 ⎟ ⎜ 3 ⎟ ⎜ 3x3 x22 + 4x4 5 x5 x x 4 10 2 4 ⎟ ⎜ M =⎜ ⎟. 5 ⎟ ⎜ 4x4 5x5 − 5 x2 x4 − x x ⎠ 8 8 2 5 ⎝ 5x5 2x2 x4

15 3 2 16 x3 x4 5 x4

+

15 16 x3 x5

4. Systems of uniformization equations If one obtains a Saito free divisor, the next question is to ask the existence of systems of uniformization equations with singularities along it. In [13], the author treated this question for the case of the reflection group of type

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D4 . In this section, we treat the cases of the groups No.28 and No.31. In particular we show examples of systems of uniformization equations with singularities along the discriminant sets for these two groups. One of our interests is to study the properties of the holonomic systems constructed in this section. We hope to discuss this subject as well as the construction of the systems of uniformization equations for the remaining cases in a future. In the sequel, the variables will not called x1 , x2 , x3 , x4 and the basic vector fields will not to be called V 1 , V 2 , V 3 , V 4 , but rather subscripts will indicate the degree of variable or weight degree of vector fields.

4.1. The case No. 28 We replace the variables y2 , y6 , y8 , y12 in §3.1 with x1 , x3 , x4 , x6 , respectively and modify the generating matrix M (F4 ) with M  (F4 ) defined by ⎛

x1 ⎜ 3x 3 ⎜ ⎜ ⎜ ⎜ 1 x4 + 4x 4 ⎜ 2 1 ⎜ ⎜  M (F4 ) = ⎜ ⎜ ⎜ ⎜ ⎜ 1 3 ⎜ − x1 x3 + 6x6 ⎜ 2 ⎝

3x1 2

⎞ 3x3 4x4 6x6 4 3 2 2 4 2 (x1 − 10x4 ) −6x1 x3 − 18x6 4x1 x3 − 4x1 x4 + 4x4 ⎟ ⎟ 1 7 ⎟ x − 32 x61 x3 ⎟ 2 1 − 92 x31 x3 ⎟ +7x1 x23 −x33 + 9x21 x3 x4 ⎟ −18x6 ⎟ 3 3 −5x1 x4 +3x1 x6 ⎟ ⎟. 1 11 x ⎟ 12 1 ⎟ 11 5 2 5 2 2 ⎟ + x x x x 3 6 1 3 1 3 6 2 ⎟ − x x 3 2 1 ⎟ −4x41 x4 − 11 x1 x23 x4 3 2 3 ⎟ −x3 + 7x1 x3 x4 11 3 2 2 ⎠ +4x4 + 3 x1 x4 −3x21 x3 x6

Let f0 = det(M  (F4 )) be the polynomial which defines a Saito free divisor. Define vector fields V 0 , V 2 , V 3 , V 5 from M  (F4 ). Then V 0 f0 = 24f0 , V j f0 = 0 (j = 2, 3, 5), [V 2 , V 3 ] = −2x31 V 2 − 3V 5 , [V 2 , V 5 ] = 4x21 x3 V 2 + (3x4 − 32 x41 )V 3 , [V 3 , V 5 ] = (2x21 x4 − 23 x23 − 13 x61 )V 2 + 72 x21 x3 V 3 + 52 x31 V 5 . In this case M(r1 , r2 , r3 , s) (r1 , r2 , r3 , s ∈ C) defined below is a system of uniformization equations with singularities along the hypersurface f0 = 0. V 0 u =su, 1

{(6r2 + r22 − 3r1 r3 )x41 + (18r1 + 3r1 r2 − r2 r3 )x1 x3 2 2 V V = 3 u, −r2 (6 + r2 )x4 } + (r1 + r3 )x21 V 2 + r2 x1 V 3

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⎡

⎧ ⎫⎤ 3(−36 + r22 − 4r1 r3 + 4r32 )x61 + 24r2 r3 x31 x3 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬⎥ +4(−18 + 3r2 + r22 − 6r1 r3 + 2r32 )x23 1 ⎥ 2 2 2 36 ⎪ ⎥ −12(18 + 9r2 + r2 + 2r1 r3 − 2r3 )x1 x4 ⎪ ⎪ ⎪ ⎩ ⎭ ⎥ u, ⎥ +72(6r1 + r1 r2 − r2 r3 )x6

⎢ ⎢ ⎢ 3 3 V V u =⎢ ⎢ ⎢ 1 ⎥ ⎣ + 3 {(r1 − r3 )(x41 + 4x4 ) − 2(−3 + r2 )x1 x3 }V 2 ⎦ + 61 (−9x31 − 2(3r1 + r3 )x3 )V 3 + (r1 − r3 )x1 V 5

⎫⎤ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎬⎥ ⎢ ⎢ 1 −4(54 + 15r2 + r2 + 6r1 r3 − 6r3 )x1 x4 ⎥ ⎢ 108 ⎥ 3 −8(18r + 3r r − r r )x x x ⎪ ⎪ ⎢ ⎥ 1 1 2 2 3 1 3 4 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ 2 2 2 2 ⎪ ⎪ +12(−18 + 3r + r − 6r r + 2r )x x ⎪ ⎪ ⎢ ⎥ 2 1 3 2 3 1 4 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ 2 ⎪ ⎪ +72(18 + 3r + 2r r − 2r )x x x ⎪ ⎪ ⎢ 2 1 3 1 3 6 3 ⎩ ⎭⎥ ⎥ u, V 5V 5u = ⎢ −72(6r1 + r1 r2 − r2 r3 )x4 x6 ⎢ ⎥  $ ⎢ ⎥ 4 − r )(x + 4x ) (r 1 3 4 ⎢ ⎥ 1 1 + 18 (x41 − 2x4 ) V2 ⎢ ⎥ −2(−9 + r2 )x1 x3 ⎢ ⎥ $  ⎢ ⎥ 4 ⎢ 3(r − r )(−3x x + 2x x + 12x x ) 1 3 3 4 1 6 1 1 3 3 ⎥ ⎢ + 18 ⎥ V ⎢ ⎥ +2(−9 + r2 )(x1 x23 − 2x31 x4 ) ⎢ ⎥  $ 4 ⎣ ⎦ (r1 − r3 )x1 − (−3 + 2r2 )x1 x3 − 61 x1 V5 +2(5r1 − r3 )x4 ⎡

⎧ 2 (−54 − 3r2 + r22 − 6r1 r3 + 6r32 )(x10 ⎪ 1 − 4x3 x4 ) ⎪ ⎪ +4r r (3x7 x + 2x x3 ) ⎪ 2 3 1 3 ⎪ 1 3 ⎪ ⎪ ⎪ +9(−36 + r22 − 4r1 r3 + 4r32 )x41 x23 ⎪ ⎨ 2 2 6

⎫⎤ (6r1 + r1 r2 − 2r2 r3 )x41 ⎬ ⎢ 13 x1 −(18r2 + 3r22 − 4r1 r3 )x1 x3 ⎥ ⎢ ⎩ ⎭⎥ 2 3 3 2 (V V + V V )u = ⎢ +4(12r1 + 2r1 r2 − r2 r3 )x4 ⎥ ⎢ ⎥ u, ⎣ ⎦ + 31 {(−6 + r2 )x31 − 4r3 x3 }V 2 +2r1 x21 V 3 + (−3 + 2r2 )V 5 ⎡

⎡

⎧ ⎨

⎫⎤ r2 (6 + r2 )(x71 + 4x1 x23 ) ⎪ ⎪ ⎬⎥ ⎢ −2(18r1 + 3r1 r2 − 8r2 r3 )x41 x3 ⎢ 1 ⎥ ⎢ 18 ⎪ ⎥ ⎪ −2(30r2 + 5r22 − 12r1 r3 )x31 x4 ⎢ ⎪ 2 5 5 2 ⎪ ⎭⎥ (V V + V V )u = ⎢ ⎩ ⎥ u, ⎢ −8r2 r3 x3 x4 + 72(6r1 + r1 r2 − r2 r3 )x1 x6 ⎥ ⎢ ⎥ ⎣ ⎦ − 13 x1 {(−12 + r2 )x1 x3 + 4r3 x4 }V 2 + 16 (−9 + 2r2 )(x41 − 2x4 )V 3 + 2r1 x21 V 5 ⎧ ⎪ ⎪ ⎨

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⎧ ⎫⎤ (6r1 + r1 r2 − 3r2 r3 )x81 ⎪ ⎪ ⎪ ⎪ 2 2 5 ⎪ ⎥ ⎢ ⎪ ⎪ −(−144 + 6r2 + 5r2 − 16r1 r3 + 16r3 )x1 x3 ⎪ ⎪ ⎢ ⎪ ⎨ ⎬⎥ ⎥ ⎢ 1 2(18r1 + 3r1 r2 + r2 r3 )(x41 x4 − x21 x23 ) ⎥ ⎢ 18 2 2 +8(−18 + 3r + r − 4r r + 2r )x x x ⎥ ⎢ ⎪ ⎪ 2 1 3 1 3 4 2 3 ⎪ ⎥ ⎢ ⎪ ⎪ ⎪ 2 ⎪ −16(6r + r r − r r )x ⎥ ⎢ ⎪ ⎪ ⎪ 1 1 2 2 3 4 ⎩ ⎭ ⎥ ⎢ −36(18 + 3r2 + 2r1 r3 − 2r32 )x21 x6 ⎥ u. (V 3 V 5 + V 5 V 3 )u = ⎢  $ ⎥ ⎢ (−6 + r2 )(x61 + 2x23 − 6x21 x4 ) ⎥ ⎢ 1 2 V ⎥ ⎢ 9 ⎥ ⎢ +6(r1 − r3 )(6x6 − x31 x3 )  $ ⎥ ⎢ 4 ⎥ ⎢ 2(r − r )x − (−27 + 2r )x x 1 3 1 2 1 3 3 ⎥ ⎢ + 1 x1 V ⎦ 6 ⎣ −4(r1 + r3 )x4 + 16 {−(−3 + 2r2 )x31 − 4(3r1 − r3 )x3 }V 5 ⎡

The system M(r1 , r2 , r3 , s) sometimes has a non-trivial quotient Dmodule. By direct computation we find the following: (1) M(r1 , 0, r3 , s) has a quotient added by a differential equation V 2 u = r1 x21 u. (2) M(r1 , 3, r3 , s) has a quotient added by a differential equation  $ 1 1 1 V 3u = (r1 − r3 )x1 V 2 + (−3r1 + r3 )x3 + (−9 − 2r12 + 2r1 r3 )x31 u. 3 3 6 (3) M(r1 , 9, r3 , s) has a quotient added by a differential equation 1 1 V 5 u = − 54 (r1 − r3 )((r1 − r3 )x31 + 6x3 )V 2 + 16 (r1 − r3 )V 3 1 1 − 3 (5r1 − 3r3 )x1 x4 + 18 (45 + (r1 − r3 )(5r1 − r3 ))x21 x3 2 1 2 + 108 (r1 − r3 )(27 + 2r1 − 2r1 r3 )x51 u. 4.2. The case No. 31 Let M be the matrix with polynomial entries defined by ⎞ ⎛ 3x3 5x5 6x6 2x2 ⎞ ⎛ ⎞ ⎛ ⎜   ⎟  −2x32 −x42 ⎜ −2x22 x5 ⎟ x −7x 2 3 ⎟ ⎜3 2 ⎠ ⎠ ⎝ ⎝ 2 −9x3 2 +15x3 x5 2 ⎟ ⎜ +50x5 +27x3 x6 ⎟ ⎜ +90x6 +51x2 x6 ⎟ ⎜ ⎟ ⎜ 3   ⎟ ⎜ x 3x 3 2 3 −8x2 6x2 x3 x5 ⎟ ⎜ 17x x 2 3 2 2 ⎟ ⎜ +36x23 2x −100x −50x x 2 5 2 5 ⎟ ⎜ −70x 2 5 +16x x ⎟ ⎜ +720x6 6 2 +432x3 x6 ⎟ ⎜ ⎟ ⎜ 4x62 − 27x32 x23 ⎟ ⎜ 8x42 x5 3 5 2 2 2 54x2 x3 8x2 − 360x2 x3 +90x2 x3 x5 ⎟ ⎜ 2 −54x2 x3 x5 ⎟ ⎜ −324x3 2 +1980x x x −600x x 2 3 5 2 5 2 3 ⎟ ⎜ −300x3 x5 ⎟ ⎜ −420x22 x5 −3000x25 −324x32 x6 2x x ⎠ ⎝ +468x 3 6 2 2 2 +6480x3 x6

−600x2 x6

+1512x3 x6 +4320x26

−1200x2 x5 x6

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and let V j (j = 0, 3, 4, 7) be vector fields on (x2 , x3 , x5 , x6 )-space defined by t

(V 0 , 150V 3 , 720V 4 , 108000V 7 ) = M · t (∂x2 , ∂x3 , ∂x5 , ∂x6 ).

Then f0 = det M is regarded as the discriminant of the group No.31 by the correspondence explained in the previous section. A simple computation shows the following identities: [V 0 , V j ] [V 3 , V 4 ] [V 3 , V 7 ] [V 4 , V 7 ]

= jV j (j = 3, 4, 7), 1 2 3 1 = − 72 x2 V − 25 x3 V 4 + V 7 , 11 1 7 3 = − 1800 x2 x5 V + 1875 (−2x32 + 9x23 + 60x6 )V 4 − 50 x3 V 7 , 1 1 7 2 7 = 14400 (x42 + 6x2 x23 − 40x3 x5 − 40x2 x6 )V 3 − 180 x2 x5 V 4 + 360 x2 V , V 0 f0 = 30f0 ,

V 3 f0 = 0,

V 4 f0 = 0,

V 7 f0 = 0.

These actually show that the hypersurface f0 = 0 is a Saito free divisor in the (x2 , x3 , x5 , x6 )-space. In this case M(r, s) (r, s ∈ C) defined below is a system of uniformization equations with singularities along the hypersurface f0 = 0. V 0 u =su D C r−2 3 2 {2rx − 3(2r + 5)x − 30(r − 2)x } − 6 2 3 3750 u, V 3V 3u = 3 4 x3 V 3 + 12(r+1) + 50 125 x2 V ⎤ ⎡ r−2 {rx42 + 3(r + 5)x2 x23 − 45(r + 2)x3 x5 32400 ⎦ u, V 4 V 4 u = ⎣ −30(r − 4)x2 x6 } r+2 1 2 4 3 − 432 (x2 x3 + 10x5 )V − 90 x2 V ⎛ 7 ⎞ ⎤ ⎡ 4 2 4rx2 − 3(40 + 17r)x2 x3 4

3

⎜ −324(5 + r)x2 x33 + 120(−1 + r)x2 x32x52 ⎟ ⎥ ⎢ ⎜ +180(44 + 7r)x3 x5 − 300(−2 + 3r)x2 x5 ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ −60(−2 + 5r)x4 x6 ⎟ ⎥ ⎢ − 28r−2 5 6 ·3 ·5 ⎜ ⎜ +360(−16 + r)x2 2 x2 x6 ⎟ ⎥ ⎢ 3 ⎢ ⎝ ⎠ ⎥ ⎥ ⎢ +3600(−4 + r)x3 x5 x6 ⎥ ⎢ 2 7 7 +7200(2 + r)x x ⎥ u, ⎢ 2 6 V V u =⎢ ⎛ 4 ⎞ ⎥ 3 3 x2 x3 + 27x2 x3 − 20x2 x5 ⎥ ⎢ ⎥ ⎢ r+5 ⎝ ⎥ ⎢ − 26 ·34 ·54 −30x23 x5 − 60x2 x3 x6 ⎠ V 3 ⎥ ⎢ ⎥ ⎢ +600x5 x6 ⎥ ⎢ 4 2 4⎦ ⎣ + 4r+5 x (x + 27x2 x3 − 120x3 x5 − 60x2 x6 )V 2 ·33 ·55 2 2 + 3+2r x x V7 1800 2 5  2−r

x2 {(20 + 3r)x2 x3 + 10(−4 + 3r)x5 } u, (V 3 V 4 + V 4 V 3 )u = 9000 − 7+2r x2 V 3 + 3+r x V 4 + (3 + 2r)V 7 360 2 25 3

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⎡

⎤ ⎞ 4rx52 + 45(5 + r)x22 x23 ⎢ 4r−2 ⎥ ⎠ ⎢ 2 ·33 ·55 ⎝ −90(14 + 3r)x2 x3 x5 ⎥ ⎢ ⎥ 2 2 3 7 7 3 +1500x5 − 60(−2 + 3r)x2 x6 (V V + V V )u = ⎢ ⎥ u, ⎢ (13+2r) ⎥ (3+r) 3 3 2 4 ⎣ − 1800 x2 x5 V − 3750 (2x2 − 9x3 − 60x6 )V ⎦ − (3+2r) x3 V 7 50 ⎡ ⎛ ⎞⎤ (20 + r)x42 x3 − 36(5 + r)x2 x33 ⎥ 3 2 ⎢ 6r−2 ⎢ 2 ·34 ·54 ⎝ +10(−4 + 5r)x2 x5 − 60(2 + r)x3 x5 ⎠ ⎥ ⎥ ⎢ +2160x2x3 x6 − 1200(2 + r)x5 x6 (V 4 V 7 + V 7 V 4 )u = ⎢ ⎥ u. ⎥ ⎢ (7+2r) 4 ⎦ ⎣ + 43200 (x2 + 6x2 x23 − 40x3 x5 − 40x2 x6 )V 3 (3+2r) 2 7 4 − (7+r) x x V + x V 2 5 2 900 360 ⎛

The characteristic exponents of the system M(r, s) along the hypersurface f0 = 0 are 1 1 1 1 (s − r + 2), (s − r + 2), (s − r + 2), (s + 3r + 8). 30 30 30 30 The system M(r, s) possibly has a non-trivial quotient D-module. By direct computation we find the following: (1) M(2, s) has a quotient defined by V 0 u = su, V 3 u = V 4 u = V 7 u = 0. 3 x3 )u = 0. (2) M(−1, s) has a quotient added by (V 3 − 50 1 2 4 (3) M(−2, s) has a quotient added by (V + 90 x2 )u = 0. 7 (4) M(−5, s) has a quotient added by (V 7 + 1800 x2 x5 )u = 0. s/30 In the case (1), the quotient has a solution u = f0 . On the other hand, in the cases (2), (3), (4), there are three fundamental solutions outside f0 (x2 , x3 , x5 , x6 ) = 0. References 1. D. Bessis and J. Michel, Explicit presentations for exceptional braid groups. Experimental Math., 13 (2004), 257-266. 2. Y. Haraoka and M. Kato, Generating systems for finite irreducible complex reflection groups. Funkcialaj Ekvacioj 53 (2010), 435-488. 3. M. Kato and J. Sekiguchi, Systems of uniformization equations with respect to the discriminant sets of complex reflection groups of rank three. Preprint. 4. P. Orlik and H. Terao. Arrangements of Hyperplanes. Grundlehren der mathematisches Wissenschaften 300, Berlin: Springer-Verlage, 1992. 5. K. Saito, On the uniformization of complements of discriminant loci. RIMS Kokyuroku 287 (1977), 117-137. 6. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields. J. Faculty of Sciences, Univ. Tokyo 27 (1980), 265-291. 7. K. Saito, Uniformization of orbifold of a finite reflection group. In “Frobenius Manifold, Quantum Cohomology and Singularities.” A Publication of the Max-Planck-Institute, Mathematics, Bonn, 265-320.

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8. J. Sekiguchi, Three dimensional Saito free divisors and deformations of singular curves. J. Siberian Federal Univ., Mathematics and Physics, 1 (2008), 33-41. 9. J. Sekiguchi, A classification of weighted homogeneous Saito free divisors. J. Math. Soc. Japan, 61 (2009), 1071-1095. 10. J. Sekiguchi, Systems of uniformization equations along Saito free divisors and related topics, in “The Third Japanese-Australian Workshop on Real and Complex Singularities”, Proceedings of the Centre for Mathematics and its Applications, 43 (2010), 83-126. 11. J. Sekiguchi, Systems of uniformization equations and dihedral groups. Kumamoto J. Math., 23 (2010), 7-26. 12. J. Sekiguchi, Systems of uniformization equations and hyperelliptic integrals. J. Math. Sci., 175 (2011), 57-79. 13. J. Sekiguchi, The dicriminant of the reflection group of type D4 and holonomic systems with singularities along its zero locus. To appear in Tohoku Math. J. 14. G. C. Shephard and A. J. Todd, Finite reflection groups. Canad. J. Math., 6 (1954), 274-304. 15. T. Yano and J. Sekiguchi, The microlocal structure of weighted homogeneous polynomials associated with Coxeter systems. I, II, Tokyo J. Math. 2 (1979), 193-219, Tokyo J. Math. 4 (1981), 1-34.

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Holonomic systems of differential equations of rank two with singularities along Saito free divisors of simple type Jiro Sekiguchi∗ Department of Mathematics, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan [email protected] We will show many examples of holonomic systems of three variables of rank two with singularities along Saito free divisors of simple type. Solutions of some of them are expressed in terms of elementary functions or hypergeometric functions. But some of the holonomic systems are difficult to solve. In the last section, we discuss the relationship between the holonomic systems obtained in this paper and algebraic solutions of Painlev´e sixth equation. Keywords: Saito free divisor, holonomic system AMS classification numbers: 34M56, 20F65

1. Introduction The notion of Saito free divisors is introduced by K. Saito around 1970’s. One of the reasons why he introduced it is to describe characteristic properties of the parameter space of versal deformations of isolated singularities. He also formulated the notion of systems of uniformization equations with logarithmic poles along Saito free divisors. These were published in a short note [4] (see also [5], [6]). After his pioneering work, a lot of studies related with Saito free divisors were done in various branches of mathematics; hyperplane arrangements, topologies of complements of Saito free divisors, flat structure or Frobenius manifold structure, etc. As to systems of uniformization equations, the existence of such a system depends on the choice of a Saito free divisor. It is A. G. Aleksandrov who pointed out this basic result (cf. [1]) and such systems were studied in [9], [3]. In this paper, we will study holonomic systems of differential equations on C3 of rank two which have singularities along Saito free divisors. These ∗ Partially

supported by Grand-in-Aid for Scientific Research (No. 2354077), Japan Society of the Promotion of Science.

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systems are rank two analogue of systems of uniformization equations but it is easier to treat compared with uniformization equations. We start the explanation of this paper with defining Saito free divisors. Let Cn be an n-dimensional affine space with coordinate system x = (x1 , x2 , . . . , xn ) and let R = C[x1 , x2 , . . . , xn ] be its coordinate ring. Let F (x) = F (x1 , x2 , . . . , xn ) be a reduced polynomial. Then the hypersurface F (x) = 0 is Saito free if the following condition (C.1) is satisfied: (C.1) There are vector fields Vi =

n 

aij (x)∂xj (i = 1, 2, . . . , n)

j=1

such that Vi F/F ∈ R (i = 1, 2, . . . , n) and det M = cF for a non-zero constant c, where M is an n × n matrix whose (i, j)-entry is aij (x). The matrix M is called a generating matrix. We now assume that F (x) is weighted homogeneous, namely, there is a n vector field E = j=1 dj xj ∂xj such that EF = dF , where d1 , d2 , . . . , dn , d are positive integers. Moreover we assume that 0 < d1 < d2 < . . . < dn . Under these assumptions, we may take Vi (i = 1, 2, . . . , n) so that V1 = E and [E, Vi ] = ki Vi with constants ki (i = 1, 2, . . . , n) such that k1 = 0 ≤ k2 ≤ · · · ≤ kn . We next introduce the notion of systems of uniformization equations. Let F (x) be Saito free and let D : F (x) = 0 be the hypersurface defined by F (x). Define 

V1 u = su, &n ' k Vi Vj u = k=2 pij (x)Vk + qij (x) u

(i, j = 2, 3, . . . , n),

(1.1)

where pkij (x), qij (x) ∈ R, s ∈ C. If (1.1) is integrable, it is called a system of uniformization equations. The system (1.1) has singularities along D. We restrict our attention to the case n = 3. Let F (x, y, z) be one of the seventeen polynomials given in [8], Theorem 1 and let M be the matrix given in [8], Appendix. In the following we write x1 , x2 , x3 in place of x, y, z. We define vector fields Vj (j = 1, 2, 3) by t (V1 , V2 , V3 ) = M t(∂x1 , ∂x2 , ∂x3 ). Let u(x1 , x2 , x3 ) be an unknown function and define two kinds of systems of differential equations. The first one is ⎧ ⎨ V1 u = su, V u = p1 (x)u, ⎩ 2 V3 V3 u = (p2 (x) + p3 (x)V3 )u,

(1.2)

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and the second one is ⎧ ⎨ V1 u = su, V V u = (p1 (x) + p2 (x)V2 )u, ⎩ 2 2 V3 u = (p3 (x) + p4 (x)V2 )u.

(1.3)

The former is called of Type I and the latter is called of Type II. In both cases, s is a complex number and p1 (x), p2 (x), . . . are polynomials of x = (x1 , x2 , x3 ). The purpose of this paper is to show the result on a classification of holonomic systems of Type I and Type II for Saito free divisors introduced in [8]. Since the Saito free divisors in [8] are regarded as 1-parameter deformations of simple curve singularities of exceptional type, they are called Saito free divisors of simple type in this paper. A basic question is whether the holonomic systems obtained in this paper are reduced to elementary ones or not. We also discuss on this question in the main text. Solutions of some of holonomic systems thus obtained are expressed by the products of complex powers of polynomials and their integrals. This kind of phenomenon reflects the solvability of groups realized as two dimensional representations of fundamental groups of complements of Saito free divisors. The fundamental groups in these cases are studied in [7]. Let D : F = 0 be a Saito free divisor and let M be one of holonomic systems treated in this paper. We note that there are two multi-valued holomorphic fundamental solutions u1 , u2 of M on C3 − D. Then the second basic question is to determine the image of the so called Schwarz map defined by ϕ : C3 − D  x = (x1 , x2 , x3 ) → (u1 (x) : u2 (x)) ∈ P1 . As an attempt to answer to this question we study the restriction of u1 , u2 to x1 = 0 and x2 = 0. In some cases the restrictions are expressed by hypergeometric functions. We explain the contents of this paper briefly. In section 2, we introduce the notion of holonomic systems of Type I and those of Type II for Saito free divisors in C3 and their integrability conditions. In section 3, we will construct holonomic systems of Type I/Type II for Saito free divisors defined as the zero sets of seventeen polynomials in [8]. In section 4, we discuss the meaning of the systems constructed in section 3 and in particular show a relationship between the holonomic systems treated in this paper and algebraic solutions of Painlev´e VI equation taking as an example the case of the holonomic system in subsection 3.9. In closing this introduction, we note that our computations in this paper are performed using the computer algebraic software Mathematica. Acknowledgement: The author thanks the referee for reading the con-

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tents carefully. In particular, being pointed out by him, the author recognized that there are many mistakes of the computations on the systems of Types I and II in the first draft and could correct them. 2. Holonomic systems of rank two Let R = C[x1 , x2 , x3 ] be a polynomial ring of three variables and let M = (mij (x1 , x2 , x3 ))i,j=1,2,3 be a 3 × 3 matrix whose matrix entries are polynomials. Let V1 , V2 , V3 be vector fields defined by M as in section 1, namely, t

(V1 , V2 , V3 ) = M t (∂x1 , ∂x2 , ∂x3 ).

We assume that m1j = dj xj (j = 1, 2, 3), where d1 , d2 , d3 are integers such that 0 < d1 < d2 < d3 . A polynomial g(x) ∈ R is weighted homogeneous of degree k if V1 g = kg. In this case we write deg g = k. We assume that each mij is weighted homogeneous. A vector field V with polynomial coefficients is weighted homogeneous of degree d if [V1 , V ] = dV . In this case, we say d the degree of V and write deg V = d. In the sequel, we assume that V1 , V2 , V3 are weighted homogeneous and that deg V1 = 0 < deg V2 < deg V3 . As is explained in the introduction, the polynomial F (x1 , x2 , x3 ) = det M plays a central role in the subsequent argument. We assume that F (x) is reduced and that V1 , V2 , V3 are logarithmic along the hypersurface D : F (x) = 0, that is, Vj F/F ∈ R (j = 1, 2, 3). As an easy consequence, there are polynomials cj (x) ∈ R such that Vj F = cj F (j = 1, 2, 3). In particular c1 is a positive integer. Under the above assumption, it follows from [5] that LM = RV1 +RV2 + RV3 is a Lie algebra over R. In particular, [V1 , Vi ] = (deg Vi )Vi (i = 2, 3), [V2 , V3 ] = w1 (x)V1 + w2 (x)V2 + w3 (x)V3 ,

(2.4)

where wj are weighted homogeneous polynomials. We now introduce two kinds of systems of differential equations. Let u = u(x1 , x2 , x3 ) be an unknown function such that V1 u = su for a complex number s. To define the first one, we consider the R-module N = Ru + RV3 u. We assume that u and V3 u are linearly independent over R and that LM N ⊂ N . Then u is a solution of the system of differential equations ⎧ ⎨ V1 u = su, (2.5) V2 u = p1 (x)u, ⎩ V3 V3 u = (p2 (x) + p3 (x)V3 )u,

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where pj (x) ∈ R. The system (2.5) is called of Type I.   u We take a solution u of (2.5) and put uI = . We rewrite (2.5) V3 u to a system for uI . Since [V1 , V3 ] = (deg V3 )V3 and since V1 u = su, we have   s 0 V1 uI = uI . 0 s + deg V3 Our next job is to compute V2 uI . Since V3 V2 u = V3 (p1 u) = p1 V3 u + [V3 , p1 ]u, it follows that V2 V3 u = [V2 , V3 ]u + V3 V2 u = (w1 V1 + w2 V2 + w3 V3 )u + (p1 V3 + [V3 , p1 ])u. Then

 V2 uI =

V2 u V2 V3 u



 =

0 p1 sw1 + p1 w2 + [V3 , p1 ] w3 + p1

 uI .

Our last job is to compute V3 uI . In this case,       V3 u 0 1 V3 u uI . V3 uI = = = V3 V3 u (p2 + p3 V3 )u p2 p 3 As a consequence, we have MI : Vi uI = Ai uI

(i = 1, 2, 3),

(2.6)

where A1 , A2 , A3 are 2 × 2 matrices of the forms   s 0 , A1 = 0 s + deg V3   0 p1 A2 = , sw + p w + [V3 , p1 ] w3 + p1  1 1 2 0 1 . A3 = p2 p3 To define the second one, we consider the R-module N = Ru+RV2 u. We assume that u and V2 u are linearly independent over R and that LM N ⊂ N . Then u is a solution of the system of differential equations ⎧ ⎨ V1 u = su, (2.7) V2 V2 u = (p1 (x) + p2 (x)V2 )u, ⎩ V3 u = (p3 (x) + p4 (x)V2 )u, where pj (x) ∈ R. The system (2.7) is called of Type II. We take a solution   u . We can compute Vi uII (i = 1, 2, 3) by u of (2.7) and put uII = V2 u

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an argument similar to the argument above. Then from (2.7), we find the system for uII to be MII : Vi uII = Ai uII where

(i = 1, 2, 3)

(2.8)

 s 0 , A1 = 0 s + deg V2   0 1 , A2 = p1 p2   p4 p3 A3 = . − sw1 + [V2 , p3 ] + p1 p4 − p3 w3 −w2 − p4 w3 + p3 + p2 p4 + [V2 , p4 ] 

Both of MI and MII are written in the form M : Vi u = Ai u

(i = 1, 2, 3)

(2.9)

where A1 , A2 , A3 are 2 × 2 matrices. With the help of the relations (2.4), we obtain the following compatibility conditions on the matrices A1 , A2 , A3 : [A1 , A2 ] + [V1 , A2 ] − (deg V2 )A2 = O, = O, [A1 , A3 ] + [V1 , A3 ] − (deg V3 )A3 [A2 , A3 ] − [V2 , A3 ] + [V3 , A2 ] + w1 (x)A1 + w2 (x)A2 + w3 (x)A3 = O, (2.10) where we note that A1 is a constant matrix. The first and the second relations of (2.10) imply that each matrix entry of A2 , A3 are weighted homogeneous. On the other hand, the third one implies non-trivial equations among the matrix entries of A2 , A3 . The relations (2.10) are rewritten in a familiar form which we are going to explain. Since t (V1 , V2 , V3 ) = M t (∂x1 , ∂x2 , ∂x3 ), we introduce 2 × 2 matrices B1 , B2 , B3 by t

(B1 , B2 , B3 ) = M −1t (A1 , A2 , A3 ).

(2.11)

Then F Bj has polynomial entries and the system (2.9) is equivalent to ∂xj u = Bj u (j = 1, 2, 3).

(2.12)

The integrability condition for (2.12) is [Bi , Bj ] +

∂Bi ∂Bj − =O ∂xj ∂xi

(∀i, j).

(2.13)

It is clear that (2.13) is also an integrability condition for M. Remark 2.1. Let u be an unknown function such that V1 u = su for a complex number s. Replacing the R-module N with different one, many

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examples of systems of differential equations are produced. In this remark, we introduce two such R-modules which are interesting to study. As the first one, we take the R-module N = Ru. If LM N ⊂ N , then u satisfies the system of equations ⎧ ⎨ V1 u = su, (2.14) V u = p1 (x)u, ⎩ 2 V3 u = p2 (x)u, where pj (x) ∈ R (j = 1, 2). On the other hand, we have V2 V3 u = V2 (p2 u) = p2 V2 u + [V2 , p2 ]u = p2 p1 u + [V2 , p2 ]u, V3 V2 u = V3 (p1 u) = p1 V3 u + [V3 , p1 ]u = p1 p2 u + [V3 , p1 ]u. Then [V2 , V3 ]u = (w2 V2 + w3 V3 )u = (w2 p1 + w3 p2 )u and as a consequence, [V2 , p2 ] − [V3 , p1 ] = w2 p1 + w3 p2 is the integrability condition of (2.14). The most interesting example of the system (2.14) is the system which has the solution u = F s/ deg F . There are a lot of studies on D-modules which govern complex powers of polynomials. As the second one, we take N = Ru + RV2 u + RV3 u, keeping the assumption V1 u = su. If LM N ⊂ N , it is easy to see that u satisfies a system of uniformization equations (cf. (1.1)). In this manner, it is easy to define various kinds of systems of differential equations related with Saito free divisors. But it is not so easy to find Saito free divisors such that there exist non-trivial systems of differential equations related with R-modules defined in this section. Our main purpose of this paper is to construct systems of Type I and/or Type II with respect to Saito free divisors of simple type. Moreover we study solutions of the systems. Some of solutions of the systems are expressed in terms of elementary functions or hypergeometric functions. But some seem not to be done by known functions. We now explain how to obtain the systems of Type I/II by taking the case of the polynomial FH,1 of §3.8 below as an example. We discuss here only the case of systems of Type I. Our effort is focused on the systems MI of (2.6), because the system (2.5) is recovered from MI . In this case, variables x1 , x2 , x3 have weights 1, 3, 5, respectively.

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To determine MI , we first introduce polynomials ϕj (j = 1, 2, . . . , 5) of x1 , x2 , x3 such that   s 0 , A1 = 0 s+4   ϕ1 0 A2 = , ϕ ϕ  2 3 0 1 , A3 = ϕ4 ϕ5 where s ∈ C. By the weight condition, it is easy to see that ϕ1 , ϕ2 , ϕ3 , ϕ4 , ϕ5 are weighted homogeneous of degree 2,6,2,8,4, respectively. Then we may take ϕ1 ϕ2 ϕ3 ϕ4 ϕ5

= = = = =

c1 x21 , c2 x1 x3 + c3 x22 + c4 x31 x2 + c5 x61 , c6 x21 , c7 x2 x3 + c8 x31 x3 + c9 x21 x22 + c10 x51 x2 + c11 x81 , c12 x1 x2 + c13 x41 ,

where c1 , c2 , . . . , c13 are constants. Next, we define B1 , B2 , B3 from A1 , A2 , A3 by the equation (2.11). Then the integrability condition (2.13) is equivalent to a system of algebraic equations among s, c1 , c2 , . . . , c13 . Solving this system, we obtain the polynomials ϕj (j = 1, 2, 3, 4, 5) for the system MI of (2.6) to be integrable. 3. Construction of holonomic systems of rank two The Saito free divisors treated in this paper are seventeen divisors defined as hypersurfaces constructed by the author [8] where he classified weighted homogeneous polynomials of three variables under certain conditions. They are named FA,1 , FA,2 , FB,j (j = 1, 2, . . . , 7), FH,j (j = 1, 2, . . . , 8) and are closely related with irreducible real reflection groups of rank three. Their types are A3 , B3 , H3 . In fact FX,1 is regarded as the discriminant of the reflection group of type X3 (X = A, B, H). Moreover they are also related with simple curve singularities of types E6 , E7 , E8 . In particular each of the hypersurfaces in [8] is regarded as a 1-parameter family of simple curve singularity whose type is one of E6 , E7 , E8 . For this reason, the divisors treated in this paper are called Saito free divisors of simple type. The purpose of this section is to construct systems of Type I/Type II with singularities along Saito free divisors which are defined by FX,j (X = A, B, H). Throughout this section, for a given polynomial FX,j defining a

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Saito free divisor, M is a generating matrix of FX,j and V1 , V2 , V3 are vector fields defined by M . Note that there are many choices of the matrix M and you find in [8] a standard one of the Saito matrices for each polynomial FX,j . Since it is possible to show that if FX,j is equal to one of FB,j (j = 5, 7), FH,j (j = 4, 5, 6, 7, 8), there is no holonomic system neither of Type I nor of Type II, we only treat the polynomials FA,j (j = 1, 2), FB,j (j = 1, 2, 3, 4, 6), FH,j (j = 1, 2, 3). 3.1. The case FA,1 The polynomial is given by FA,1 = −4x31 x22 − 27x42 + 16x41 x3 + 144x1 x22 x3 − 128x21 x23 + 256x33 and its generating matrix is ⎞ ⎛ 3x2 4x3 2x1 ⎠. M = ⎝ 3x2 −x21 + 4x3 − 21 x1 x2 1 1 2 4x3 − 2 x1 x2 4 (8x1 x3 − 3x2 ) Note that FA,1 is regarded as the discriminant of the reflection group of type A3 . There are a holonomic system of Type I and a holonomic system of Type II. System of Type I ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ V2 u = 0, 1 ⎪ {−(r1 + r2 + 1)(r1 − r2 + 1)x21 + 12(2r1 + r22 − 1)x3 }u V V u = 36 ⎪ ⎩ 3 3 1 + 3 (1 + r1 )x1 V3 u. (3.15) In this case, solutions of the system (3.15) are expressed in terms of hypergeometric functions. We explain this result obtained by K. Saito (cf. [4]) briefly. Put 1 1 8 2 P = 4x3 + x21 , Q = (x22 − x1 x3 + x31 ). 3 2 3 27 Then det M = −P 3 + 27Q2 . Let V1 , V2 , V3 be vector fields defined by M . Putting V3 = V3 − 16 x1 V1 , we have 1 V3 P = −6Q, V3 Q = − P 2 . 3

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The system (3.15) turns out to be ⎧ ⎪ ⎨ V1 u = r1 u, V2 u = 0, ⎪ ⎩ V  V  u = r22 −1 P u 3 3 12 and

 u = (P 3 − 27Q2 )r1 /12 F

1 + r2 1 − r2 2 P3 , , ; 3 12 12 3 P − 27Q2



is one of its solutions, where F (a, b, c; x) is a hypergeometric function. System of Type II ⎧ ⎨ V1 u = r1 u, V u = 16 (r1 − 1)x1 u, ⎩ 3 V2 V2 u = − 12 x1 u.

(3.16)

In spite that it is not clear whether there is a solution of the system (3.16) which is expressed in terms of elementary functions or special functions, it can be shown that the restriction of any solution of the system (3.16) to the hypersurface x1 = 0 is expressed by hypergeometric functions. In fact, there is a solution of (3.16) such that   1 1 1 7 3 27x42 4 3 12 (r1 −1) 4 F − , , ; u|x1 =0 = x3 (−27x2 + 256x3 ) . 12 12 4 256x33 On the other hand, we obtain differential equations   ⎧ u ⎪ ⎪ ∂ ⎪ x1 V u = ⎪ ⎪ 2 ⎪ ⎨   u ⎪ ⎪ ∂ = ⎪ x 3 ⎪ V2 u ⎪ ⎪ ⎩

  (2r1 + 1)x1 u 0 V2 u 0 (2r1 + 1)x1 ⎛ ⎞ (r1 − 1)x21   0 ⎜ −12r1 x3 ⎟ u 1 ⎜ ⎟ 2 2 12x3 (x1 −4x3 ) ⎝ (r1 + 2)x1 ⎠ V2 u 0 −12(r1 + 1)x3 

1 6(x21 −4x3 )

(3.17)

by restricting the system (3.16) to x2 = 0. As a consequence, we obtain a solution 1

u|x2 =0 =c1 x312 1

(V2 u)|x2 =0 =c2 x312

(r1 −1)

(4x3 − x21 ) 12 (2r1 +1) ,

(r1 +2)

(4x3 − x21 ) 12 (2r1 +1)

1

1

for some constants c1 , c2 of the differential equations (3.17).

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3.2. The case FA,2 The polynomial is FA,2 = 2x61 + 18x31 x22 + 27x42 − 3x41 x3 − 18x1 x22 x3 + x33 and its generating matrix is ⎞ ⎛ 3x2 4x3 2x1 ⎠. M = ⎝ 3x2 21 (x3 − x21 ) 6x1 x2 3 2 4x3 −2x1 x2 16x1 + 24x2 − 8x1 x3 There is a holonomic system of Type I but no holonomic system of Type II. System of Type I ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ V2 u = 0, ⎪ V V u = 43 {−(3r12 + 6r1 − r22 + 1)x21 − (6r1 + r22 − 1)x3 }u ⎪ ⎩ 3 3 −4(r1 + 1)x1 V3 u.

(3.18)

In this case, it is easy to solve the system (3.18) by an argument parallel to that for the case FA,1 . We explain it briefly. First put P = x21 − x3 ,

Q=

1 (3x22 − x1 x3 + x31 ). 3

Then det M coincides with P 3 −27Q2 up to a constant factor. Let V1 , V2 , V3 be vector fields defined by M . Putting V3 = V3 + 2x1 V1 , we have V3 P = −24Q,

4 V3 Q = − P 2 . 3

The system (3.18) turns out to be ⎧ ⎨ V1 u = r1 u, V u = 0, ⎩ 2  V3 V3 u = 43 (r22 − 1)P u, and

 u = (P 3 − 27Q2 )r1 /12 F

is one of its solutions.

P3 1 + r2 1 − r2 2 , , ; 3 12 12 3 P − 27Q2



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3.3. The case FB,1 The polynomial is FB,1 = x3 (−x21 x22 + 4x32 + 4x31 x3 − 18x1 x2 x3 + 27x23 ) and its generating matrix is ⎛

⎞ x1 2x2 3x3 M = ⎝ 2x2 x1 x2 + 3x3 2x1 x3 ⎠ . 3x3 2x1 x3 x2 x3 Note that FB,1 is regarded as the discriminant of the reflection group of type B3 . There are three holonomic systems of rank two. One is of Type I and the remaining two are of Type II. System of Type I ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ V2 u = 16 (2r1 + 6r2 − 1)x1 u, ⎪ V3 V3 u = 14 {−4(r2 − r3 )(r2 + r3 )x22 + (8r2 − 12r32 − 1)x1 x3 }u ⎪ ⎩ +2r2 x2 V3 u.

(3.19)

In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.19) are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.19) such that 1

r −r3 − 53

2 u|x1 =0 = (4x32 + 27x23 ) 6 (1+r1 −3r2 +3r3 ) x3

x22   5 5 5 4x32 ×F . , r3 + , ; − 6 6 3 27x23

We next treat the case x2 = 0. There is a solution u of (3.19) such that 1

1

(r +3r −3r −1)

u|x2 =0 = (4x31 + 27x3 ) 6 (1+r1 −3r2 +3r3 ) x36 1 2 3   6r3 + 1 6r3 + 7 2 4x31 ×F . , , ;− 12 12 3 27x3

Systems of Type II There are two holonomic systems of Type II.

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Type (II.1) ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ V2 V2 u = 14 {(r2 − r3 )(r2 + r3 )x21 + (−1 + 4r3 − 3r22 )x2 }u ⎪ +r3 x1 V2 u, ⎪ ⎩ V3 u = − 16 (1 + 2r1 − 3r3 )x2 u.

(3.20)

In this case, solutions of the system (3.20) are expressed in terms of hypergeometric functions. We will explain the result briefly. We first construct 2 × 2 matrices Aj (j = 1, 2, 3) from (3.20) as we explained the way similar to do (2.8) from (2.7). Then we obtain a system (3.21) Vj u = Aj u (j = 1, 2, 3),  u . We next define 2 × 2 matrices Bj (j = 1, 2, 3) by the where u = V2 u equation similar to (2.11). Then we obtain a system 

∂xj u = Bj u

(j = 1, 2, 3).

(3.22)

We thirdly change the variables: 1 1 (3y2 + y12 ), x3 = (27y3 + 9y1 y2 + y13 ). 3 27 Then (3.22) turns out to be x1 = y1 , x2 =

∂yj u = Cj u

(j = 1, 2, 3),

(3.23)

where Cj (j = 1, 2, 3) are 2 × 2 matrices defined by using Bj (j = 1, 2, 3). In particular C2 = B2 + 13 x1 B3 , C3 = B3 . Putting − 16 (−2r1 +3r3 −1)

v = x3

1

(27y32 + 4y23 )− 12 (2+4r1 −3r2 −3r3 ) u,

we find that v is a solution of   6(r2 − 2)y22 (3r2 − 1)(3r2 − 7)y2 2 ∂y + v=0 ∂y2 − 3 4y2 + 27y32 2 4(4y23 + 27y32 ) from ∂y2 u = C2 u and is also a solution of   27(r2 − 2)y3 3(3r2 − 1)(3r2 − 5) 2 ∂y3 − v=0 ∂y + 4y23 + 27y32 3 4(4y23 + 27y32 ) from ∂y3 u = C3 u. As a consequence, we obtain a solution   1 1 − 3r2 5 − 3r2 1 27y32 4 (3r2 −1) , , ;− 3 v = y2 F 12 12 2 4y2 of the two differential equations. Then

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u = x36

(−2r1 +3r3 −1)

1

f012

(2+4r1 −3r2 −3r3 )



×F

1

(3x2 − x21 ) 4 (3r2 −1)

1 − 3r2 5 − 3r2 1 (27x3 − 9x1 x2 + 2x31 )2 , , ; 12 12 2 4(x21 − 3x2 )3



is a solution of the system (3.20), where f0 = (−x21 x22 + 4x32 + 4x31 x3 − 18x1 x2 x3 + 27x23 ). Type (II.2) ⎧ ⎨ V1 u = r1 u, V V u = − 41 r2 {(2 + r2 )x21 − 4x2 }u + (1 + r2 )x1 V2 u, ⎩ 2 2 V3 u = 16 {−3r2 r3 x21 + (2 − 2r1 + 3r2 − 6r3 )x2 }u + r3 x1 V2 u.

(3.24)

In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.24) are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.24) such that 1

1

u|x1 =0 = (4x32 + 27x23 ) 12 (2+4r1 −3r2 −6r3 ) x36

(−2−2r1 +3r2 +6r3 )



×F

1 2 5 4x3 , −r3 + , ; − 22 6 3 3 27x3

 .

We next treat the case x2 = 0. There is a solution u of (3.24) such that 1

1

(−2+3r +6r )

2 3 u|x2 =0 = (4x31 + 27x3 ) 12 (2+4r1 −3r2 −6r3 ) x312   4x31 −3r3 + 1 −3r3 + 4 1 . , , ;− ×F 6 6 3 27x3

3.4. The case FB,2 The polynomial is FB,2 = x3 (2x32 − 4x31 x3 − 18x1 x2 x3 − 27x23 ) and its generating matrix is ⎛ ⎞ x1 2x2 3x3 M = ⎝ 2x2 − 32 (2x1 x2 − 9x3 ) −4x1 x3 ⎠ . 3x3 − 32 (x22 + 3x1 x3 ) −2x2 x3 There are four holonomic systems of rank two. One is of Type I and the remaining three are of Type II. In this subsection, we always put f0 = 2x32 − 4x31 x3 − 18x1 x2 x3 − 27x23 .

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System of Type I ⎧ ⎨ V1 u = r1 u, V2 u = r2 x1 u, ⎩ r2 V3 V3 u = − 12 {(−4 + 3r2 )x22 + 12x1 x3 }u + 13 (−4 + 3r2 )x2 V3 u.

(3.25)

In this case, fundamental solutions are expressed in terms of elementary functions and their integrals. In fact, we find that − 16 (2r1 +3r2 )

u = x3

1

f012

(4r1 +3r2 )

(c1 z + c2 )

(3.26)

is a solution of the system (3.25), where c1 , c2 are constants and z is a function such that (∂x1 z, ∂x2 z, ∂x3 z) 2/3

=

−x3

2/3

(2x1 x2 +9x3 ) (2x21 +3x2 )x3 , f0 f0

,

(−2x21 x2 −6x22 +9x1 x3 ) 1/3

3x3

f0



(3.27)

.

We are going to explain the outline of the argument how to find the solution (3.26) of the system (3.25). Putting   r1 0 , A1 = 0 2 + r1   0 r2 x1 , A2 = 1 r (2x1 x2 + 9x3 ) 13 (−8 + 3r2 )x1 3 2  0 1 A3 = , 1 r2 (−4x22 + 3r2 x22 + 12x1 x3 ) 13 (−4 + 3r2 )x2 − 12 we see that (3.25) turns out to be Vi u = Ai u

(i = 1, 2, 3),

(3.28)

where u = t (u, V3 u). Using the formula (2.11), we define 2 × 2 matrices B1 , B2 , B3 from A1 , A2 , A3 . Then (3.28) turns out to be ∂xi u = Bi u

(i = 1, 2, 3).

To solve (3.29), we introduce v by v=

− 1 (4r +3r ) 1 (2r +3r ) f0 12 1 2 x36 1 2



1 0 −2/3 −2/3 r2 − 2 x2 x3 x3

The system of differential equations for v is then   0 ϕi ∂xi v = v (i = 1, 2, 3), 0 0

(3.29)  u.

(3.30)

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where (ϕ1 , ϕ2 , ϕ3 )   2/3 2/3 −x3 (2x1 x2 + 9x3 ) (2x21 + 3x2 )x3 (−2x21 x2 − 6x22 + 9x1 x3 ) = . , , 1/3 f0 f0 3x f0 3

If v = (v1 , v2 ), then (3.30) is equivalent to the system  ∂xi v1 = ϕi v2 (i = 1, 2, 3), ∂xi v2 = 0 (i = 1, 2, 3). t

(3.31)

At this moment, we note that ∂xi ϕj = ∂xj ϕi

(i, j = 1, 2, 3). (4r +3r )/12 −(2r +3r )/6

On the one hand, we find that u = f0 1 2 x3 1 2 v1 . On the other hand, from the second equation of (3.31), it follows that v2 is a constant. In this manner we have the required result. Systems of Type II There are three holonomic systems of Type II. Type (II.1) ⎧ ⎨ V1 u = r1 u, 1 V2 V2 u = − 12 r2 {(−8 + 3r2 )x21 − 12x2 }u + 13 (−4 + 3r2 )x1 V2 u, ⎩ 1 V3 u = 4 r2 x2 u.

(3.32)

In this case, we find that 1 − 12 (4r1 +3r2 )

u = x3

1

f024

(8r1 +3r2 )

(c1 z + c2 )

is a solution of the system (3.32), where c1 , c2 are constants and z is a function such that (∂x1 z, ∂x2 z, ∂x3 z)  1/3 1/3 (3.33) −x3 (−x22 +3x1 x3 ) −x3 (2x1 x2 +9x3 ) (x1 x22 +3x21 x3 +9x2 x3 ) . = , , 2/3 f0 2f0 3x3

f0

The proof is similar to that for the case of Type I. Type (II.2) ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ 1 V2 V2 u = 12 {−(64 − 32r2 + 3r22 )x21 + 12r2 x2 }u ⎪ + 31 (−16 + 3r2 )x1 V2 u, ⎪ ⎩ 1 V3 u = 36 {2(8 − 3r2 )x21 + 3(−16 + 3r2 )x2 }u + 13 x1 V2 u.

(3.34)

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Solutions of (3.34) are expressed in terms of elementary functions. In particular, 1

u1 = (2x21 + 3x2 )x312

(16−4r1 −3r2 )

1

f024

1 12 (12−4r1 −3r2 )

u2 = (2x1 x2 + 9x3 )x3

(8r1 +3r2 −24)

,

1 24 (8r1 +3r2 −24)

f0

are fundamental solutions of (3.34). Type (II.3) ⎧ ⎨ V1 u = r1 u, 1 V V u = − 12 r2 {(−16 + 3r2 )x21 − 12x2 }u + 13 (−8 + 3r2 )x1 V2 u, ⎩ 2 2 1 V3 u = 12 r2 (2x21 + 3x2 )u − 13 x1 V2 u.

(3.35)

In this case, solutions of the system (3.35) are expressed by elementary function. In fact, we find that 1 − 12 (4r1 +3r2 )

u1 = x2 x3

1 − 12 (4r1 +3r2 )

u2 = x3

1

f024

(8r1 +3r2 −8)

,

1 24 (8r1 +3r2 )

f0

are fundamental solutions of the system (3.35). Remark 3.1. The structure of the fundamental group G of the complement of the Saito free divisor FB,2 = 0 in C3 is studied in [7]. In particular, it is shown there that there are three generators a, b, c of G such that a two dimensional representation of G is realized by     2  11 1  0 , , ρ(c) = u ρ(a) = u , ρ(b) = v 01 0 −1 0 1 where u, v ∈ C× and 6 = 1. The group ρ(G) generated by ρ(a), ρ(b), ρ(c) is solvable. This is a reason why solutions of the holonomic systems in this subsection are expressed by elementary functions. 3.5. The case FB,3 The polynomial is FB,3 = x3 (2x32 − 9x1 x2 x3 − 45x23 ) and its generating matrix is ⎛ ⎞ x1 2x2 3x3 M = ⎝ 2x2 − 53 (x1 x2 − 5x3 ) − 56 x1 x3 ⎠ . 3x3 − 35 x22 − 56 x2 x3

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There are three holonomic systems. One is of Type I and the remaining two are of Type II. In these cases, solutions of the holonomic systems are expressed by elementary functions. System of Type I ⎧ ⎨ V1 u = r1 u, V u = r2 x1 u, ⎩ 2 V3 V3 u = − 15 r2 (5r2 − 3)x22 u + 25 (5r2 − 3)x2 V3 u.

(3.36)

In this case, solutions of (3.36) are expressed by elementary functions. In fact, we find that − 13 (3r1 +10r2 )

1

u1 = x3

(−2x32 + 9x1 x2 x3 + 45x23 ) 3 (2r1 +5r2 −2) (−x22 + 3x1 x3 ),

u2 = x3

(−2x32 + 9x1 x2 x3 + 45x23 ) 3 (2r1 +5r2 )

− 13 (3r1 +10r2 )

1

are fundamental solutions of the system (3.36). Systems of Type II There are two holonomic systems of Type II. Type (II.1) ⎧ ⎨ V1 u = r1 u, 1 r2 {(5r2 − 6)x21 − 20x2 }u + 15 (5r2 − 3)x1 V2 u, V V u = − 20 ⎩ 2 2 1 V3 u = 2 r2 x2 u.

(3.37)

In this case, solutions of (3.37) are expressed by elementary functions. In fact, we find that − 13 (3r1 +5r2 )

u 1 = x 2 x3

− 13 (3r1 +5r2 )

u 2 = x3

1

(−2x32 + 9x1 x2 x3 + 45x23 ) 6 (4r1 +5r2 −2) , 1

(−2x32 + 9x1 x2 x3 + 45x23 ) 6 (4r1 +5r2 )

are fundamental solutions of the system (3.37). Type (II.2) ⎧ ⎨ V1 u = r1 u, 1 r2 {(5r2 − 12)x21 − 20x2 }u + 15 (5r2 − 6)x1 V2 u, V2 V2 u = − 20 ⎩ 1 3 V3 u = 20 r2 (3x21 + 10x2 )u − 10 x1 V2 u.

(3.38)

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In this case, solutions of (3.38) are expressed by elementary functions. In fact, we find that − 13 (3r1 +5r2 )

1

(−2x32 + 9x1 x2 x3 + 45x23 ) 6 (4r1 +5r2 −8) ×(−2x42 + 12x1 x22 x3 − 9x21 x23 + 60x2 x23 )

u1 = x3

− 13 (3r1 +5r2 )

u2 = x3

1

(−2x32 + 9x1 x2 x3 + 45x23 ) 6 (4r1 +5r2 )

are fundamental solutions of the system (3.38). Remark 3.2. The structure of the fundamental group G of the complement of the Saito free divisor FB,3 = 0 in C3 is studied in [7]. In particular, it is shown there that G  Z2 . This is a reason why solutions of the holonomic systems in this subsection are expressed by elementary functions. 3.6. The case FB,4 The polynomial is FB,4 = x3 (9x21 x22 − 4x32 + 18x1 x2 x3 + 9x23 ) and its generating matrix is ⎛ ⎞ x1 2x2 3x3 M = ⎝ 2x2 3(3x1 x2 + x3 ) 6x1 x3 ⎠ . 3x3 0 −3x2 x3 In this case, there are three holonomic systems of rank two. One is of type I and the remaining two are of Type II. In this subsection, we always put f0 = 9x21 x22 − 4x32 + 18x1 x2 x3 + 9x23 . System of Type I ⎧ ⎨ V1 u = r1 u, (3.39) V2 u = r2 x1 u, ⎩ V3 V3 u = −(3r1 − r2 )(3r1 − r2 − 3)x22 u − (6r1 − 2r2 − 3)x2 V3 u. In this case, 1

u = x33

(3r1 −r2 )

1

f06

(−2r1 +r2 )

(c1 z + c2 )

is a solution of the system (3.39), where c1 , c2 are constants and z is a function such that   (5x1 x2 + 3x3 ) 2(x21 − x2 ) 9x21 x2 − 4x22 + 3x1 x3 . (∂x1 z, ∂x2 z, ∂x3 z) = , ,− 1/6 1/6 1/6 x3 f 0 x3 f0 3x23 f0 (3.40)

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The proof is similar to that in subsection 3.4. Systems of Type II There are two holonomic systems of Type II. Type (II.1) ⎧ ⎨ V1 u = r1 u, V V u = 14 {(9 − r32 )x21 + (−13 + 4r3 + r22 )x2 }u + r3 x1 V2 u, ⎩ 2 2 V3 u = − 12 (6r1 − r3 + 3)x2 u.

(3.41)

A construction of a solution of (3.41) is accomplished by an argument similar to the case of Type (II.1) in subsection 3.3. To construct solutions of this system, we first put f0 = 9(x3 + x1 x2 )2 − 4x32 . Then the singularities of the system are x3 = 0 and f0 = 0. We next define v by 1

v = x36

(−6r1 +r3 −3)

1

f012

(4r1 −r2 −r3 +2)

u.

By direct computation, we obtain differential equations for v:   3(r2 + 6)x2 (x1 x2 + x3 ) (r2 + 1)(r2 + 5)x22 2 ∂x1 + ∂x1 + v = 0, f0 4f0   3(r2 + 6)(x1 x2 + x3 ) (r2 + 1)(r2 + 5) v = 0. ∂x3 + ∂x23 + f0 4f0 Then it is easy to show that   r2 + 1 r2 + 5 1 9(x3 + x1 x2 )2 , , ; v = c(x2 )F 12 12 2 4x32 is a solution of the two equations for a function c(x2 ) of x2 . By the weight − 14 (r2 +1)

condition V1 u = r1 u, c(x2 ) coincides with x2 As a consequence, 1

u = x36

(6r1 −r3 +3)

1

f012

up to a constant factor.

(−4r1 +r2 +r3 −2) − 14 (r2 +1) x2



×F

r2 + 1 r2 + 5 1 9(x3 + x1 x2 )2 , , ; 12 12 2 4x32



is a solution of the system (3.41). Type (II.2) ⎧ ⎨ V1 u = r1 u, V2 V2 u = − 41 r2 {(18 + r2 )x21 − 4x2 }u + (r2 + 9)x1 V2 u, ⎩ V3 u = 12 {−3r2 x21 + (−6r1 + r2 )x2 }u + 3x1 V2 u.

(3.42)

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In this case, solutions of (3.42) are expressed by elementary functions and their integrals. In fact, 1

u = x36

(6r1 −r2 )

1

f012

(−4r1 +r2 )

(c1 z + c2 )

is a solution of the system (3.42), where c1 , c2 are constants and z is a function such that (∂x1 z, ∂x2 z, ∂x3 z) 1/6 1/6 −2 3 = ((15x21 x2 − 2x22 + 9x1 x3 )f0 x−2 3 , (6x1 − 5x1 x2 + 3x3 )f0 x3 , (3.43) 1/6 −3 3 2 2 −(9x1 x2 − 4x1 x2 + 3x1 x3 + 2x2 x3 )f0 x3 ). The proof is similar to that in subsection 3.4. 3.7. The case FB,6 The polynomial is FB,6 = 9x1 x42 + 6x21 x22 x3 − 4x32 x3 + x31 x23 − 12x1 x2 x23 + 4x33 and its generating matrix is ⎛ ⎞ x1 2x2 3x3 ⎠. M = ⎝ 2x2 3x1 x2 + 52 x3 29 x22 + 15 2 x1 x3 3 2 3x3 4 (15x2 + x1 x3 ) 18x2 x3 In this case, there are three holonomic systems of rank two. One is of Type I and the remaining two are of Type II. System of Type I ⎧ ⎨ V1 u = r1 u, 1 (20r1 − 7)x1 u, V u = 12 ⎩ 2 1 V3 V3 u = − 16 (20r1 − 7){2(10r1 + 1)x22 − 3x1 x3 }u + 10(r1 + 1)x2 V3 u. (3.44) In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.44) are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.44) such that   1 1 2 1 x3 1 (5r +8) u|x1 =0 = (x32 − x23 ) 45 (5r1 −4) x345 1 F , − , ; 22 . 15 15 3 x3 We next treat the case x2 = 0. There is a solution u of (3.44) such that   1 1 x3 1 13 2 (10r1 +4) u|x2 =0 = (x31 + 4x3 ) 45 (5r1 −4) x345 F − , , ;− 1 . 6 30 3 4x3

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Systems of Type II There are two holonomic systems of Type II. Type (II.1) ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ 1 V2 V2 u = 144 {−(20r1 + 1)(20r1 + 19)x21 + 12(40r1 + 11)x2 }u ⎪ + 35 (2r1 + 1)x1 V2 u, ⎪ ⎩ 1 V3 u = 4 (20r1 + 1)x2 u.

(3.45)

In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.45) are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.45) such that   1 1 1 8 2 x32 3 2 45 (5r1 −2) 45 (5r1 +4) u|x1 =0 = (x2 − x3 ) x3 F − , , ; 2 . 15 15 3 x3 We next treat the case x2 = 0. There is a solution u of (3.45) such that   1 1 1 1 x31 1 (10r1 +2) . u|x2 =0 = (x31 + 4x3 ) 45 (5r1 −2) x345 , − , ; − F 6 30 3 4x3 Type (II.2) ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ 1 {−(20r1 − 17)(20r1 + 37)x21 + 96(5r1 − 2)x2 }u V2 V2 u = 144 ⎪ + 35 (2r1 + 1)x1 V2 u, ⎪ ⎩ 1 V3 u = 16 {3(−20r1 + 17)x21 + 16(5r1 − 2)x2 }u + 94 x1 V2 u.

(3.46)

In this case, the restrictions to x1 = 0 and x2 = 0 are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.46) such that   1 1 1 2 2 x32 3 2 45 (5r1 −11) 45 (5r1 +22) x3 F − , , ; 2 . u|x1 =0 = (x2 − x3 ) 15 15 3 x3 We next treat the case x2 = 0. There is a solution u of (3.46) such that   1 1 1 13 1 x3 (10r1 +11) F u|x2 =0 = (x31 + 4x3 ) 45 (5r1 −11) x345 ,− , ;− 1 . 6 30 3 4x3 3.8. The case FH,1 The polynomial is FH,1 = −8x91 x22 − 20x61 x32 − 230x31 x42 − 135x52 + 8x71 x2 x3 + 120x41 x22 x3 +450x1 x32 x3 + 8x51 x23 − 100x21 x2 x23 − 100x33

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and its generating matrix is ⎞ ⎛ 3x2 5x3 x1 ⎠. M = ⎝ 3x2 2x3 + 2x21 x2 7x1 x22 + 2x41 x2 1 2 4 3 4 3 2 5x3 7x1 x2 + 2x1 x2 2 (15x2 + 4x1 x3 + 18x1 x2 ) Note that FH,1 is regarded as the discriminant of the reflection group of type H3 . In this case, there are three holonomic systems of rank two. One is of Type I and the remaining two are of Type II. System of Type I ⎧ V1 u = r1 u, ⎪ ⎪ ⎪ 2 ⎪ + 2)x21 u, ⎪ ⎨ V2 u = 15 (r1  V3 V3 u = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2 − 225

8(2 + r1 )2 x81 + 10(2 + r1 )(−7 + 4r1 )x51 x2 +25(−4 + r1 )(−5 + 2r1 )x21 x22 −300(1 + 2r1 )x31 x3 − 375(2 + r1 )x2 x3

 u

(3.47)

4 + 15 (2 + r1 )x1 (2x31 + 5x2 )V3 u.

In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.47) are expressed in terms of hypergeometric function. We first treat the case x1 = 0. There is a solution u of (3.47) such that   1 2 7 3 27x52 −2 , , ;− u|x1 =0 = (27x52 + 20x33 ) 15 (r1 +2) x3 5 F . 15 15 5 20x33 We next treat the case x2 = 0. There is a solution u of (3.47) such that   1 2 2 4 2x51 1 (2r −2) . , , ; u|x2 =0 = (2x51 − 25x3 ) 15 (r1 +2) x315 1 F 5 5 5 25x3 Systems of Type II There are two systems of Type II. Type (II.1) ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ 4 x1 {(r12 + 2r1 − 44)x31 − 45(r1 + 12)x2 }u V2 V2 u = − 225 4 ⎪ + 15 (r1 + 1)x21 V2 u, ⎪ ⎩ 2 V3 u = 15 x1 {2(r1 + 1)x31 + 5(r1 + 4)x2 }u.

(3.48)

In this case, the restriction to x1 = 0 of any solution of (3.48) is expressed in terms of hypergeometric functions. In particular, there is a solu-

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tion u of (3.48) such that 2

1

u|x1 =0 = (27x52 + 20x33 ) 15 (r1 −2) x35 F

  2 8 4 27x52 . − , , ;− 15 15 5 20x33

On the other hand, we obtain differential equations  ∂x1  ∂x3

u V2 u u V2 u



 =

1 3(2x51 −25x3 )

 =

2(r1 − 2)x41 0 + 4)x x 2(r + 4)x41 −20(r 1 3 1 ⎛ 1 

1 15x3 (2x51 −25x3 )

⎜ ⎝

4(r1 + 1)x51 −75r1 x3



0



(r1 + 1)x51 −75(r1 + 2)x3

20(r1 + 4)x21 x3

 u V⎞ 2u   u ⎟ ⎠ V2 u



by restricting the system (3.48) to x2 = 0. Putting 2

h = x315

(r1 +1)

1

(25x3 − 2x51 ) 15 (r1 −2) ,

we find that if u is a solution of (3.48), then   2 2 2 5 c1 (r1 + 4)x1 + c2 (2x1 − 25x3 ) 5 h, u|x2 =0 = c1 h, (V2 u)|x2 =0 = 15 where c1 , c2 are constants. Type (II.2) ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ V u = − 2 x {2(r + 1)2 x3 − 45(2r − 3)x }u V2 2 1 1 2 1 225 1 4 2 ⎪ + (r + 1)x V u, 1 2 1 ⎪ 15 ⎩ 1 V3 u = 15 x1 {2(r1 + 1)x31 + 5(2r1 − 1)x2 }u + x21 V2 u.

(3.49)

In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.49) are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.49) such that u|x1 =0 =

(27x52

+

1 −1 20x33 ) 15 (r1 +1) x3 5 F



1 11 4 27x52 , , ;− 15 15 5 20x33

 .

We next treat the case x2 = 0. There is a solution u of (3.49) such that 1

1

u|x2 =0 = (2x51 − 25x3 ) 15 (r1 +1) x315

(2r1 −1)

 F

1 1 2 2x51 , , ; 5 5 5 25x3

 .

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3.9. The case FH,2 The polynomial is FH,2 = 100x31 x42 + x52 + 40x41 x22 x3 − 10x1 x32 x3 + 4x51 x23 − 15x21 x2 x23 + x33 and its generating matrix is ⎛ ⎞ x1 3x2 5x3 ⎠. M = ⎝3x2 36x21 x2 + 6x3 90x1 x22 + 90x21 x3 10 50 3 3 2 5x3 − 3 (12x1 − 55x2 )x1 x2 − 3 (6x1 x2 − x32 + 6x41 x3 − 18x1 x2 x3 ) In this case, there are two holonomic systems of rank two. One is of Type I and the other is of Type II. System of Type I ⎧ V1 u = r1 u, ⎪ ⎪ ⎪ ⎪ V2 u = 3(4r1⎛− 1)x21 u, ⎪ ⎪ ⎪ ⎨ V3 V3 u = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎞

4(−1 + 4r1 )(17 + 4r1 )x81 ⎟ −12(−27 + 100r1 + 40r12 )x51 x2 25 ⎜ −9 ⎝ ⎠u +15(−7 + 20r1 + 60r12 )x21 x22 3 +8(12r1 − 1)x1 x3 − 9(10r1 − 1)x2 x3

(3.50)

3 − 20 3 (2 + r1 )x1 (4x1 − 15x2 )V3 u.

In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.50) are expressed in terms of hypergeometric functions. We first treat the case x1 = 0. There is a solution u of (3.50) such that   1 1 3 3 x52 1 5 3 30 (2r1 −1) 10 x3 F − , , ; − 3 . u|x1 =0 = (x2 + x3 ) 30 10 5 x3 We next treat the case x2 = 0. There is a solution u of (3.50) such that   1 1 1 8 4 4x5 (4r1 +1) u|x2 =0 = (4x51 + x3 ) 30 (2r1 −1) x330 F − , , ;− 1 . 15 15 5 x3 System of Type II ⎧ ⎨ V1 u = r1 u, V2 V2 u = −9(4r1 + 7)x1 {(4r1 + 1)x31 − 2x2 }u + 24(r1 + 1)x21 V2 u, ⎩ x2 V u. V3 u = 53 (30r1 + 7)x1 x2 u − 10 9 1 2 (3.51) In this case, the restrictions to x1 = 0 and x2 = 0 of solutions of (3.51) are expressed in terms of hypergeometric functions. We first treat the case

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x1 = 0. There is a solution u of (3.51) such that   3 1 1 17 4 x5 u|x1 =0 = (x52 + x33 ) 30 (2r1 −3) x310 F − , , ; − 23 . 10 30 5 x3 We next treat the case x2 = 0. There is a solution u of (3.51) such that   1 1 2 2 4x51 1 (4r1 +3) 5 (2r −3) 30 1 x3 F − , , ;− . u|x2 =0 = (4x1 + x3 ) 30 15 15 5 x3 3.10. The case FH,3 The polynomial is FH,3 = 8x31 x42 + 108x52 − 36x1 x32 x3 − x21 x2 x23 + 4x33 and its generating matrix is ⎛ 3x2 x1 1 2 ⎝ M = 3x2 10 (x1 x2 + 2x3 ) 5x3 5x1 x22

⎞ 5x3 23 3 2 x x2 + 20 x1 x3 ⎠ . 10 1 2 15 2 x (2x2 + x1 x3 ) 2 2

In this case, there are two holonomic systems of rank two. One is of Type I and the other is of Type II. System of Type I ⎧ ⎨ V1 u = r1 u, 1 (16r1 − 13)x21 u, V u = 600 ⎩ 2 16r1 −13 V3 V3 u = − 144 x2 {(16r1 + 17)x21 x2 − 60x3 }u + 83 (r1 + 2)x1 x2 V3 u. (3.52) In this case, the restriction to x1 = 0 of any solution of (3.52) is expressed in terms of hypergeometric functions. In particular, there is a solution u of (3.52) such that   1 1 1 3 27x52 1 5 3 30 (2r1 −5) 2 . ,− , ;− 3 u|x1 =0 = (27x2 + x3 ) x3 F 6 6 5 x3 On the other hand, we obtain differential equations by restricting (3.52) to x2 = 0:    1   0 5x3 u u ∂x1 = , V3 u V 0 0 3u      r1 x1 − 25x u u 2 5x3 3 = . ∂x3 1 +4 V3 u V3 u 0 r5x 3

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Solving these equations, we find that if u is a solution of (3.52), then r1

1

u|x2 =0 = c1 x35 + c2 x1 x35

(r1 −1)

,

1

(V3 u)|x2 =0 = 5c2 x35

(r1 +4)

,

where c1 , c2 are constants. System of Type II ⎧ V1 u = r1 u, ⎪ ⎪ ⎨ 1 V2 V2 u = − 360000 x1 {(16r1 + 1)(16r1 + 31)x31 − 3600(16r1 + 3)x2 }u 4 + 75 ⎪ (r1 + 1)x21 V2 u, ⎪ ⎩ 1 V3 u = 12 (16r1 + 1)x1 x2 u. (3.53) In this case, the restriction to x1 = 0 is expressed in terms of hypergeometric functions. There is a solution u of (3.53) such that   1 1 19 4 27x5 1 u|x1 =0 = (27x52 + x33 ) 30 (2r1 −1) x310 F − , , ; − 3 2 . 30 30 5 x3 On the other hand, we obtain differential equations by restricting (3.53) to x2 = 0:      u 0 0 u ∂x 1 = 16r1 +1 , V u V2 u 300 x1 0  2     u r1 0 u = 5x1 3 . ∂x 3 1 +1 2 V2 u − 16r300 V2 u x1 r1 + 2 Solving these equations, we find that r1

r1 1 16r1 + 1 (r +2) c1 x21 x35 + c2 x35 1 600 is the restriction of a solution of (3.53) to x2 = 0, where c1 , c2 are constants.

u|x2 =0 = c1 x35 , (V2 u)|x2 =0 =

4. Relationship with algebraic solutions of Painlev´ e sixth equation The polynomial FA,2 was found by M. Sato about forty years ago. This polynomial looks like the discriminant of the reflection group of type A3 . The motivation of the paper [8] is to find weighted homogeneous polynomials of three variables like FA,2 . As a result, the author found seventeen polynomials including discriminants of irreducible real reflection groups of rank three. The most important property of them is that they define Saito free divisors. As the next stage of the study following the paper [4], the author investigated systems of uniformization equations with singularities

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along the Saito free divisor of simple type. The result was summarized in [9]. The existence of such a system is deeply related with that of three dimensional representation of the fundamental group of the complement of the Saito free divisor. The construction of such a system is reduced to an integrability condition which one needs a simple but complicated computation to solve. At this moment, it is underlined that in [4], Saito used a suggestive argument to construct solutions of a special case of systems of uniformization equations. In the special case he treated, the system has a quotient in the sense of D-module and the quotient is a holonomic system of rank two and is of Type I in our terminology. This leads the author to the study on the construction of holonomic systems of Type I and Type II. It is a basic problem whether holonomic systems obtained in the previous section are reduced to elementary systems or classically known systems or not. For example, take holonomic systems given in subsection 3.4. Their solutions are expressed by elementary functions, namely by product of complex powers of polynomials and their integrals. On the other hand, take the holonomic system given in subsection 3.2. In this case, their solutions are expressed by Gaussian hypergeometric functions. This means that the holonomic system in subsection 3.2 is reduced to an ordinary differential equation of hypergeometric type. But in some of holonomic systems obtained in the previous section, it seems hard to show whether solutions of them are expressed by elementary functions or special functions or not. We treat the holonomic system of Type I in subsection 3.9 and show that it induces an ordinary differential equation of second order with three singular points and an apparent singular point. This is an evidence that its solutions are not expressed by neither elementary functions nor hypergeometric functions. Put F = FH,2 and consider the system (3.50) (cf. subsection 3.9). Then by direct computation, we obtain a differential equation for u with respect to x3 : ∂x23 u +

P1 (x) P2 (x) u = 0, ∂x3 u + (x3 − as )F (x3 − as )F 2

(4.54)

where P1 , P2 are polynomials of x3 with rational coefficients of x1 , x2 and as is a rational function of x1 , x2 . We now put u = F (2r1 −1)/30 y and obtain a differential equation for y from (4.54). Then  ∂x23 y +

1 2 ∂x3 F − 3 F x3 − as



 ∂x3 y +

c0 c0 x23 + c1 x3 + c2 − F x3 − as

 y = 0,

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where c0 = c1 = c2 = as =

x1 , 15x2 (12x31 −x2 ) 6 80x1 −456x31 x2 +33x22 , 300x2 (12x31 −x2 ) x21 (528x61 −3224x31 x2 +825x22 ) − , 900(12x31 −x2 ) 3(4x31 x2 −x22 ) − . 2x1

Let z0 , z1 , z2 be solutions of F = 0 as a cubic polynomial of x3 , namely, F = (x3 − z0 )(x3 − z1 )(x3 − z2 ). Using as , z0 , z1 , z2 , we put t=

z2 − z 0 , z1 − z 0

w=

as − z 0 . z 1 − z0

Then both t, w are algebraic functions of m = m=

x2 . x31

Putting

s2 (s + 1)2 , (s2 + s + 1)2

we find that t=

s5 (s + 2)(2s2 + 3s + 3)2 , (2s + 1)(3s2 + 3s + 2)2

w=

s2 (s + 2)(s2 + 1)(2s2 + 3s + 3) . 2(s2 + s + 1)(3s2 + 3s + 2)

The pair (t, w) is an algebraic solution of Painlev´e sixth equation given in [2], p.23. This suggests that solutions of the differential equation (3.50) are not expressed by special functions. For the details of the argument above, see [10]. Remark 4.1. The author thanks M. Kato of Univ. of Ryukyus for explaining him an idea of the argument on deriving an algebraic solution of Painlev´e sixth equation from the systems of equations obtained in this paper. References 1. A. G. Aleksandrov, Moduli of logarithmic connections along Saito free divisor, Contemp. Math., 314 (2002), 2-23. 2. P. Boalch, The fifty-two icosahedral solutions to Painlev´e VI, J. Reine Angew. Math., 596, (2006), 183-214. 3. M. Kato and J. Sekiguchi, Systems of uniformization equations with respect to the discriminant sets of complex reflection groups of rank three. Preprint. 4. K. Saito, On the uniformization of complements of discriminant loci. RIMS Kokyuroku 287 (1977), 117-137.

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5. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields. J. Faculty of Sciences, Univ. Tokyo 27 (1980), 265-291. 6. K. Saito, Uniformization of orbifold of a finite reflection group. In “Frobenius Manifold, Quantum Cohomology and Singularities.” A Publication of the Max-Planck-Institute, Mathematics, Bonn, 265-320. 7. K. Saito and T. Ishibe, Monoids in the fundamental groups of the complement of logarithmic Saito free divisors in C3 , J. Algebra 344 (2011), 137-160. 8. J. Sekiguchi, A classification of weighted homogeneous Saito free divisors, J. Math. Soc. Japan, 61 (2009), 1071-1095. 9. J. Sekiguchi, Systems of uniformization equations along Saito free divisors and related topics. In “The Third Japanese-Australian Workshop on Real and Complex Singularities”, Proceedings of the Centre for Mathematics and its Applications, 43 (2010), 83-126. 10. J. Sekiguchi, Free divisors, holonomic systems and algebraic solutions of Painlev´e sixth equation. In preparation.

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Parametric local cohomology classes and Tjurina stratifications for µ-constant deformations of quasi-homogeneous singularities Shinichi Tajima Division of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan [email protected] Local cohomology classes attached to semi-quasihomogeneous hypersurface isolated singularities are considered. A new effective method to compute Tjurina stratifications associated with μ−constant deformation ft (x) = F (x, t), t ∈ T of weighted homogeneous isolated singularities is proposed. The proposed method also computes on each stratum, via Grothendieck local duality, a parametric standard basis of the relevant ideal quotient Jt : ft , where Jt stands for the Jacobi ideal of the function ft in the local ring of germs of holomorphic functions. The key idea in this approach is the use of parametric local cohomology classes. Keywords: standard basis, Grothendieck local duality, algebraic local cohomology AMS classification numbers: 32S25, 32S30, 32C36

1. Introduction Let X be an open neighborhood of the origin O of the n-dimensional complex space Cn with coordinates x = (x1 , x2 , ..., xn ) and let OX be the sheaf on X of holomorphic functions. Let f be a holomorphic function defined on X with an isolated singularity at the origin O and let Jf denote the ∂f ∂f Jacobi ideal ( ∂x , ∂f , ..., ∂x ) in OX,O generated by the partial deriva1 ∂x2 n ∂f ∂f ∂f tives ∂x1 , ∂x2 , ..., ∂xn , where OX,O is the stalk at O of the sheaf OX . Let ∂f ∂f (f, Jf ) denote the ideal (f, ∂x , ∂f , ..., ∂x ) in OX,O generated by f and 1 ∂x2 n ∂f ∂f ∂f , , ..., . ∂x1 ∂x2 ∂xn The Milnor number μf = dimC (OX,O /Jf ) is a topological invariant and the Tjurina number τf introduced by Tjurina in the context of deformation theory, which is defined to be the dimension of the vector space OX,O /(f, Jf ), is an analytic invariant of the singularity. Tjurina numbers have been extensively studied ([17], [18], [28], [39]). It turned out that the Tjurina number is closely related with several complex analytic properties

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of singularities ([4], [12], [29], [37], [39]). In 1989, B. Martin and G. Pfister ([22]) constructed an algorithm of computing parameter dependency of Tjurina numbers of μ-constant deformations of quasi-homogeneous hypersurface isolated singularities. The algorithm has been successfully applied to study of semi quasi-homogeneous singularities ([6], [7]). In this paper, we propose an alternative approach, in a context of computational algebraic analysis, to compute Tjurina stratifications of μconstant deformations. The resulting algorithm has already been implemented in a computer algebra system Risa/Asir ([27]). A description of the algorithm will appear elsewhere. Note that the method is extendable to handle μ-constant deformations of Newton non-degenerate hypersurface isolated singularities. Note also that the framework we adopted is expected to provide an effective method to study certain families of holonomic DX -modules attached to μ-constant deformations of hypersurface singularities ([37]). 2. Local cohomology and non-quasihomogeneity n n (ΩX ) be the local cohomology supported at the origin O, where Let H{O} n n n ΩX is the sheaf on X of holomorphic differential n-forms. Let H[O] (ΩX ) denote the algebraic local cohomology supported at the origin O defined by n n n H[O] (ΩX ) = lim ExtnOX (OX /(x1 , x2 , ..., xn )k , ΩX ), k→∞

where (x1 , x2 , ..., xn ) is the maximal ideal generated by x1 , x2 , ..., xn . Then, n n (ΩX ) naturally has a structure of a Fr´echet-Schwartz topothe space H{O} n n logical vector space and the space H[O] (ΩX ) has a structure of the dual Fr´echet-Schwartz topological vector space ([14], [31], [38]). A theory of Functional Analysis asserts that the topological vector space n n H{O} (ΩX ) and the space OX,O of convergent power series at the origin are mutually strong dual via the local residue pairing ([14], [15], [23]). The n n X,O of (ΩX ) and the space O same duality holds between the space H[O] formal power series at the origin ([16], [19]). Set ∂f ∂f ∂f n n (ΩX )| ω= ω = ··· = ω = 0}. WJf = {ω ∈ H{O} ∂x1 ∂x2 ∂xn Since f has isolated singularity at the origin, Wf coincides with the space Ef of the algebraic local cohomology classes defined by W EJ = {ω ∈ Hn (Ω n ) | ∂f ω = ∂f ω = · · · = ∂f ω = 0}. W f X [O] ∂x1 ∂x2 ∂xn

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Note that an efficient algorithm for computing a basis of the vector EJf is described in [34], [35]. space W ∂f ∂f Let Jf (resp. Jf ) denote the Jacobi ideal generated by ∂x , ∂f . . . ∂x 1 ∂x2 n X,O ). Then, the local duality theorem ([13], in the local ring OX,O (resp. O [15]) asserts that there is a non-degenerate pairing resO ( , ) : OX,O /Jf × WJf −→ C. An efficient method to evaluate the pairing above is also given in [34], [35]. The non-degeneracy of the pairing implies in particular the fact that, a convergent power series h ∈ OX,O is in the ideal Jf if and only if resO (h, ω) = 0 for every ω ∈ WJf . Thus, the Jacobi ideal Jf is comEJ ) of (algebraic) local cohomolpletely determined by the space WJf (= W f ogy classes via local residues ([26], [33], [34]). n ), the non-degenerate pairing Note that, since WJf = ExtnOX (OX /Jf , ΩX above coincides with the classical pairing n )) −→ C OX,O /Jf × ExtnOX (OX /Jf , (ΩX

induced from Yoneda pairing in homological algebra. The corresponding non-degeneracy coincides with an analytical version of the Grothendieck local duality theorem on residues ([8], [9], [10], [11], [16], [19]). Now we introduce another vector space WTf defined to be the set of n n (ΩX ) that are annihilated by the ideal local cohomology classes in H{O} generated by f and

∂f , ∂f ∂x1 ∂x2

∂f . . . ∂x in the local ring OX,O : n

n n WTf = {ω ∈ H{O} (ΩX ) | fω =

∂f ∂f ∂f ω= ω = ··· = ω = 0}, ∂x1 ∂x2 ∂xn

that is also equal to ∂f ∂f ∂f n ET = {ω ∈ Hn (ΩX ) | fω = ω= ω = ··· = ω = 0}. W f [O] ∂x1 ∂x2 ∂xn It follows directly from the definitions that WTf = {ω ∈ WJf | f ω = 0}. Note also that the dimension of the vector space WJf is equal to the Milnor number μf defined to be μf = dimC (OX,O /Jf ) and the dimension of the vector space WTf is equal to the Tjurina number τf = dimC (OX,O /(f, Jf )) Let ϕ : WJf −→ WJf be a map defined by ϕ(ω) = f ω and set WQf = {f ω | ω ∈ WJf }. Since Kerϕ = WTf and Imϕ = WQf , we have the following result. Lemma 1 The sequence 0 −→ WTf −→ WJf −→ WQf −→ 0

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of vector spaces defined by ϕ is exact. Theorem 2 (i) dim WQf = μf − τf . (ii) AnnOX,O (WQf ) = {h ∈ OX,O | hf ∈ Jf }. Proofs (i) Trivial. (ii) See the next section. Now let U0 = X and Uj = {x = (x1 , . . . , xn ) ∈ X | xj = 0} for j = 1, . . . , n. Consider the pair (X, X − {O}) of open sets and its relative covering (U, U  ) where U = {U0 , U1 , . . . , Un } and U  = {U1 , . . . , Un }. Then, any local cohomology class in the space WJf , which is actually a subspace n n of H[O] (ΩX ), can be represented as a finite sum of the form C D   1  1 cλ λ dx = c(l1 ...ln ) l1 dx , x x1 . . . xlnn λ where cλ ∈ C with λ = (l1 , . . . , ln ) ∈ Nn+ , dx = dx1 ∧ dx2 ∧ · · · ∧ dxn and ˇ cohomology class ([26], [30], [34], [35]). [ x1λ dx] is a relative Cech   We also use the notation ( cλ [ x1λ ])dx or [ cλ x1λ ]dx for representing n n algebraic local cohomology classes in H[O] (ΩX ). Note that  1 [ xλ−κ dx], li > ki , i = 1, . . . , n 1 xκ [ λ dx] = x 0, otherwise, where κ = (k1 , . . . , kn ) ∈ Nn , λ = (l1 , . . . , ln ) ∈ Nn+ , and λ − κ = (l1 − k1 , . . . , ln − kn ). Example 1 (E12 singularity) Let f (x, y) = x3 + y 7 + axy 5 . The Milnor number μf is 12 and the following 12 algebraic local cohomology classes constitute a basis of the vector space WJf ([33]). [

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ ], [ 2 2 ] ], [ ], [ xy xy 2 x2 y xy 3 x y [

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ 2 3 ], [ ], [ 2 4 ], xy 4 x y xy 5 x y ([

1 a 1 ] − [ 3 ])dx ∧ dy, xy 6 3 x y

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([

1 5a 1 5a 1 ] − [ 7 ] + [ 3 2 ])dx ∧ dy, x2 y 5 7 xy 21 x y

and ([

5a 1 a 1 1 5a2 1 ] − ] − ])dx ∧ dy. [ [ ] + [ x2 y 6 7 xy 8 3 x4 y 21 x3 y 3

It is easy to see that the first 11 local cohomology classes in the list above belong to WTf . We have by a direct computation WQf = SpanC {a[

dx ∧ dy ]}, xy

which implies in particular that τf = 12 − 1 = 11, if a = 0. Example 2 Let f (x, y) = x3 + y 7 + bxy 6 . The following 12 algebraic local cohomology classes constitute a basis of the vector space WJf . [

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ ], [ 2 2 ], ], [ ], [ xy xy 2 x2 y xy 3 x y

[

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ 2 3 ], [ ], [ 2 4 ], xy 4 x y xy 5 x y

[

dx ∧ dy dx ∧ dy ], [ 2 5 ], xy 6 x y

and ([

1 x2 y 6

]−

6b 1 2b2 1 [ 7]+ [ ])dx ∧ dy. 7 xy 7 x3 y

One can easily see that WQf = {0}. According to a result of K. Saito [29], we find that f is quasihomogeneous. Remark Let w = (7, 3) and let degw ( ) denote the weighted degree w.r.t. the weight vector w. Then x3 +y 7 is a weighted homogeneous function of weighted degree 21. The weighted degree of the monomial xy 5 is 22 and that of the monomial xy6 is 25. The monomials xy 5 and xy 6 are upper monomials. The monomial xy 5 satisfies the condition degw (xy 5 ) ≤ 2 × 21 − 2(7 + 3), whereas the monomial xy 6 does not satisfy the condition degw (xy 6 ) ≤ 2 × 21 − 2(7 + 3).

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3. µ-constant deformations Let w = (w1 , w2 , ..., wn ) ∈ Zn be a weight vector and let f be a weighted homogeneous function (w.r.t. the weight vector w) with an isolated singularity at the origin O ∈ X. Let μ = dimC (OX,O /Jf ) denote the Milnor number of f . Assume that a μ-constant deformation ft (x) = F (x, t), t ∈ T of f with f0 (x) = f (x) is given, where F (x, t) is a holomorphic function defined on X × T, X ⊂ Cn and T ⊂ C are open neighbourhood of O ∈ Cn and of O ∈ C . ∂ft ∂ft ∂ft , , . . . ∂x ), the Jacobi ideal of ft in the local ring OX,O . Let Jt = ( ∂x 1 ∂x2 n Then, the Milnor number μt = dimC (OX,O /Jt ) of ft , which is independent of t ∈ T from the assumption, is equal to μ. We consider the following sets of parametric local cohomology classes defined by n n WJt = {ω ∈ H{O} (ΩX )|

∂ft ∂ft ∂ft ω= ω = ··· = ω = 0}, ∂x1 ∂x2 ∂xn

and n n (ΩX ) | ft ω = WTt = {ω ∈ H{O}

∂ft ∂ft ∂ft ω= ω = ··· = ω = 0}. ∂x1 ∂x2 ∂xn

Note that the dimension of the vector space WTt , which is equal to the Tjurina number τt = dimC (OX,O /(ft , Jt )), depends on the parameter t ∈ T . Let ϕt : WJt −→ WJt be a map defined by ϕt (ω) = ft ω. Set WQt = Imϕt . Lemma 3 The sequence 0 −→ WTt −→ WJt −→ WQt −→ 0 of vector spaces defined by ϕt is exact. Theorem 4 (i) dim WQt = μ − τt . (ii) AnnOX,O (WQt ) = {h ∈ OX,O | hft ∈ Jt } Proof (ii) Since WQt = Imϕt , we have AnnOX,O (WQt ) = {h ∈ OX,O | h(ft ω) = 0, ∀ ω ∈ WJt } = {h ∈ OX,O | (hft )ω) = 0, ∀ ω ∈ WJt }.

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Grothendieck local duality theorem implies AnnOX,O (WJt ) = Jt , which also yields AnnOX,O (WQt ) = {h ∈ OX,O | hft ∈ Jt }. Example 3 (E18 singularity) Let f (x, y) = x3 + y 10 and let f(t1 ,t2 ) (x, y) = F (x, y, t1 , t2 ) = x3 + y 10 + t1 xy 7 + t2 xy 8 . The weight vector is w = (10, 3) and the Milnor number is equal to 18. The following 18 parametric local cohomology classes constitute a basis of the vector space WJt . [

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ ], [ ], [ ], [ ], [ ], [ ], xy xy 2 xy 3 xy 4 xy 5 xy 6 xy 7 [

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ 2 2 ], [ 2 3 ], [ 2 4 ], [ 2 5 ], [ 2 6 ], x2 y x y x y x y x y x y ([

([

([

ω(2,8) = ([

1 t1 1 ] − [ 3 ])dx ∧ dy, 8 xy 3 x y

t1 1 t2 1 1 ] − [ 3 2 ] − [ 3 ])dx ∧ dy, xy 9 3 x y 3 x y

1 7t1 1 7t21 1 7t1 t2 1 ] − ] + ])dx ∧ dy, [ [ 3 3] + [ 2 7 10 x y 10 xy 30 x y 30 x3 y 2

7t1 1 1 t1 1 7t21 1 4t2 1 [ ] − [ [ [ ] − ] + ]− ] x2 y 8 3 x4 y 10 xy 11 30 x3 y 4 5 xy 10 +

4t2 1 t 1 t2 1 [ 3 3 ] + 2 [ 3 2 ])dx ∧ dy, 2 x y 15 x y

and ω(2,9) = ([

1 t1 1 7t1 1 7t21 1 t2 1 ] − ] − ] + [ [ [ 3 5] − [ 4 ] 2 9 4 2 12 x y 3 x y 10 xy 30 x y 3 x y −

4t2 1 t 1 t2 1 4t2 1 [ 11 ] + [ 3 4 ] + 2 [ 3 3 ])dx ∧ dy, 5 xy 2 x y 15 x y

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One can readily see that first 16 local cohomology classes in the list above belong to the space WTt . Therefore, from t1 dx ∧ dy [ ], 30 xy t1 1 2t2 1 [ ])dx ∧ dy, ϕ(t1 ,t2 ) (ω(2,9) ) = (− [ 2 ] − 30 xy 15 xy

ϕ(t1 ,t2 ) (ω(2,8) ) = −

we have WQf = SpanC {t1 [

1 1 1 ]dx ∧ dy, (t1 [ 2 ] + 4t2 [ ])dx ∧ dy}. xy xy xy

4. An Example We give an example to illustrate the proposed method to compute Tjurina stratification of the parameter space and the corresponding Tjurina numbers associated with a μ-constant deformation. Let f (x, y) = x5 + xy 5 and w = (5, 4). Set f(t1 ,t2 ,t3 ) (x, y) = x5 + xy 5 + t1 x3 y 3 + t2 y 7 + t3 y 8 . The Milnor number is equal to 21. We first compute, by using an algorithm described in [25], a basis local cohomology classes of the vector space WJt . We obtain the following 21 parametric local cohomology classes. [

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ ], [ 2 2 ], ], [ ], [ ], [ ], [ xy xy 2 xy 3 xy 4 xy 5 x2 y x y

[

dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy dx ∧ dy ], [ 3 2 ], [ 3 3 ], [ ], [ 4 2 ], ], [ 2 4 ], [ x2 y 3 x y x3 y x y x y x4 y x y ([

([

1 1 1 1 ] − 5[ 6 ])dx ∧ dy, ([ 3 4 ] − 3t1 [ 6 ])dx ∧ dy, x5 y xy x y xy

3t1 1 1 1 1 1 [ ]− ])dx ∧ dy, ([ 5 2 ] − 5[ 7 ] + 7t2 [ 2 5 ])dx ∧ dy, x4 y 3 5 x2 y 5 x y xy x y ω(4,4) = ([

ω(5,3) = ([

3t1 1 1 12t1 1 [ ]− [ ]− ])dx ∧ dy, x4 y 4 25 x6 y 5 x2 y 6

1 1 3t1 1 7t2 1 1 [ [ ] − 7t2 [ 2 6 ] ] − 5[ 8 ] − ]+ x5 y 3 xy 5 x3 y 5 5 x6 y x y +(

1 63 2 t t2 + 8t3 )[ 2 6 ])dx ∧ dy, 25 1 x y

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and ω(5,4) = ([

1 x5 y 4

]−5[

3t1 1 1 12t1 1 7t2 1 1 [ 7 ]− [ 3 6 ]− [ 6 2 ]−7t2 [ 2 7 ] ]− 9 xy 25 x y 5 x y 5 x y x y +

49t22 1 63 2 8 1 9t21 1 [ 8]− [ 3 5]+( t1 t2 − t3 )[ 6 ] 5 xy 5 x y 125 5 x y +(

1 147 2 1 63 2 t1 t2 + 8t3 )[ 2 6 ] − t 1 t2 [ 7 ] 25 x y 5 xy

+(−

72 2 1029 3 1 t1 t2 − t1 t2 )[ 2 4 ])dx ∧ dy. 25 25 x y

It is easy to see that the first 18 cohomology classes in the list above belong to the space WTt . In order to obtain a set of generators of the vector space WQt , we compute three local cohomology classes ϕt ((ω(4,4) ), ϕt (ω(5,3) ) and ϕt (ω(5,4) ). From ϕt (ω(4,4) ) = (−

2 1 3 1 t1 [ ])dx ∧ dy, ϕt (ω(5,3) ) = ( t2 [ ])dx ∧ dy, 25 xy 5 xy

and ϕt (ω(5,4) ) = (−

3 2 1 1 27 2 7 1 t1 [ 2 ] + t2 [ 2 ] + (− t1 t2 + t3 )[ ])dx ∧ dy, 25 x y 5 xy 125 5 xy

we find that the space WQt is spanned by t1 [

1 1 1 1 1 ]dx ∧ dy, t2 [ ]dx ∧ dy, (−2t1 [ 2 ] + 15t2 [ 2 ] + 35t3 [ ])dx ∧ dy. xy xy x y xy xy

By using a parametric version of the standard basis computation presented in [35], we obtain the following stratification. (0) if t1 = t2 = t3 = 0, then μ − τ = 0, WQ0 = {0}, f ∈ J. (1) if t1 = t2 = 0, t3 = 0, then μ − τt = 1, WQt = Span{[

dx ∧ dy ]}, Jt : ft = (x, y). xy

(2) if t1 = 0, or t2 = 0, then μ − τt = 2. (i) if t1 = 0, t2 = 0, then WQt = Span{[

dx ∧ dy dx ∧ dy ], [ ]}, Jt : ft = (x, y 2 ). xy xy 2

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(ii) if t1 = 0, t2 = 0, then WQt = Span{[

dx ∧ dy dx ∧ dy ], [ ]}, Jt : ft = (x2 , y). xy x2 y

(iii) if t1 = 0, t2 = 0, then WQt = Span{[

dx ∧ dy 1 1 ], (−2t1 [ 2 ] + 15t2 [ 2 ])dx ∧ dy}, xy x y xy

and Jt : ft = (2t1 y + 15t2 x, xy, y 2 ). Some remarks are in order. Remarks (1) The existing algorithm due to B. Martin and G. Pfister [22] is based on the deformation theory and utilize the Kodaira-Spencer maps. The proposed method is free from the deformation theory. (2) The proposed method can be extendable to handle the case where even the quasi-homogeneous part contain parameters. (3) The method is also applicable to compute the Hilbert function of the Tjurina algebra OX,O /(ft , Jt ) (cf. [7]). References 1. V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiabla Maps, I, Birkhauser 1985. 2. C. B˘ anic˘ a et O. St˘ an˘ a¸sil˘ a, M´ethodes Algebriques dans la Th´eorie Globale des Espaces Complexes, Gauthier-Villars, 1974. 3. J. Brian¸con, M. Granger and Ph. Maisonobe, Le nombre de modules du germe de courbe plane xa + y b = 0, Math. Ann. 279 (1988), 535–551. 4. P. Cassou-Nogu`es, Etude du comportement du polynˆ ome de Bernstein lors d’une deformation a ` μ-constant de xa +y b avec (a, b) = 1 , Ann. Inst. Fourier, Grenoble 36 (1986), 1–30. 5. A. M. Dickenstein and C. Sessa, Duality methods for the membership problem, Progress in Math. 94 (1991), Effective Methods in Algebraic Geometry, 89–103, Birkh¨ auser. 6. G. -M. Greuel and G. Pfister, On moduli space of semiquasi-homogeneous singularities, Progress in Math. 134 (1996), 171–185, Birkh¨ auser. 7. G. -M. Greuel and C. Hertling, Moduli spaces of semiquasihomogeneous singularities with fixed principal part, J. Algebraic Geom. 6 (1997), 169–199. 8. P. Griffiths and J. Harris, Principles of Algebraic Geometry, WileyInterscience Pub. 1976.

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9. A. Grothendieck, Th´eor`emes de dualit´e pour les faisceaux alg´ ebriques coh´erents, S´eminaire Bourbaki 149 (1957). 10. A. Grothendieck, Local Cohomology, Lecture Notes in Math. 41 (1967), Springer. 11. R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20 (1966), Springer. 12. C. Hertling and C. Stahlke, Bernstein polynomial and Tjurina number, Geometriae Dedicata 75 (1999), 137–176. 13. M. Kashiwara, On the maximally overdetermined system of linear differential equations, I, Publ. Res. Inst. Math. Sci. 10 (1975), 563–579. 14. H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366–383. 15. H. Komatsu, Relative cohomology of sheaves of solutions of differential equations, Lecture Notes in Math. 287 (1973), 192–261. 16. E. Kunz, Residues and Duality for Projective Algebraic Varieties, AMS. 2008. 17. O. A. Laudal and G. Pfister, Local Moduli and Singularities, Lecture Notes in Math. 1310, 1988. 18. O. A. Laudal, B. Martin and G. Pfister, Moduli of plane curve singularities with C ∗ -action, Banach Center Pub. 20 (1988), 255–278. 19. J. Lipman, Dualizing sheaves, Differentials and Residues on Algebraic Varieties, Ast´erisque 117 (1984) 20. Lˆe Dung Trang and C. P. Ramanujan, The invariance of Milnor number implies the invariance of the topological type, Amer. J. Math. 98 (1973), 67–78. 21. M. G. Marinari, H. M. M¨ oller and T. Mora, Gr¨ obner bases of ideals given by dual bases, ISSAC 1991, ACM (1991), 55–63. 22. B. Martin and G. Pfister, The kernel of the Kodaira-Spencer map of the versal μ-constant deformation of an irreducible plane curve with C ∗ -action, J. Symbolic Comp. 7 (1989), 527–531. 23. M. Morimoto, An Introduction to Sato’s Hyperfunctions, Translations of Mathematical Monographs, 129 (1993), AMS. 24. B. Mourrain, Isolated points, duality and residues, J. Pure and Appl. Alg. 117 & 118 (1997), 469–493. 25. K. Nabeshima and S. Tajima, On the computation of algebraic local cohomology classes associated with semi-quasihomogeneous singularities, to appear in Adv. Studies in Pure Math. 26. Y. Nakamura and S. Tajima, On weighted-degrees for algebraic local cohomologies associated with semiquasihomogeneous singularities, Adv. Studies in Pure Math., 46 (2007), 105–117. 27. M. Noro and T. Takeshima, Risa/Asir - a computer algebra system, ISSAC 1992 (ed. P. S. Wang), ACM (1992), 387–396. 28. R. Peraire, Tjurina number of a generic irreducible curve singularity, J. Algebra 196 (1997), 114–157. 29. K. Saito, Quasihomogene isolierte Singularitaten von Hyperfl¨ achen, Invent. Math. 14 (1971), 123–142.

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Author Index Akahori, T., 1 Ando, K., 13 Fukui, T., 31 Harris, A., 51 Hasegawa, M., 31 Hiratuka, J. T., 61 Isaev, A. V., 75 Ishikawa, G., 87 Koike, S., 115 Kuo, T. C., 115 Miyajima, K., 51 Paunescu, L., 115 Saeki, O., 61 Sekiguchi, J., 141, 159 Tajima, S., 189

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