Advanced Courses in Mathematics CRM Barcelona
Michèle Audin Ana Cannas da Silva Eugene Lerman
Symplectic Geometry of Integrable Hamiltonian Systems
Birkhä user Verlag Basel . Boston Berlin
Authors' addresses:
Michéle Audin Institut de Recherche Mathematique Avancée Université Louis Pasteur et CNRS 7 rue René Descartes 67084 Strasbourg Cedex France
[email protected]
Ana Cannas da Silva Departamento de Matemática Institute Superior Técnico Av. Rovisco Pais 1049-001 Lisboa Portugal
[email protected]
Eugene Lerman
Department of Mathematics University of Illinois at Urbana-Champaign Urbana. IL 61820 USA
[email protected]
2000 Mathematical Subject Classification 14J32. 14M25, 14R05, 52B20, 53C22, 53010. 53012
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.
ISBN 3-7643-2167-9 Birkhauser Verlag, Basel — Boston — Berlin
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Contents Preface
ix
A Lagrangian Submanifolds Michèle Audin Introduction I
Lagrangian and special Lagrangian imxnersions in C'1 1.1 Symplectic form on C", symplectic vector spaces I.1.a Symplectic vector spaces I.1.h Symplectic bases 1.1 .c The symplectic form as a differential form Li .d The symplectic group 1.1 .e Orthogonality, isotropy 1.2 Lagrangian subspaces I.2.a Definition of Lagrangian subspaces 1.2.b The symplectic reduction 1.3 The Lagrangian Grassmannian 1.3.a The Grassmannian A,, as a homogeneous space L3.b The manifold A, 1.3.c The tautological vector bundle I.3.d The tangent bundle to A, I.3.e The case of oriented Lagrangian snbspaces I.3.f The determinant and the Maslov class 1.4 Lagrangian submanifolds in C" 1.4.a Lagrangian submanifolds described by functions I.4.b Wave fronts 14.c Other examples I.4.d The Gauss map
1 3 5
5
5
6 7 8 9 9 9 10 11
ii 11
13 14
15 15 16 16
20 25 28
Contents
vi
Special Lagrangian submanifolds in I.5.a Special Lagrangian subspaces I.5.b Special Lagrangian submanifolds Graphs of forms 1.5.c I.5.d Normal bundles of surfaces From integrable systems T.5.e I.5.f Special Lagrangian submanifolds invariant under SO(n) Appendices 1.6 I.6.a The topology of the symplectic group I.6.b Complex structures I.6.c Hamiltonian vector fields, integrable systems Exercises
29
1.5
H Lagrangian and special Lagrangian submanifolds in symplectic and Calabi-Yau manifolds 11.1 Symplectic manifolds 11.2 Lagrangian submanifolds and immersions II.2.a In cotangent bundles 11.3 Tubular neighborhoods of Lagrangian submanifolds I1.3.a Moser's method 113.h Tubular neighborhoods II.3.c "Moduli space" of Lagrangian submanifolds 11.4 Calabi-Yau manifolds II.4.a Definition of the Calabi-Yau manifolds I1.4.b Yau's theorem II.4.c Examples of Calabi-Yau manifolds II.4.d Special Lagrangian submanifolds 11.5 Special Lagrangians in real Calabi-Yau manifolds I1.5.a Real manifolds II.5.b Real Calabi-Yau manifolds II.5.c The example of elliptic curves II.5.d Special Lagrangians in real Calabi-Yau manifolds 11.6 Moduli space of special Lagrangian submanifolds 11.7 Towards mirror symmetry? 1L7.a Fibrations in special Lagrangian submanifolds I1.7.b Mirror symmetry
29 32 33 35 36 .
.
38
39 39 41 41 44
49 49 51 51
53 53 57 59 60 61
62 63 66 66 66 67 68
68
Exercises
70 73 74 75 76
Bibliography
81
Contents
vii
B Symplectic Toric Manifolds Ana Cannas da Silva
85
Foreword
8T
I
Symplectic Viewpoint Symplectic Toric Manifolds 1.1 Symplectic Manifolds 1.1.1 1.1.2 Hamiltonian Vector Fields 1.1.3 Integrable Systems Hamiltonian Actions 1.1.4 Hamiltonian Torus Actions 1.1.5 Symplectic Toric Manifolds 1.1.6 1.2 Classification Delzant's Theorem 1.2.1 Orbit Spaces 1.2.2 Symplectic Reduction 1.2.3 1.2.4 Extensions of Symplectic Reduction 1.2.5 Delzant's Construction 1.2.6 Idea Behind Del.zant's Construction Moment Polytopes 1.3 Equivariant Darboux Theorem 1.3.1 1.3.2 Morse Theory Homology of Symplectic Tone Manifolds 1.3.3 Symplectic Blow-Up 1.3.4 Blow-Up of Toric Manifolds 1.3.5 Symplectic Cutting 1.3.6
II Algebraic Viewpoint 11.1
11.2
Toric Varieties 11.1.1 Affine Varieties 11.1.2 Rational Maps on Affine Varieties 11.1.3 Projective Varieties 11.1.4 Rational Maps on Projective Varieties 11.1.5 Quasiprojective Varieties 11.1.6 Toric Varieties Classification 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.2.6
Spectra Tone Varieties Associated to Semigroups Classification of Affine Toric Varieties Fans Toric Varieties Associated to Fans Classification of Normal Toric Varieties
89 89 89 90 91
94 96 98 101
101 103
104 106 109 113 118 118 119 120 122 124 126
129 129 129 131 132
135 136 137 140 140 142 143 144 148 153
Contents
VII'
11.3 Moment Polytopes 11.3.1 Equivariantly Projective Tone Varieties 11.3.2 Weight Polytopes 11.3.3 Orbit Decomposition 11.3.4 Fans from Polytopes 11.3.5 Classes of Tone Varieties 11.3.6 Symplectic vs. Algebraic
154 154 155 156 158 160 161
Bibliography
165
Index
169
C
Geodesic Flows and Contact Toric Manifolds Eugene Lerman
Foreword I
175 177
From toric integrable geodesic flows to contact tone manifolds Introduction 1.1 1.2 Symplectic cones and contact manifolds
179 179 184
II Contact group actions and contact moment maps
193
III Proof of Theorem 1.38 111.1 Homogeneous vector bundles and slices 111.2 The 3-dimensional case 111.3 Uniqueness of sympleet,ic toric manifolds 111.3.1 Cech cohomology 111.4 Proof of Theorem 1.38, part three 111.4.1 Morse theory on orbifolds
197
Appendix
Hypersurfaces of contact type
Bibliography
198 202 205 211 213 216
221
223
Preface
This book contains an expanded version of the lectures delivered by the authors at the Euro Summer School Symplectic Geometry of Integrable Hamiltonian
Systems. The summer school took place at the Centre de Recerca Matemàtica (Barcelona) from the 10th to the 15th of July 2001. It consisted of three main courses of 7.5 hours each and of some complementary talks dealing with integrable Hamiltonian systems.
Thus the book has three parts. 'I'he first part by Michèle Audin is devoted to special Lagrangian submanifolds, the second part by Ana Cannas da Silva deals with syrnplec tic tomc manifolds, and the last part by Eugene Lerman centers on contact tone manifolds. Hamiltonian systems arose, as their name suggests, from the formulation (by Hamilton) of classical niechaiiics: a Hamiltonian system is a dynamical system that describes the motion of a mechanical system whose total energy is conserved. This is where symplectic geometry comes from. It has since
developed into an area of mathematics in its own right, but it remains a point of contact between physics and geometry. Among all the Hamiltonian systems, the integrable ones -- those which have many conserved quantities — have special
geometric properties; in particular, their solutions are very regular and quasiperiodic. Moreover, their study has been central in symplectic geometry, as it stands at a crossroads between dynamical systems, algebraic geometry, and group representation theory (to name a few areas of mathematics involved). The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads, in a natural way, to symplectic toric manifolds (part B of this book), which are examples of extremely symmetric Hamiltonian systems. Physics makes a surprising come-back in part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact tone manifolds (part C of this book). Along the way, tools from many different areas of mathematics are brought to bear on the questions at hand. Thus actions of Lie groups in symplectic and contact manifolds, the Delzant theorem, Morse theory, sheaves and Cech cohomology, and aspects of Calabi-Yau manifolds make an appearance in this book.
We hope that this book can serve as an introduction to syinpiectic and contact geometry for graduate students and that it. can also be useful to research mathematicians interested in integrable systems.
x
Preface
Acknowledgments
This Euro Summer School was supported by the European Commission under contract number HPCF-CT-2000-001 10 of the Improving Human Research Potential Programme, by the Spanish Ministry of Science and Technology under contract number PGC2000-2266-E, and by the Department of Universities, Research and Information Society of the Catalan Government under contract number ARCSO1-152.
We would like to thank the Centre de Recerca Matemàtica and its director Manuel Castellet for sponsoring and hosting this Advanced Course, the CRM administrative staff Maria Julia and Consol Roca for smoothly working out innumerable details, and Caries Curràs and Eva Miranda for the mathematical organisation of the course and for making it such a pleasant experience. Thanks are also due to the participants of the course for their many comments on the material. It was indeed a very stimulating audience.
Part A
L agrangian S ubmanifolds Michèle Audin
Introduction This text is an introduction to Lagrangian and special Lagrangian submanifolds. Special Lagrangian submanifolds were invented twenty years ago by Harvey and Lawson [18]. They have become very fashionable recently, after the work of McLean [25], leading to the beautiful speculations of Strominger, Yau and Zaslow [32] and the remarkable papers of Hitchin [19, 20] and Donaldson [11]. My aim here is mainly to present as many examples as possible. I have taken some time to explain why we know so many Lagrangian and so few special Lagrangian submanifolds and immersions. There are mainly two reasons:
• To be Lagrangian is, eventually, a linear property. On the other hand, the property to be special Lagrangian is, in dimension 3 and morc, non linear. • The moduli space of Lagrangian submanifolds that are close to a given one is an infinite dimensional manifold, while the corresponding moduli space of special Lagrangian submanifolds is finite dimensional. This will be apparent in the number and nature of the examples I describe in these notes.
To prepare these lectures, in addition to the papers mentioned above, I have used standard textbooks on manifolds and vector fields as [22], on symplectic geometry as [4, 7, 24, 30] and on complex manifolds and Hodge theory as [8, 15].
I have used standard notation but, although this text pretends to be written in English, I have kept a preference for (transparent) French standards, for instance
P'1 (K) for the projective space of dimension n over the field K and tA for the transpose of a matrix A. I thank Etienne Mann, Edith Socié, Thomas Vogel and Jean-Yves Welschinger for their comments and their help during the preparation of these notes. Special thanks to MiMi Damian, Alicia Jurado and Sébastien Racanière.
Chapter I
Lagrangian and special Lagrangian immersions in C n S
S
To In this chapter, 1 define Lagrangian and special Lagrangian immersions in is the standard real vector space endowed with a begin with, I explain that non degenerate alternated bilinear form 1.1) and use this "symplectic structure" 1.2, 1.3 and 1.4). Later, I use to define Lagrangian subspaces and immersions the complex structure as well, to define special Lagrangian immersions 1.5).
1.1
I.1.a
symplectic vector spaces
Symplectic form on Symplectic vector spaces
with the Hermitian form
Consider the vector space
(Z,Z') = (note that it is anti-linear in the first entry and linear in the second). Decompose
it in real and imaginary parts:
(Z, Z') = (Z, Z') —
Z').
The real part is the standard scalar product (Euclidean structure) of C'1 = R'
(Z, Z') =
+
= X X' + Y . .
x
InCa
6
a symmetric non degenerate (real) bilinear form. The imaginary part defines a (real) bilinear form
that is alternated, this meaning that w(Z, Z) =
0
for all Z. Equivalently, w is
skew-symmetric, that is,
w(Z',Z) = —w(Z,Z'). To write these formulas, I have decomposed the complex vectors of
as
Z=X+iY, and I have used the scalar product X• Y of
The form w is non degenerate
too, as
X =0.
w(X, Y) =0 for all Y
More generally, on a real vector space E, a symplectic form is a non degenerate alternated bilinear form. A vector space endowed with a symplectic form is said to be a symplectic vector space.
I.1.b Symplectic bases Fix a complex unitary basis (ej,.
.
.
(e1,.. is
a basis of the real vector space
,e,,) of Ctm. Put .
.
=
—ie3,
so that
.
Compute w on the vectors of this basis:
= Im(e;,ej) =
1m513
=
0,
also
w(f2,f3) =
= Im(e2,e3) =
0
and eventually
=
—ie3) =
=
Inspired by these properties, we say that a basis (ej,. symplectic vector space is a symplectic basis if
=
and
=
. .
, e,,,
fi,.
.
.
, f,,) of a
0 for all i and j.
There are symplectic bases in all symplectic spaces, thanks to the following proposition.
Li. Symplectic form on C", symplectic vector spaces Proposition 1.1.1. Let to
be
a symplectic form on a finite dimensional vector space of E such that and =
E. There exists a basis (ei,... w(ej,ej) = = 0.
Proof As to is non degenerate, it is not identically zero so that one can find two vectors e1 and Ii such that w(ej, 11) = 1. One then checks that the restriction of to to the orthogonal complement (with respect to to) of the plane (ei, is non degenerate. One eventually concludes by induction on the dimension once noticed that an alternated bilinear form on a 1-dimensional vector space is zero. 0
In particular, the dimension of P is an even number and this is the only invariant of the isomorphism type of (E, to). If E has dimension 2n, then P with its symplectic form is isomorphic to C" with the form to. This result can be called a "linear Darboux theorem", in reference with the forthcoming (Darboux) theorem 11.3.6.
More generally, an alternated bilinear form has a rank, that is the dimension of the largest subspace on which it is non degenerate, and is an even number. Matrices
In a symplectic basis, the matrix of the symplectic form is
(
0
Id
Notice that the matrix J satisfies
J2 =
— Id.
As the matrix of an endomorphism, this is a complex structure. In the symplectic basis of C" associated with the canonical (complex) basis (ej,... e4, J is nothing other that the matrix of multiplication by i. ,
I.1.c
The symplectic form as a differential form
One can write to
as
a differential form to
A
dxi.
= This is an exact differential form (the differential of a degree 1-form): to
=
= d(Y• X).
The form A = Y dX is called Liouvifle form (see § 11.1 below). .
8
I.1.d The syinpiectic group This is the group of isometries of satisfies
A transformation g of
w(gZ,gZ') =w(Z,Z')
for
is symplectic if it
all Z,Z' E C".
Call Sp(2n) the symplectic group of the space C" of dimension 2n. Consider all the groups O(2n), GL(n; C), U(n) and Sp(2n) as subgroups of GL(2n; R).
Proposition 1.1.2. The foUov,ing equalities hold
Sp(2n) fl O(2n) = Sp(2n) fl GL(n; C) = O(2n) fl GL(n; C) = U(n). Proof. Let us characterize our subgroups of GL(2n; R):
(1) g e GL(n; C) if and only if g is C-linear, that is, if and only if
g(iZ) = ig(Z) for all Z. For a matrix A, this is to say that AJ = JA. (2) g E Sp(2n) if and only if g preserves w, that is, if and only if w(gZ, gZ') = w(Z, Z') for all Z and Z'. For a matrix A, this is
tAJA =
J.
(3) g E O(2n) if and only if(gZ,gZ') = (Z,Z'). For a matrix A, this is LAA = Id. One
then checks that two of these conditions imply the third:
• (2) and (3) imply that (gZ, gZ') = (Z, Z')
thus that g
U(n) C GL(n; C).
• (3) and (1) imply that
gZ') = w(gZ, —ig(iZ')) = (gZ, g(iZ'))
(Z, iZ') = w(Z, Z')
thus that g E Sp(2n). • in the same way, (1) and (2) imply (3). In matrix terms, the intersection Sp(2n) fl O(2n) is the set of matrices
;j')
GL(n; C) C GL(2n;
such that
I
= Id. This is exactly the condition that U + iV be a unitary matrix.
1.2.
9
Lagrangian subspaces
I.1.e
Orthogonality, isotropy and F° for
Write F' for the Euclidean orthogonal of the real subspace F of
its symplectic (that is, with respect to w) orthogonal. As w is non degenerate, one has dimF + dimF° = 2n = dimRC". and (F°)° = F
Notice however that a subspace and its orthogonal may have a non trivial intersection. The restriction of the non degenerate form w to a subspace is not always a non degenerate form, in contradiction with what happens in the Euclidean case (which is due to the positivity of the scalar product). In other words, all the subspaces of a symplectic space do not have the same behaviour with respect to the symplectic form. See Exercises 1.6 and 1.7.
One says that a subspace F is isotropic if F C F°, co-isotropic if F F°. For instance, a (real) line is always isotropic, as it lies in its orthogonal which is a (real, co-isotropic) hyperplane. Notice that F is isotropic if and only if F° is co-isotropic. Notice also that the dimension of an isotropic subspace is at most equal to n, half the dimension of 1.2
Lagrangian subspaces
I.2.a Definition of Lagrangian subspaces The isotropic subspaces of maximal dimension n are Lagrangian. For instance, C is a Lagrangian subspace. More generally, a subspace generated by "one half" of a symplectic basis is Lagrangian. Conversely, if F is an isotropic subspace of dimension k n, it is possible to complete any basis (e1 of F in a symplectic basis and thus to obtain Lagrangian subspaces containing F. Let us use now the complex multiplication in C" to state: Lemma 1.2.1. A real subspace P of C" is Lagrangian if and only if P' = iP. Proof. This is a straightforward computation:
w(Z, Z') =
0
Im(Z, Z') =
Re(Z,iZ') = (Z,iZ') = 0.
0 0
U
Lemma 1.2.2. Let P be a Lagrangian subspace of C" and let (x1,. be an orthonormal basis of this real subspace. Then n,,) is a complex unitary basis of C". Conversely, if (xi,. .. Zn) a unitary basis of C", the real subspaee it spans is Lagrangian. . .
,
,
InCa
10
an orthonormal basis of the Lagrangian P, the previous is an orthonormal basis of the . lemma says that the basis (x1,. ,
Proof. If (xi,... ,
is
,
- .
thus that (x1,...
real space
(XZ,X3)
is a complex basis of
= (x,,x1) —
=
Moreover, one has
—0,
[J
thus this is a unitary basis. The converse is even more obvious.
I.2.b The symplectic reduction This is a simple but useful operation, essentially contained in the next lemma. Lemma 1.2.3. Let P be a Lagrangian stthspace and F be a co-isotropic subspace of slAch that
Then the restriction of the projection
PnFcF is injective, the space F/F° is symplectic and the image of P fl F is a Lagrangian subspace.
Proof The symplectic form of clearly induces a non degenerate form on F/F°, as F° is the kernel of the restriction of w to F. The kernel of the composition
PnFcF is
P n F fl F° = P fl F° F being co-isotropic, F D F° = (P° + F)°, since (A + B)° = A° fl B°,
=(P+F)° asPisLagrangian,P=P° =(C'1)° =0
as w is non degenerate.
The map is thus injective. Eventually P fl F is isotropic and has dimension dim P fl F = dim P + dim F - dim(P + F) = dim F
- n,
half the dimension of the symplectic space F/F°, that is
dim F/F° = dimF
—
(2n
—
dim F)
2(dimF — n).
0 See more generally Exercise 1.9.
1.3.
The Lagrangian Grassmannian
11
The Lagrangian Grassmannian
1.3
We consider now the set A,, of all Lagrangian subspaces of
I.3.a The Grassinannian
as a homogeneous space
Look again at lemma 1.2.2. If P1 and P2 are two Lagrangian subspaces of choose an orthonormal basis for each. We thus have two unitary bases of C11. There exists a unitary transformation (an element of the unitary group U(n)) that maps the basis of P1 on that of P2... and thus a fortiori the Lagrangian P1 on the Lagrangian P.2.
In other words, the group U(n) acts transitively on the set of Lagrangian The stabilizer of the Lagrangian subspaces of is the group 0(n) of orthonormal basis changes in R'7. We have defined this way a bijection
U(n)/0(n)
A,,
with the help of which we identify the two sets. Notice that this provides A,, with a topology, namely that of U(n)/ 0(n), the quotient topology of the topology of the matrix group U(n). Example 1.3.1. As all lines are isotropic, the space A1 is the space of real lines in C = R2, namely the projective space P' (R). The unitary group U( 1) is a circle and the orthogonal group 0(1) is the group with two elements {±1 }. As
the unitary group U(n) is compact (being closed and bounded in the
space of matrices) and path-connected (exercise), the space A,, is a compact pathconnected topological space.
I.3.b
The manifold
Let us firstly describe a neighbourhood of P E
(4. = {Q
A,,
in A,,. Put
A,, jQn(iP) =CJ}.
This is an open subset: using a unitary matrix, one can assume that P =
but then is the image in A,, of the (saturated) open subset of U(n) consisting of all the unitary bases the real parts of whose vectors form a basis of This is, clearly, a neighbourhood of P.
Lemma 13.2. The open set Up is homeomorphic to the real vector space of all symmetric endomorphisms of P.
Proof. The subspaces Q that intersect iP only at 0 are the graphs of the linear P iP. It is more convenient to call icc the linear map, so that is a linear map from P to itself. Write now that Q is Lagrangian, namely that maps
Vx,y
P,
w(x + içc(x),y + iço(y)) =
0.
InCa
12
We have = — Im(x + iço(x), y + iw(y)) = w(x, y) + 'p(y)) +
w(x + iço(x), y +
= (w(x),y)
y) — (x,
—
P being Lagrangian. The subspace Q is Lagrangian if and only if the last expression vanishes for all x and y in P, namely if and only if is symmetric'. We have thus defined a bijection that maps 0 to P
End Sym(P)
>
Up
W'
0
a homeomorphism. Remark 1.3.3. Consider for instance the "vertical" Lagrangian C is a disjoint union = U where is the set of all Lagrangians that are not transversal to identified with the space of n x n real symmetric matrices.
We see
that
and
is
We intend to prove now that the open sets Up define the structure of a x R"
Notice firstly that any n-dimensional subspace Q of manifold on may be represented by a rank-n matrix Z
=
fx\
with 2n lines and n columns,
the column vectors of which form a basis of Q. Two matrices Z and Z' describe the same subspace if and only if there exists an n x n invertible matrix g E GL(n; R), such that Zg = Z'.
Lemma 1.3.4. The subspace Q is Lagrangian if and only if the two matrices X and
Y are such that tXY Proof. Let u, u' E
tYX.
and let z, z' be the corresponding vectors in Q:
(x\ Note that Xu, YtL,
,
fx\,
Xii and Yu' are vectors of
We compute:
z') = ,((Xu, Yu), (Xv!, Yu')) = (Xii) . (Yu') — (Yu) (Xu') (scalar product in Ri') = tUtXTuI — = (try —
(as
U• V = tUV)
0 'See also Exercise 1.8.
1.3.
13
The Lagrangian Grassmannian
it can be represented Remark 1.3.5. If Q is the graph of a linear map R'2 —' The relation in lemma 1.3.4 simply expresses the fact that by a matrix Z = the matrix A is symmetric. , n} and the Lagrangian subspace Consider more generally a subset J of { 1, by Uj (for simplicity). Denote Pj of R" x RTh spanned by Any element of Ui is described by a unique matrix Z such that, if we extract from .
. .
Z the matrix containing the lines j (for j J) and j + ii (for j J), we get the Moreover, as we have said open sets Uj clearly cover identity matrix. The
it, each of them can be identified with the subspace Sym(n; R) of n x n symmetric matrices. The Uj's, with their identification with Sym(n; R) are coordinate charts. Change of coordinates are given by Sym(n; R)
A'
U,j fl Uj'
'Zj(A)=Z.r(B)i
where Zj (A) is the matrix obtained from
by
Sym(n; R) B
mapping the first n lines on
the lines j (for j E J) and j + n (for j J). The matrix Zj'(B) is obtained by multiplying Zj (A) by the inverse matrix of the (invertible!) matrix of the lines corresponding to J' in Zj (A). The coordinate change A '—' B is clearly smooth (it is actually rational, thus analytic).
Proposition 1.3.6. The Grassmannian dimension
n(n+1)
is a compact and connected manifold of
0
2
I.3.c The tautological vector bundle Consider the space
E P}. Together with its projection on this is a rank-n vector bundle over The a reason why this fiber of E bundle is qualified as "tautological". The property expressed in Lemma 1.2.1, namely pi = iP, is translated, in terms of the bundle in the fact that ®R C, the complexified bundle, is trivial (has a canonical trivialization). The (global) trivialization is the isomorphism of complex vector bundles
(P,x®(a+ib)) I
' (P(a+ib)x).
InC"
14
I.3.d The tangent bundle to The canonical identification of the open subset Up with the space of synunetric with the bundle endomorphisms of P allows to identify the tangent bundle of It is also possible to describe this bundle from the tangent bundle End of U(n). The group U(n) is described as a submanifold of the space of all complex matrices by the equation LAA = Id, so that we have TA U(n) = {X E GL(n; C) I tAX + tXA = O}.
Call u(n) the vector space Tjd U(n) of skew-Hermitian matrices. There is an isomorphism u(n) TA U(n) >
identifying the tangent bundle T U(n) with the trivial bundle U(n) x u(n) — as image in any Lie group, U(n) is parallelizable. Consider the Lagrangian of the identity matrix Id. One can write
TRn(U(fl)/O(n)) = u(n)/o(n), this is the quotient of the vector space of anti-Hermitian matrices by that of skewsymmetric real matrices. We thus identify
= iSym(n;R), as the real part of a skew-Hermitian matrix is skew-symmetric and its imaginary part is symmetric. Let P be any Lagrangian subspace. Choose a unitary matrix A such that P=A. As we have identified the quotient u(n)/o(n) with the subspace i Sym(n; R) of u(n), we identify the quotient T(A1An with a subspace of TA U(n):
iSym(n;R)
T(AAfl
I
u(n)
We derive an isomorphism
iSym(n;R) —-f X Remark 1.3.7. This isomorphism depends on the choice of A, this is why it does not follow that is parallelizable (it is actually not, as soon as n 2).
1.3.
The Lagrangian Grassmannian
15
I.3.e The case of oriented Lagrangian subspaces of oriented Lagrangian subspaces. Replacing One can also consider the space "orthonormal basis" by "positive orthonormal basis" in what precedes, we get an identification of with U(n)/ SO(n).
I.3.f The determinant and the Maslov class The "determinant" mapping det : U(n)
St
descends to the quotient by SO(n) and, in the same way, its square
S'
det2 U(n)
to the quotient by 0(n). This allows to compute the fundamental groups of and
Proposition 1.3.8. The fundamental group of (resp. is isomorphic to Z. The covering shows as an index-2 subgroup in Proof. Recall first that the group SU(n) is simply connected. This can be proved by induction on n: SU(1) is a point and SU(n + 1) acts transitively on the unit + 1 of sphere with stabilizer SU (n), so that the exact sequence ir1 SU(n)
1r1 SU(n + 1)
gives the result. As the determinant mapping
S'
det : U(n)
is a fibration with fiber SU(n), it induces an isomorphism :
in(S1).
7r1 U(n)
The fiber of the determinant mapping simply connected, thus
S' is SU(n)/ SO(n), which is in1 S'
in1
is an isomorphism. What is left to prove is a consequence of the fact that the diagram
det
det2
SI is
commutative.
j
o
InC'2
16
"The" generator of class" the cohomology class
is called the Maslov class. One also calls "Maslov
that it defines by duality. Using the notation of Remark 1.3.3, it can be shown that (see [1, 12]). is the dual class to the integral homology class represented by
1.4
Lagrangian submanifolds in
We are going now to globalize the notion of Lagrangian subspace, considering submanifolds of C'2 whose tangent space at any point is Lagrangian. We will not really need actual submanifolds, but maps
f:V from some n-dimensional manifold to
C" the tangent mapping of which
is an injection for any point x of V, with image a Lagrangian subspace. It is then said that f is a Lagrangian immersion. For instance, any immersion of a curve (real manifold of dimension 1) in C is a Lagrangian immersion. Any product of Lagrangian immersions is a Lagrangian immersion (into the product target space), we thus obtain Lagrangian immersions of tori (products of circles). Our next aim is to describe examples of Lagrangian submanifolds and immersions in C" and to give a necessary (and sufficient) condition for a given manifold to have a Lagrangian immersion into C".
I.4.a Lagrangian submanifolds described by functions We consider firstly graphs.
Proposition 1.4.1. The graph of a map F: R" —' (i)R" is a Lagrangian submanifold if and only if F is the gradient of a function f : R" R. Proof. The tangent space to the graph at the point (x, F(x)) is the graph of the differential of F at the point x. This graph is a Lagrangian subspace if and only if is a symmetric endomorphism (see the proof of Lemma 1.3.2). The matrix is symmetric for all x if and only if the differential form over R'2 is closed or, equivalently, exact:
F = Vf.
1.4.
Lagrangiari submanifolds in C" See, more generally, Proposition 11.2.1. The Lagrangian submanifolds obtained as graphs have a very specific prop-
erty: the projection of the Lagrangian submanifold on R" is a diffeomorphism. We would like to consider more general Lagrangian immersions, for instance immersions of compact manifolds. Here is a way to construct Lagrangian immersions using the reduction process of § 1.2.h. We start from a Lagrangian submanifold2 We want to construct a Lagrangian immersion into C". To write C" LC
as F/F°, we choose the co-isotropic subspace F = C" Rk, the orthogonal of which is F° = 0 Rk. We suppose that the submanifold L is "transversal to F" in the sense that, for all x,
+F= The Lagrangian subspace L thus satisfies the assumption of the reduction lemma (Lemma 1.2.3). Hence the composition
is the injection of a Lagrangian subspace.
Consider now the intersection V of the submanifold L with F. With the transversality assumption we have made on L, V is an n-dimensional submanifold of F (a consequence of the inverse function theorem) whose tangent space V is the intersection of with F. Thus, the reduction lemma asserts, at the level of each tangent space, that, for all x in V = L fl F, we have the injection of a Lagrangian subspace
In other words, the composition
V=LnFCF is a Lagrangian immersion. Remark 1.4.2. Even if one starts from a Lagrangian submanifold, what we get in general is only an immersion.
Generating functions We generalize the "graph" construction, using the reduction process as explained. Let us start with a nice and useful example. Example 1.4.3 (The Whitney immersion). Consider the unit sphere in
R and the map
a Lagrangian immersion.
1x112
+ a2 =
f:S" (x,a)'
20r
f
'(l+2ia)x.
i}
18
The tangent space to the sphere is + aa =
x RI x
= {(e,a) E
O}
and the tangent mapping to f is T(x,a)f :
C'2
a) S'2
+ 2i(ae + ax).
I
is injective for all (x,a) e S'2: if = 0, then = 0 if x = 0, we have a = ±1 and the equality x + aa = 0 gives a = 0. Thus we have = 0 and a = 0, so that f is an immersion. Moreover, we have The map
and ax =
+
0;
+ dx)) =
+ ax),e' +
(at' + dx) —
+ ax))
so that the image of is an isotropic subspace of dimension n, a Lagrangian subspace. In conclusion, the map f is a Lagrangian immersion. It has a unique double point (North and South poles of the sphere are mapped to 0). In dimension 1, this is a "figure eight". Below (in § I.4.b) we will draw pictures in dimensions 1 and 2.
Obviously, the Whitney sphere is not the graph of a map from to R'2. Let us show that it can nevertheless be described from the graph of a. map defined on a larger space. We start from a function
As we have seen it above, the graph of Vf is a Lagraugian subspace of We reduce as in § I.2.b using the co-isotropic subspace F = C'2 Rc. Here we intersect the graph of Vf with F, namely we consider 0a1
j
Oak
The transversality assumption above is equivalent to the assumption that V is a submanifold of R'2 x in other words that the map
R'1 x Rc (x,a) I is
(Of '
Of Oak
a submersion along V. In terms of partial derivatives, this is to say that the
matrix
(82f\ I
1.4.
19
Lagrangian subrnanifolds in
has maximal rank k. In terms of tangent subspaces, this is to say that the Lagrangian subapaces that are tangent to the graph of Vf are transversal to the co-isotropic subspace F. The reduction lemma 1.2.3 says that the map V
(x,a) is
Of Ix'—
\
-,
I
Ox1
of —
a Lagrangian immersion.
Example 1.4.4 (The Whitney immersion, again). With k =
1
and f(x, a) =
a3
2
2ax gives the x R, and Of/Ox C equation which describes the sphere Whitney map. Example 1.4.5 (Unfolding). Unfoldings are deeply related with Lagrangian submanifolds (see (2]). I will not explain here the general theory but rather show an example. Let P E R.[X] be a degree-(n + 1) polynomial an
P(X) =
+
+
R. These coefficients are going to vary, this is the reason where Xj,. . , the polynomial corresponding to x = why they are named as variables. Call and consider the map (x1,. . .
.
(xj,.
. .
I
P1(a)
to which we apply the previous techniques. The manifold V is
V=
a)
a) E
O}
(zeroes of its derivative when x varies. The this is the set of critical points of condition that V actually be a submanifold is that the matrix of partial derivatives
(02f has
rank 1. But
=
= (ii +
+ (n —
1)xia"2 +... +
InCtm
20
so
that
is
identically 1. Thus V is indeed a submanifold. The La-
grangian immersion is V
(x,a) I
/ 1
8Px
—(a),
.
.
(a)
ax1
instance, starting from the family
For
X4+xiX2+x2X, we
get
V = {(xi,x2,a) E R3 4a3 + 2x1a + x2 = 0) and the Lagrangian immersion from V into R2 x R2 is the map I
(xl,x2,a) Figure
(x1,x2,a2,a).
1.1 shows V with its projection on the plane R2 of coefficients (x1, X2). The
cusp curve is the discriminant of the family of degree-3 polynomials, the set of points x such that has a multiple root. It is obtained here as the set of critical values of the projection V —, R2. Over such a point x in the space of coefficients are the (one or three) roots of the polynomial
Figure 1.1: The discriminant of degree-3 polynomials
I.4.b
Wave fronts
Exact Lagrangian iininersions
If /
V —, is a Lagrangian immersion, the 2-form f*w is zero, so that d(f*A) = 0 and f* A is a. closed 1-form on V. If, for some reason, for instance because Hj,R(V) = 0, this form is exact, there exists a function
F:V
1.4.
21
Lagrangian subrnanifolds in
such that f*A = dF. The immersion f is qualified as exact Lagrangian immersion. The mapping
Fxf:V
C"xR
has the property3
=0.
(1 xF)* Wave fronts
Instead of looking at the Lagrangian immersion f, consider the projection
fxF
R'1xR (X+iY,z)
(X,z).
We will assume here that, at a general point of the Lagrangian, the tangent space is transversal to the subspace of coordinates Y. The image of the Lagrangian imx R. This hypersurface is the wave front. mersion is then a hypersurface of
Of course, it will in general be singular. Precisely, at a point of V where V is X is singular. However, as the projection X + iY not a graph over at every point of the wave front, there is a y3dxj) = 0, (f x F)* (dz — tangent hyperplane, the hyperplane in the space
z
R of (x, z) coordinates
=
at the point image of (X, Y, z). Notice that, as the coefficient of z in this equation is non zero, the hyperplane is always transversal to the z-axis. Conversely, R has at every point a tangent hyperplane if a singular hypersurface of that is transversal to the z-axis, this hyperplane has a unique equation of the Y3x3 and it is possible to reconstruct a (maybe singular) Lagrangian form z = submanifold from the "slopes" Y3. We begin with an example of dimension 1, that of the Whitney immersion
again. Notice that this is indeed an exact Lagrangian immersion: the restriction of the Liouville form ydx to the curve is exact because f ydx 0 (the area surrounded by the curve is zero). A primitive of ydx is easily found. The curve is parametrized by t '—i (cost, sin 2t) and
ydx = —2sin2tcost = 3The manifold C" x R is a "contact manifold" and F x f is a "Legendrian immersion" lifting the Lagrangian immersion F.
InCa
22
Figure 1.2: Eye
Figure 1.3: Crossbow
A map to the (x, z) space is thus t'—' (x,z) = This is depicted on Figure 1.2, in an old-fashion "descriptive geometry" mood. It can be seen that the singular points of the (x, z) curve correspond to the tangents to the (x, y) curve that are vertical, and that the double point of the latter corresponds to the two tangents to the wavefront (the "eye") at points with the same x coordinate that are parallel. Figure 1.3 represents an example in which we start from the wave front (a "crossbow") to reconstruct the Lagrangian. From the wave front, it is seen that the Lagrangian curve has two double points and two "vertical" tangents.
Figure 1.4: Flying saucer
Figure 1.5: Cylinder
One could wonder what it is useful for to replace an immersed curve by a singular one. Notice that, in higher dimensions, the wave front is a hypersurface in x RTh. Even for R and it replaces a submanifold of the same dimension n in n = 2, this is very useful as this allows to represent exact Lagrangian surfaces of R4 by (singular) surfaces in a dimension-3 space. Here are some beautiful examples. Rotate the eye (Figure 1.2) about the z-a.xis to get the flying saucer depicted on
Figure 1.4. The corresponding Lagrangian surface in R2 x It2 is a Lagrangian
1.4.
Lagrangian submanifolds in
23
immersion of the dimension-2 sphere in C2 with a double point. In Exercise 1.14,
one checks that this is, indeed, the Whitney immersion... eventually drawn in dimension 2! Figurc 1.5 represents a cylinder constructed on the eye, namely a Lagrangian immersion of a cylinder, product of a figure eight with an interval, with two whole lines of singular points.
Singularities
Wave fronts are, as we have said it, singular hypersurfaces. We have seen, in dimension 1, cusps, in dimension 2, lines of cusps, but this can be more complicated, as Exercise 1.15 shows it.
Wave fronts of non exact Lagrangian linmersions
Wave fronts are so nice that it is a pity not to have them for all Lagrangian immersiolls. In dimension 1, the problem is to represent by wave fronts curves that do not surround a zero area. Consider for instance the standard (round) circle in C. As f ydx 0, it seems that nothing can be done. Look, however, at the parametrization (cost,sint).
t
It gives
ydx =
—
sin2
tdt =
d
(sin2t
—
z
Figure 1.6: Wave front of the circle
24
Nothing forbids us to represent the Lagrangian (non exact) immersion of the circle by a piece of the (non closed) wave front4 parametrized by
tt
I
sin2t
t
and depicted on Figure
Figure 1.7
If we rotate the (unbounded) wave front of Figure 1.6 around a line parallel to the z-axis that does not intersect the wave front, we get the wave front of a Lagrangian torus, the one depicted on Figure 1.7. One can then use the cylinder represented on Figure 1.5 to perform connected sums of wave fronts. This way, Figure 1.8 represents (the wave front of) a genus-2 Lagrangian surface. In the same way, one constructs Lagrangian immersions of all orientable surfaces in C2.
These figures are copied from Givental's paper [13], that contains many other examples.
Figure 1.8: A genus-2 surface
Remark 1.4.6. Except for the torus, all the surfaces depicted here have double points, that show up in the wave fronts as points having the same projection on the horizontal plane and parallel tangent planes. It is rather easy to prove that the torus is the only orientable surface that can be embedded as a Lagrangian submanifold in C2. As for non orientable surfaces, they can be embedded as Lagrangian surfaces when (and only when) their Euler characteristic is divisible by 4 (with the exception of the Klein bottle). See the pictures in [13]. As for the Klein bottle, it has long been unknown whether it had or had not a Lagrangian embedding. Mohnke [27J has recently proved that it has not. 4This is a place where one can really appreciate the difference between closed and exact
1-forms.
5Notice that wave fronts are defined only up to a "vertical" translation, the actual constant
C used in Figure 1.6 is (ir + 1)/4.
1.4.
Lagrangian submanifokis in C"
25
Exact Lagrangian embeddings
Notice that, in all the examples of exact Lagrangian immersions we have given, there are double points. This is obviously necessary in dimension 2 (n = 1), due to Jordan theorem: an embedded curve cannot surround a zero area. This is also true in higher dimensions, due to a (hard) theorem of Gromov [16]: there is no exact Lagrangian submanifold in C".
I.4.c
Other examples
Here are a few other examples.
Consider the map
Sym(n;C)
U(n) A
from the group U(n) to the complex vector space of symmetric matrices. tAA defines a Lagrangian immersion
Proposition 1.4.7. The map A
Sym(n;C).
Proof As tAA = Id when A 0(n), the map 4 is well defined. Call [A] the class of a unitary matrix A in We have seen in I.3.d that the tangent space to at the point [A] can be identified with = {AH H I
It is mapped into Sym(n; C) par AH
iSym(n;R)}.
as follows
'tA (AHLA + AtHtA) A.
The matrix AHtA+AtHtA has the form K-k for K = AWA =
in u(n) and this describes all the matrices in the vector space i Sym(n; R) when H varies in i Sym(n; R). The image of the tangent mapping TEAl 4' is, thus, the subspace p(A) . i Sym(n; R) where
p:U(n) is the representation of U(n) operating on complex symmetric matrices. This image is, indeed, a Lagrangian subspace, being the image of the real part of the complex vector space i Sym(n; C) by a unitary matrix. 0
In C" Tori,.intpgrabte systems
systems (mechanical systems with many conserved quantities) yield iii'any Lagrangian tori. We use here a few standard symplectic notions: Hamilt9nian vector fields, Poisson bracket, commuting functions. See if necessary Ap= is a pendix I 6.c. Recall for instance that an integrable system on —+ R" whose components ft,... , f,, are functionally independent map f : commuting functions.
which is locally free at the regular This defines a local R''-action on are points of the system (the points at which the derivatives of the functions actually independent). Call X1, . , X,., the Hamiltonian vector fields associated These vector fields commute: with the functions . .
=
=
0.
The R'2-action is given by integration: t x=
E R'' is close to 0 (in order and t = (t1,.. denotes the flow of be defined). This local action is indeed locally free on the open set of that give independent tangent vectors at regular points because the vector fields where
. ,
these points. are complete, namely that the Assume moreover that the vector fields flows are defined for all values of t. We then have a locally free action of R'' on the whole set of regular points. The vector fields being tangent to the common this action preserve the level sets. The connected components level sets of the
of the regular level sets of f are thus homogeneous spaces, quotients of R" by discrete subgroups. The discrete subgroups of R'' are the lattices Z' in the linear suhspaces of dimension k. The connected components of the regular level sets are thus diffeomorphic to Rn—k x Tk for some k such that 0 k n. In particular, the compact connected components are tori T" and these tori are Lagrangian6, they are called the Liouviile tori. The next proposition is the easiest part of the Arnold-Liouville theorem (see for instance [2, 6]). Proposition 1.4.8. Compact connected components of the regtdar common level sets of an integrable system are Lagrangian tori. 0
There are many examples of integrable systems and thus of Lagrangian tori, coming from mechanical systems (spinning top, pendulum. ..)7. The most classical example is that of the standard action of the torus
T''={(ti,...,tn)EC'2IItjI=1, i=1,...,n} 6Notice that on a compact connected component, the flows are complete. 7See for instance (6].
1.4.
on
Lagrangian submanifolds in by
(21,.... z,,) (tjzi,. . the orbits of which are the common level sets of the functions (t1,.
.
.
,
.
,
1
1
2
91 =
,9n =
'
2 IZnt
x 51 indeed, for the regular values of the gz's (namely every g2 non tori S1 x zero). We will come back to these examples in § I.5.e.
Normal bundles
be any immersion of a k-dimensional manifold into R". Let now I : V —+ Consider the total space of its normal bundle
Nf = {(s,v)
x R" Is
V
It is naturally mapped into
x
vE
V,
by
f: Nf
(f(x).v).
(x,v)
The manifold Nf has dimension k + n — k = n,
and I
is
clearly an immersion.
Moreover, it is Lagrangian. More precisely, we have:
Lemma 1.4.9. If A is the Liouville form on
x
one has f*A = 0.
Nf. Use the commutative diagram
Proof. Consider a vector X E
If
NI
irj
J.
V
I
f
to compute
(j*A) (xv)
(X) = =
(T(X.L)J(x)) 0
= V (T1f 0 T(XV)lr(X)) = 0 since v is orthogonal to
0 This method allows to construct many (non compact) examples and can be generalized by replacing x R't = T* by the cotangent bundle T* M of a manifold and V —+ by an immersion into M. See § 1I.2.a.
InCtm
28
I.4.d
The Gauss map
Let f : V —' C" be a Lagrangian immersion. Its tangent space at any point is a Lagrangian subapace of C". One can globalize the data consisting of all these tangent spaces to define the "Gauss map" V
,y(f)
xI
By definition of the tautological bundle
I.3.c), one has
= TV. In particular, the tangent bundle to V must have the same properties as Proposition 1.4.10. For a manifold to have a Lagrangian immersion into C's, it is necessary that the complexification of its tangent bundle be trivializable. 0 The converse is true, but less easy to prove. This is an application of Gromov's h-principle [17], see also [23]. (1) Spheres. We have seen examples of Lagrangian immersions of spheres in C" (in I.4.a). One deduces that TS" C is a trivial complex bundle. Notice however that it is not true that the tangent bundle TS" itself is trivial (except for n = 0, 1, 3 and 7).
Examples 1.4.11.
(2) Surfaces. All orientable surfaces and half the non orientable surfaces have Lagrangian immersions in C2 (as we have seen it in §I.4.b). This is not the case, neither for the real projective plane nor for the connected sums of an odd number of copies of this plane.
(3) Normal bundles. This is a case where the tangent bundle itself is trivial (before complexification):
=
e TZV,U
and this is canonically isomorphic to the ambient space
(4) Grassmannians. The Gauss map
of the Lagrangian immersion 4)
satisfies of course
= End
1.5.
29
Special Lagrangian subinanifolds in
The Maslov class
Every Lagrangian immersion has a Maslov class: use the Gauss map
to pull back
to a class
E
p(f) E H1(V;Z). One can also, with the notation of Remark 1.3.3, define see [24] for example. ogy class dual to
1.5
as
the cohomol-
Special Lagrangian submanifolds in
whose tangent space at each Lagrangian submanifolds are submanifolds of point is a Lagrangian subspace. They have a Gauss map into the Grassmannian namely into U(n)/ 0(n). We look now at the submanifolds whose Gauss map takes = SU(n)/ SO(n). These are the special Lagrangian submanifolds, values in invented by Harvey and Lawson [18}.
I.5.a Special Lagrangian subspaces is said to be special Lagrangian if it has a positive An oriented subspace P of orthonormal basis that is a special unitary basis of C'2. For instance, if n = 1, as C has a unique special unitary basis (the group SU(1) is the trivial group), there is only one special Lagrangian subspace in C, the line R c C... this will not be a very interesting notion in dimension 1. Fortunately, for n ? 2, this is more exciting. Identify the space C2 with the skew-field H of quaternions:
Z=(zi,z2) =X+iY =(x1
+iy1,x2+iy2)
= (x1 +iy1)+j(x2+iy2) = (Xi + jX2) + i(yi — jy2). The 2 x 2 matrices that are in SU(2) are the matrices of the form
(zi
with ziI2 +
)z212
= 1.
Thus the special Lagrangian planes are those who have an orthonormal basis (Z, Z') with Z and Z' of the form
JZ=(xj +iy1)+j(x2+iy2) —iy1).
30
Notice that
Z' = [(xi +iy1)+j(x2 = Zj. Thus a basis (Z, Z') of C2 is special unitary if and only if Z' = Zj. Now use multiplication by j to give H the structure of a complex vector space. One has: Proposition 7.5.1. The special Lagrangian subspaces of C2 are the complex lines with respect to the complex structure defined by the multiplication by j. The Grassmannian SA2 is a complex projective line. 0
Remark 1.5.2. Notice also that Sit2 = SU(2)/SO(2)
S3/S'. This is indeed a
dimension-2 sphere.
To distinguish the special Lagrangian subspaces among all the Lagrangian subspaces or the special unitary matrices among all the unitary matrices, one uses the (complex) determinant. To globalize the notion of special Lagrangian subspace and define special Lagrangian submanifolds, it will be practical (and natural) to describe the linear objects by differential forms. The form corresponding to the complex determinant is Il = dxi A ••. A Expressing the definition of the determinant, namely
= (dot A)e1A
(Aei)A
A
e GL(n; C), we have indeed (detA)1l. Hence
detA =
1
A*fl
In order to work with real subspaces, we need an additional notation: call the two degree n real forms:
and
a=Refl, For instance, in dimension 1, 11 = dx,
= dx and L9 =
dy. In dimension 2,
= dzj A dz2 (dx1 + idy1) A (dx2 + idy2) =dx1 Adx2 —dy1 Ady2+i(dyi Adx2 +dx1 Ady2), that is
f
dx1 A dx2 — dy1 A dy2
Adx2+dx1 Ady2.
Proposition 7.5.3. Let P be an oriented (real) vector subspace of dimension n in The number A ... A depends only on P and not on the positive orthonormal basis (x1,. , of P used to express it. .
.
1.5.
31
Special Lagrangian subinanifolds in
x,,, ix1.... , ix,,) and the linear mapping Proof. Consider the 2n vectors (x1 defined by the images of the vectors of the canonical basis: A C" —+ C" x3.
A(e3)
A(ie3) = ix)
(so that A is complex linear). Then
cl(xiA ... A x,,) = detc A.
If (gx1,... ,gx7,) is a positive orthonormal basis of P (that is, if g
SO(n)), one
gets cI(gx1
(since g
Agx,,) = detc(gA) = detcgdetcA = detRgdetcA = detc A = cl(x1A •••A x,,)
0
SO(n) C GL(n; R) C GL(n; C)).
We will thus denote 11(P) the number 11(x1 A- . Ax,,). Similarly, denote ci(P) its real and imaginary parts. and Remark 1.5.4. Notice that 11(P) is non zero if and only if the 2n vectors .
(x1,.. .
,ix,,)
form a basis of C" over R, that is, if and only if P fl iP = {O} or P does not contain any complex line. These subspaces are said to be totally real. This is in particular the case for Lagrangian subspaces.
Proposition 1.5.5. A real subspace P of C" has an orientation for which it is a special Lagrangian subspace tf and only if P is Lagrangian and j3(P) = 0. Proof Let P be a Lagrangian subspace. Choose (x1 an orthonormal basis which is the image of the canonical basis of C" by a unitary matrix A. Thus
S'.
11(P) = detc A
For P to have a positive basis that is special unitary, it is necessary and sufficient
that detc A be equal to ±1, that is, that 1m11(P) =
0.
0 Here
is a last elementary remark on linear suhspaces:
Proposition 1.5.6. Let Q C C" be an oriented isotropic linear subspace of dimension n — 1. There exists a unique special Lagrangian subspace that contains Q. Proof. Choose a positive orthonormal basis (x1,. . , x,, of Q. In the complex line that is the orthogonal, with respect to the Hermitian form, of the complex subspace spanned by the xi's, there is a unique vector such that the basis is a special unitary basis of C". 0 .
InCa
32
I.5.b Special Lagrangian submanifolds A Lagrangian immersion
f:V
is a special Lagrangian is special if of an oriented manifold into C subspace for every x. The Gauss map then takes values in (1) In dimension 1, the tangent space must be the unique special Lagrangian R C C for all x. If V is connected, f must thus be the immersion of an open subset of R by t '—* t + ia. We have already noticed that this dimension will not be very exciting.
Examples 1.5.7.
must be a j-complex line for all x, is thus the (2) In dimension 2, immersion of a j-complex curve into C2. This gives quite a lot of examples. is Remark 1.5.8. The Maslov class of a special Lagrangian immersion into zero. Of course, as the examples above show it, there are much more Lagrangian immersions with zero Maslov class than there are special Lagrangian immersions.
In terms of forms, to say that the immersion
f:V is special Lagrangian is to say that it satisfies
• firstly f*w =
0
(it is Lagrangian)
• secondly
=
0
(it is special).
Proposition 1.5.9. If f is a special Lagrangian immersion, f"T is a volume for,n
onV. Proof. The complex form has type (n, 0) and defines an n-form on V, which is real since its imaginary part vanishes on V. Let x be a point in V and let One has (f*fl)
=
because of Remark 1.5.4 and since V is Lagrangian. Thus
never vanishes. 0
In dimensions 1 and 2, the special Lagrangian submanifolds are non compact (in dimension 2, Liouville's theorem forbids complex curves in C2 to be compact). This is actually always the case, a straightforward application of Proposition 1.5.9: Corollary 1.5.10. There is no special Lagrangian immersion from a compact manifold into C't.
___ 1.5.
33
Special Lagrangian submanifolds in
C" is a special Lagrangian immersion, is an exact complex form:
Proof. If f: V V. But
Decompose z1 dz2 A
is a volume form on
into its real and imaginary parts to get
A
a. =
d Re(z 1dz2 A
and eventually
... A
= dil
= f*a. =
The manifold V thus has an exact volume form, and this prevents it of being
0
compact. Let us give now examples of special Lagrangian submanifolds in from the examples of Lagrangians constructed in section 1.4.
I.5.c
starting
Graphs of forms
Let us begin with Proposition 1.4.1. Let f: R" —i R be a function. We require the graph of Vf, a Lagrangian submanifold, to be a special Lagrangian submanifold. The n = 1 case is not interesting. For n = 2, the Lagrangian immersion associated with the function f is
F:(x,y)'-
(
ofof
and the form /3 is /3 =
dy1 A dx2
+ dx1 A dy2.
Then
= We thus have:
Proposition 1.5.11. Let U be an open subset of R2 and f : U R a function of class e2. The graph of Vf is a special Lagrangian submanifold of C2 if and only 0 if f is a harmonic function. Notice that the condition is linear. Starting from dimension 3, this is no more the case. The function f must satisfy a complicated non linear partial differential
34
equation, expressed in Proposition 1.5.12 below. Let us begin by a notation. Denote
by Hess(f) the Hessian matrix of f, namely the matrix = and by crk(Hess(f)) the k-th elementary symmetric functions of its eigenvalues. More generally, for an n x n real matrix A, write det (A —
X Id) =
example, 01 (Hess(f)) is the trace of the Hessian matrix, the Laplacian of f. R a function of Proposition 1.5.12. Let U be an open subset of if and only graph of Vf is a special Lagrangian submanifold of The class e2. if f satisfies the partial differential equation For
= 0. k>O
Examples 1.5.13. For n = 1, the differential equation is f"(t) = 0 or f'(t) constant, and this is precisely the differential equation of the special Lagrangian submanifolds. For ii = 2, again, only oi appears in the (linear) relation, which expresses the fact that the function f must be harmonic. For n = 3, the relation is
01(Hess(f)) = cr3(Hess(f)) or
= det(Hess(f)). Remark 1.5.14. The only order-i term (in dl) in this partial differential equation is so that the "linear part" of this equation is = 0. this should be compared with McLean's theorem (Theorem 11.6.1 below).
Proof of the proposition. The tangent space to the graph of Vf at the point This is the image of the plane RTh under the linear map Id (x, is a special Lagrangian subspace if and only if 0. Im (detc(Id We still must check that, for any real symmetric matrix A, one has
Im(detc(Id+iA)) = k>O
Since A is real symmetric, it is diagonalizable in an orthonormal basis. It is clear that the two sides of the relation to be proved are invariant under conjugation
by matrices in 0(n). One may thus assume that the matrix A is the diagonal (A1,. .. , The left hand side is then Imfl,(1 + iA3) and it clearly coincides with the right hand side.
0
Special Lagrangian submanifolds in
1.5.
35
I.5.d Normal bundles of surfaces Let f V —+ R" be an immersion of a dimension-k manifold into :
We
know (see I.4.c) that its normal bundle has a natural Lagrangian immersion into
x W'. Look now for the conditions under which this is a special Lagrangian immersion. For the sake of simplicity, suppose here that k = in R3). There is a more general discussion in [18].
2
and n =
3
(case of surfaces
Fix a point x0 in V, a unit normal vector field n = n(x) on a neighbourhood of x0. The restriction of the normal bundle
NJ = {(x,v) x E V,v E R3,v I to this neighborhood is isomorphic with V x R by (x, JL) '—* (x, jin(x)). We map
Nf to C3 by iin(x) + if(x) (notice that, this time, the immersion f appears in the second copy of R", that of purely imaginary vectors). Let us now choose an orthonormal basis V. Assume that this , e2) of basis is orthogonal with respect to the second fundamental form, that is, to the symmetric bilinear form defined on V by (x,
I1(X,Y) = We have
= where A1 and A2 are the two "principal curvatures" of V at x0. Consider now the tangent space to Nf at (x0, v) where v = R. n(xo). The tangent mapping to our immersion is —A1e1,
P0
=
=
—'
e
x
+
t
The images of the basis vectors are e1 e2
i
I
ni
(—IL)12e2.e2)
(n,O).
Thus
fZ(P0) = (dz1 A dz2 A dz3) (((i
= (i so
—
pA1)(i
-
1zAi)ci) A ((i
that P0 is a special Lagrangian if and only if p(A1 + A2) = is zero. In other words, we have shown:
the trace of
An)
—
—
0.
This is to say that
InC"
36
Proposition 1.5.15. The immersion of the normal bundle of
f:V
R3
into C3 is a special Lagrangian immersion if and only if / is a minimal immersion.
0
For more information on minimal surfaces, see, for example, the beautiful surveys in [29] and the references quoted there. Remark 1.5.16. It is true that we have already mentioned Riemannian metrics
in these notes, but up to now, they have had only an auxiliary role. The result presented here is a genuine Riemannian one.
I.5.e From integrable systems Being compact, Lagrangian tori obtained as "Liouville tori" cannot be special Lagrangian submanifolds in C". One can try to replace them by special Lagrangian
submanifolds with the help of the remark included in Proposition 1.5.6: the idea is to consider a (necessarily isotropic) subtorus in a Liouville torus 1"' and to add a direction to construct another Lagrangian submanifold, which will be special. Here is an example, coming from [18], of such a construction. Start from an orbit L of the standard action of T'2 on C" (see § I.4.c), namely a common level set of the functions
gi(z)=
1
1
2
2
9i = none of the as's being zero, so that L is a Lagrangian torus. Choose a subtorus ofT": say
Let V be an orbit of this subtorus, an isotropic torus of dimension n — 1. Consider the Hamiltonian vector fields Y1,. , Y,, associated to the functions gj: . .
J z = (zj,... , z,,) be a point of V. The tangent space to L at z is spanned by values of the 1' 's, the tangent space to V is the hyperplane consisting of the vectors >2 X,Y, satisfying >2 A, = 0. It is spanned by the values at z of the vector Let
the
fields
X1 = Yn,. . ,Xn_i = Yn_.i — that are the Hamiltonian vector fields of the functions .
11
=91
9n,...,fn—i
Yn,
—gn.
37
Special Lagrangian submanifolds in
1.5.
We are looking now for an n-th function f such that the subspace spanned by and X1 is a special Lagrangian at each point where the the vectors X1,. . . , is isotropic and has vectors are independent. The subspace F = (X1,.. , .
dimension ri —
1.
such that:
We look for X1 as a linear combination Xf =
• The vector field Xj is in the subspace orthogonal to (X1,. . the Hermitian form), that is, (Xf,Xk) = 0 for 1 k n — must have the form = 0. Thus )'k —
,
(for
1.
This gives
.
2 I
• The determinant iz1
0
0•..
0
... is real. This allows to determine the function p. from the last vector, this Subtracting the linear combination A1X1+• Zn. so that the determinant is t.\1 + vector becomes +
We are thus looking for functions f and p such that is real.
and
For any index j, we must have:
9f
=
--
Of
and z /.i(Z1,.. ,zn)zi
= )t3z3,
.
.
.
ER.
The functions
give a solution when take
E R, namely when n is even. When n is odd, we rather
Proposition 1.5.17. The functions fi,..
fj(zi,..
.
=
.
,
fn defined by
—
.
.
=
1z12)
and
— form an integrable system on x = which are special Lagrangian cylinders
un is odd all the regular common level sets of
x R.
InC'2
38
Proof. The only thing that is left to prove is that the regular levels are "cylinders" T'2' x R. As we are dealing with an integrable system, we know that the levels are endowed with an R'2-action. Here the n — 1 first vector fields are periodic and in particular complete; the last one is complete too, because the level is a closed submanifold of C". The action is thus an action of T'2' x R and this is a free 0 action, as the level, being special Lagrangian, cannot be compact. Exercise 1.19 describes essentially the same construction.
I.5.f Special Lagrangian subinanifolds invariant under SO(n) The next and sporadic examples also come from [18]. Start from a smooth curve
r
C =C x {O} cCx C'2' = C"
and "rotate" it with the help of the diagonal SO(n)-action, namely
g. (X +iY) = g• X +ig• Y forgE SO(n) and X,Y ER". If we assume the curve does not pass through 0, we get a submanifold of C":
V = {(x+ iy)u x+ iy E F,u
R",u = g(ei) for some g€ SO(n)}
(notice that u describes a sphere S'2' C R"). The tangent space to V at (x + iy)u with is spanned by the vectors (x+iy)U with U TUS'21 and the as is easily tangent to r at x + iy. The submanifold V is always checked:
w((x + iy)U, (x + iy)U') = xy(U
U' — U'
U) =
0,
a basis of TUS"1,
It is special Lagrangian if and only if, denoting (U1,. ..
or to + i?7)detc(Uj, . . But this determinant is equal to (x + (x + iy)'2_i + iii) detR(Ul,. .. , Un_i, u) since these vectors are in R" c C". The condition is thus that -
(x +
+ iii)
R for any tangent vector
+
to I'.
We get eventually:
Proposition 1.5.18. The Lagrangian submanifold of C"
V = {(x+iy)uI (x+iy)
r,u€
R"}
is special Lagrangian if and only if, on F, the function Im ((x + iy)'2) is constant.
0
1.6.
39
Appendices
iVL Figure 1.9
Remark 1.5.19. This method gives essentially one special Lagrangian submanifold in any dimension, which is not much! Remark 1.5.20. Any connected component of r is diffeomorphic to It, the special x R. Lagrangian submanifolds obtained are (unions of) copies of
To draw a picture of the special Lagrangian submanifold, one draws first the
curve F (in the (x,y) plane), then its wave front (in the (x, z) plane). One then notices that the Liouville form = Y• dX is, on V:
= (yu) ((dr)u + xdu) = ydx .
(since udu =
=
0)
{(xu,z) E
so that the wave front of V is x
(x,z) is a point of the wave front of F}.
For example, for n = 2, the curve F is a hyperbola xy constant, its wave front is the curve z = log x and the wave front of the special Lagrangian submanifold is the surface of revolution obtained by rotating the graph of the logarithm function about the z-a.xis (Figure 1.10).
1.6
Appendices
I.6.a The topology of the symplectic group Proposition 1.6.1. The manifold Sp(2n) is diffeomorphic to the Cartesian product of the group U(n) with a convex open cone of a vector space of dimension n(n +1). Corollary 1.6.2. The symplectic group Sp(2n) is path connected. The injection of an isomorphism U(n) in Sp(2n)
Z=
ir1
U(n)
ir1 Sp(2n).
In z
Figure 1.10
Proof of the proposition. Let A E Sp(2n). As any invertible transformation of A can be written in a unique way as a product
A=Sfl and fl is the orthogwhere S is the positive definite symmetric matrix S = onal matrix fl = S'A. As A is symplectic, the matrix S is also symplectic: tA and AtA are symplectic, the matrix AtA is symmetric, positive definite, thus it is diagonalizable in an orthonormal basis and S is the matrix that, in this basis, is the diagonal of the square roots of the eigenvalues of AtA, so that S is indeed symplectic as is AtA. One deduces that
= and thus that
S'A
Sp(2n) fl O(2n) = U(n)
is a unitary matrix. We have thus obtained a bijection Sp(2n)
)
U(m) x S
A' where S denotes the set of positive definite symmetric matrices that are symplectic. We still have to prove that this space is an open convex cone in a vector space of
dimension n(n + 1). Write the matrices as block matrices in a symplectic basis. Let S E 5, we have
s=
with A and C positive definite symmetric and tSJS = J.
The last condition, that expresses the fact that S is symplectic, is equivalent to BA is symmetric and C =
A'(Id +B2).
The mapping S
Si
Sym(n;R) x (BA,A)
1.6.
Appendices
41
(n; R) of all positive definite is the desired diffeomorphism. The open set symmetric real matrices is obviously an open convex cone in the vector space Sym(n; R) of all symmetric matrices, the product is an open convex cone of the product space, that has dimension
2n(n± 1)
Proof of the corollary. The convex cone Sym(n; R) x
R) is contractible.
Remark 1.6.3. There is another beautiful proof of this type of contractibility results, due to Sévennec, in
I.6.b Complex structures If E is a vector space endowed with a symplectic from w, it is said that an endomorphism J of E is a complex structure calibrated by w if J2 = — Id (J is a complex structure),
w(Jv,Jw)=w(v,w) (J is symplectic) and
g(v,w) =c,i(v,Jw) is a scalar product (namely a positive definite bilinear form) on E.
I.6.c Hanilltonian vector fields, integrable systems In this appendix, denote for simplicity symplectic manifold W (see § 11.1).
=
by W. It can be replaced by any
Hamiltonian vector fields
To any function H : W R, the symplectic form allows to associate a vector field, a kind of gradient, the Hamiltonian vector field (sometimes called the "symplectie gradient" H). This is the vector field defined by the relation for all YE
= or by
tXHL) = —dH.
In coordinates, one has
Xjj(xj
=
(OH
— \Oyi I
OH OH —,—-—— Oy,, Ox1
OH Ox,,
Notice that the vector field XH vanishes at x if and only if x is a critical point of the function H:
Xy(x) =0
(dH)1 =0.
InCtm
42
In particular, the singularities (or zeroes) of a Hamiltonian vector field are the critical points of a function. Notice also that the function H is constant along the trajectories, or integral is skew symmetric, we have (dH) (XH) = 0 curves, of the vector field XH: as
or XH H =0. The Poisson bracket
Assume now that f and g {f,g} by the formula
are
two functions on W. Define their "Poisson bracket"
{f,g} = X1 g = dg(Xj). In coordinates, one has
c9gaf Notice
that X1 •g = dg(X1) = w(X1,X9) = —w(X9,Xj) = —df(X9) = —X9 .f,
— {g, 1). This shows that the Poisson bracket is skew-symmetric in f and g. By definition, this is also a derivation (in both entries); in other words, the Poisson bracket satisfies the Leibniz identity
so that {f, g} =
{f,gh} = Using
{f,g}h+g{f,h}.
the general relation LXtY — wCx = t[X,y1
and Cartan formula
Lx =
+ Lxd,
we get
= Lx,tx9w — = dtX1LX9W +
—
— tx9tx1dw
—dtx1tx9w—d(w(X9,X1)) = —d{f,g}, in other words
[X1,X9J
X{f,9}.
We also have
[X1,X9].h={{f,g},h}. From this, we deduce that the Poisson bracket satisfies the Jacobi identity
{f,{g,h}}+
{g,{h,f}} + {h,{f,g}} =0
43
Appendices
1.6.
and thus defines a Lie algebra structure on C°°(W), e°°(W)
the
mapping
X(W)
f'
(W) (with the Poisson bracket) into the being a morphism of Lie algebras from Lie algebra of vector fields (with the Lie bracket of vector fields). Proof of the Jacobi identity. Apply the definition of the bracket of vector fields: [Xj,X91 h = X1• (X9 h) — X9 . (X1
.
and the equality above to get
{{f,g}
,h} =
[Xj,X9J
h
=Xf(X9.h)—X9'(Xfh) = X1 .{g,h} — X9 .{f,h} = {f,{g,h}} —{g.{f,h}}.
taking into account the skew-symmetry of the Poisson bracket, is equivalent to the Jacobi identity. U This,
Integrable systems As any vector field does it, the Hamiltonian vector field XH defines a differential system on W, namely, = XH(X(t)),
the Hamiltoniari system associated with H. The function H is constant along the trajectories of this system, in other words
XH•H=OordH(XH)=O. It is said that H is a first integral of the system. More generally, a function R that is constant along the integral curves of a vector field X is f :W called a first integral of X. In the case of a Hamiltonian vector field XH, the
f
equality XH = 0 is equivalent to {f, H) = 0, we say that the functions I and H commute. It is said that a Hamiltonian system is integrable if it has "as many commuting first integrals as possible". Let us explain this: .
• Let Ii,.
..
=
0
be commuting first integrals of the system XH, so that for all i and j. Each one is constant on the trajectories of
the Hamiltonian system associated to each other one.
InC"
44
• The expression "as many as possible": at any point x of W, the subspace of W spanned by the Hamiltonian vector fields of the functions is isotropic:
=
=
0.
Its dimension is thus at most n = dim W. It is required that, at least for x in an open dense subset of W, this subspace has maximal dimension n.
• Notice that the vectors forms
are independent at x if and only if the linear
are independent.
Definition 1.6.4. The function H or the Hamiltonian vector field XH on W is qualified as integrable if it has n independent commuting first integrals. Examples 1.6.5. Every function depending only of the coordinates
is integrable: the functions are independent commuting first integrals. Every Hamiltonian system on C is integrable. Similarly, a Hatniltonian system on C2 is integrable if and only if it has a "second first integral".
Exercises Exercise Li. Let V be a real vector space and V* be its dual. Check that the form by (w,13))
ct(w) — /3(v)
is a symplectic form
Exercise 1.2 (Relative linear Darboux theorem). Let F be a vector subspace of a symplectic vector space E. Assume that the restriction of the symplectic form to F has rank 2r. Show that there exists a symplectic basis (ei, . , er,, fl,... , of . .
Esuchthat (ej,...,ef,er+1,...,er+k,fl,...,fr)isabasisofF(kistheinteger defined by 2r + k = dim F).
Exercise 1.3. Show that the symplectic group of C is isomorphic with the special linear group SL(2;R).
Exercise 1.4. Prove directly that the symplectic group Sp(2) is diffeomorphic to the product of a circle by an open disk.
Exercise 1.5. Let A E Sp(2n). Check that the matrices tA and A' are similar8. Show that A is an eigenvalue of A if and only if A' is also an eigenvalue, and that both occur with the same multiplicity. 8Thus A and
A' are similar too.
Exercises
45
Exercise 1.6. Check that a non zero vector of a symplectic space can be mapped to any other non zero vector by a symplectic transformation (in other words, the symplectic groups acts transitively on the set of non zero vectors). Show that, for n> 1, the symplectic group does not act transitively on the set of real 2-dimensional subspaces of Exercise 1.7. Let n> 1 be an integer. Let P be a real plane (dimension-2 subspace)
Show that P is either isotropic or symplectic. What are the orbits of the action of the symplectic group on the set of planes in Exercise 1.8. Let V be a vector space and V* be its dual. Endow V V* with V* be a linear map. the symplectic form defined in Exercise 1.1. Let A V Prove that the graph of A is a Lagrangian subspace if and only if the bilinear form defined by A on V is symmetric. in
Exercise 1.9. Let E be a vector space endowed with a symplectic form
and let
F be (any) subspace of E. Prove that w induces a symplectic structure on the quotient F/F fl F°. Exercise 1.10. Let E be an even dimensional vector space and let w, w' be two symplectic forms on E. Prove that the symplectic groups Sp(E, w) and Sp(E, w') are conjugated subgroups of GL(E). Let be the space of all symplectic forms on the vector space E. Prove that the linear group of E acts on this space by (g . w)(X, Y)
= w(9X,9Y).
Deduce that is in one-to-one correspondence° with the homogeneous space GL(E)/Sp(E), where Sp(E) is the symplectic group Sp(E,wo) for a given form w0 onE. Exercise 1.11. Prove that, on any symplectic vector space, there are complex structures. Prove that a complex structure is an isometry and that it is skew-symmetric for the scalar product it defines. Exercise 1.12. Let V be a real vector space. Using a scalar product on V, construct a complex structure calibrated by the standard symplectic form on V and
such that
(J(v),w) =
v
w for all v,w E V.
Exercise 1.13. Assume that the wave front t
(x(t),z(t))
has an ordinary cusp for t = 0 with a tangent line transversal to the z-axis. Prove that this is the wave front of a Lagrangian immersion of j — ct[ into R2. 9This is actually a homeomorphism.
inCa
46
Exercise 1.14. Prove that the wave front of the Whitney immersion by image of the sphere the hypersurface in
/
(x,a)u
2
—'
is
a3
(using the notation of Example 1.4.3). Find the singular points of this wave front
and draw it in the cases n =
1
(this is the eye, Figure 1.2) and n =
2
(this is the
flying saucer, Figure 1.4).
Exercise 1.15 (The swallow tail). Determine.., and draw the wave front of the Lagrangian immersion described in § 1.4.5 and on Figure 1.1. Exercise 1.16. Prove that the Maslov class of the standard (Lagrangian) embedding
of the circle is ±2. What is that of the Whitney immersion? Of the immersion defined by the crossbow10? is endowed with its LiExercise 1.17 (Lagrangian cobordisms [3]). The space ouville form A and its symplectic form dA. it is said that a Lagrangia.n immersion is "cobordant to zer&' if there exists an oriented manifold V of dif: L mension n +1, with boundary, whose boundary is L, and a Lagrangian immersion
f:V transversal to the co-isotropic subspace F = C" ® iR C
such that
f'(FnV)_—OV=L and such that the cornposiUon L
F
is the immersion 1. 1.4.3) is cobordant to zero.
(1) Prove that the Whitney immersion S"
f*A? —, C is cobordant to zero. What can be said of (2) Assume that I : Prove that, if a Lagrangian immersion St —* C is cobordant to zero, it is
exact.
(3) Consider an exact Lagrangian immersion
f:S1
C
and its wave front in R2. Assume the singularities of the wave front are ordinary cusps. The tangent line to the front at any point is transversal '0Hint: orient the circle and notice that the unit tangent vector to the Whitney immersion does not take all the values in the circle. For the crossbow, notice that this immersion of the circle into C may be deformed, among immersion, into the standard embedding.
47
Exercises
to the z-axis. The circle S' is oriented. Count the cusps of type (a) with a + sign, those of type (b) with a — sign (Figure Lii) and get a number N(f) E Z. What is the value of N(f) for the Whitney immersion? For the crossbow (Figure 1.3)?
(b)
(a)
Figure 1.11
(4) The Lagrangian immersion f : S'
C has a Gauss map 'y(f), taking its
values in the Grassmannian A1 of oriented Lagrangians in C, that is a circle S'. Call a the closed 1-form "dO" on this circle. Prove that1'
N(f) (5) Consider the mapping
j:A, PtIt can
be shown (this is an additional question, use § I.3.f) 3*
: H1(A2)
is an isomorphism. Prove that if f
'
S'
that
H'(A,) C is cobordant to zero, then
N(f) = 0. Does there exist a Lagrangian immersion of a disk into C2 whose boundary is the crossbow?
"This is tn say that N(f)
the Maslov class of the immersion f.
48
Exercise 1.18 (&om (x,y) to
Writing
dz=dx+idy, one gets a couple of relations between the expressions of the vector fields in coorProve for instance that dinates (x,y) or
Of 0
0 i
(*91
— 2
*9
a;
Of
*9
a; 0z3
Exercise 1.19. Consider the vector field X given on C2 by
X(zi,z2) = (icxizi,icx2z2) (a1
and a2 being two real parameters).
(1) Check that
X(zj,z2) and show that the form
*9
8z1
._zzi___) +a2 *921
*9
*9
822
022
is holomorphic.
(2) Show that X preserves w and find a function H such that X = Xy. (3) Under which condition does the vector field X preserve Il? Assume now that this condition holds. Find two functions g and h from C2 to R such that
Consider fl h'(b). Show that, if a is a regular value of H, this is a special Lagrangian submanifold.
(4) Describe the special Lagrangian submanifolds H1 (a) i-curves, that is, by equations.
fl
(b) as complex
(5) Check that they are diffeomorphic to S' x R. Hint: they are conies.
Chapter II
Lagrangian and special Lagrangian submanifolds in symplectic and C alabi-Yau manifolds 11.1
Symplectic manifolds
we must understand how a In order to deform a Lagrangian submanifold in tubular neighbourhood looks like. We prove here that a Lagrangian submanifold has a neighbourhood which is diffeomorphic to a neighbourhood of the zero section in its cotangent bundle. To be precise and explicit, we need to define a symplectic structure on the cotangent bundles and more generally to say what a symplectic structure on a manifold is. A symplectic manifold is a manifold W endowed with a non degenerate 2form w, namely, a non degenerate alternated bilinear form on each tangent space T1W, which is required to be closed, (dw = 0). Notice that a symplectic manifold is even dimensional. (1) The first example is of course C" with the symptectic form we have used so far, considered as a differential form:
Examples 11.1.1.
w=
(where (x1 +
.
A
+ iyn) stands for the complex coordinates in C").
II. In symplectic and Calabi-Yau manifolds
50
One also has: w(Z, Z')
-
=
X Y'.
And this is an exact, hence closed, form: w = d(Eyjdx3). x = (2) The next example is that of cotangent bundles. Think that = T*Rfl and simply replace by any manifold V. On x then that W = T1'V, there is a canonical 1-form, the Liouville form A, defined by the "compact" formula:
= in which x denotes a point of V, an element of TV (namely a linear V of the form on the tangent space TXV), and ir the projection T*V cotangent bundle. If (x1,.. . are local coordinates on V and (yj,... the cotangent coordinates, then ,
,
A
=
The 2-form dA is both closed (!) and non degenerate.
(3) Surfaces. On a surface W, any 2-form is closed. Moreover, in dimension 2, to say that a 2-form is non degenerate means that it nowhere vanishes, in other words that this is a volume form: all the orientable surfaces may be considered as symplectic manifolds. (4) The sphere. Consider, in particular, the unit sphere S2 in R3, whose tangent space at a point v is the plane orthogonal to the unit vector v. Put
Y) = v• (X A Y) = det(v, X, Y). This is a non degenerate 2-form and thus a symplectic form.
(5) The projective space is a symplectic manifold. The nicest thing to do is to define its syrnplectic form starting from that of and using the symplectic reduction process. To define we factor out the unit sphere of by the S'-action (multiplication of coordinates): =
Lagrangian submanifo)ds and immersions
11.2.
51
At each point x of the sphere the tangent space is the Euclidean orthogonal of x and the kernel of the restriction of the symplectic from is the line generated by ix. This line is also the tangent space to the circle through x on the sphere. The symplectic form of defines a non degenerate alternated bilinear form w on Its pull-back on the sphere is closed, so that is closed. It is actually a (the standard) Kähler form on (6) Complex submanifolds of the projective space are symplectic. The compatibility of w with the complex structure gives that w(X, iX) > 0 for any vector X that is tangent to the submanifold, so that w is indeed non degenerate on this submanifold. (7) More generally, all Kähler manifolds are symplectic. We will come back to this remark.
Notice that, on cotangent bundles, as on the symplectic form is exact. This cannot be the case on a compact symplectic manifold. Proposition 11.1.2. On a compact manifold, there exists no 2-form that is both non degenerate and exact.
Proof Let w be a non degenerate 2-form on the 2n-dimensional manifold W. To is a volume form. But then, if
say that w is non degenerate is to say that w
is
dcv,
also exact, thus W cannot be compact.
Haniiltonian vector fields XH for functions H: W R are defined exactly as in Appendix I.6.c and so is the Poisson bracket of two functions on W. Exercise 11.3 explains why it is required that a symplectic form be dosed.
11.2
Lagrangian submanifolds and immersions
An immersion f: L —* W into a symplectic manifold is Lagrangian if and dimW = 2dimL.
II.2.a In cotangent bundles All what was done in
in § I.4.a works as well in a cotangent bundle.
=
0
II. In symplectic and Calabi-Yau manifolds
52
Grapbs Proposition 1.4.1 generalizes as: T* L be a section of a cotangent bundle. Its image L Proposition 11.2.1. Let is a Lagrangian submanifold if and only if the 1-form is closed.
Proof. The most elegant thing to do is to state first a property of the Liouville form (which explains why it is called the "canonical" 1-form): for any form has
one
= is considered as a section of the cotangent bundle in the left In this equality, hand side and as a form in the right hand side. One has indeed:
= = =
o
by definition of by definition of becanse is a section.
Eventually, = 0 if and only if d(a*A) = 0, thus the graph of Ck is a Lagrangian 0 submanifold if and only if is closed.
Remark 11.2.2. In particular, the zero section of L C T*L is a Lagrangian submanifold. What we plan to do next is to show that L C T* L is a model for all Lagrangian embeddings of L into a symplectic manifold (Theorem 11.3.7). Generating functions A function
allows to construct a Lagrangian submanifold (the graph of dF) into T* M x Ck and then, by reduction, a Lagrangian immersion into T*M. Wave fronts
Exact Lagrangian immersions into T*M define wave fronts in M x ft and conversely.
Conorinal bundles Let
f:V
be any immersion. The conorrnal bundle is the subbundle of the pull back bundle
f*T*M =
)
x
Tf*(X)M}
v
11.3.
Tubular neighborhoods of Lagrangian submanifolds
53
defined by
N*f =
= o} = 0}.
E f*T*M I
=
E
f*T*M
I
Map N*f into T*M by )(f(x),co).
F:(x,co)u This is an immersion, since
= It is Lagrangian, as we have F*A = 0. Indeed, calling ir the two projections T*M M et N*f V, we get
= =
0
= cp
=0 as
vanishes on the vectors that are tangent to V.
0
One should check that the proof given for the normal bundle in § I.4.c for is identical to the one given here, the orthogonality used the case where M = there being an ersatz of the duality used here.
11.3
Tubular neighborhoods of Lagrangian submanifolds
Let us now present a method, invented by Moser [28], which allows to describe a symplectic manifold in the neighbourhood of a point (they are all the same) or a neighbourhood of a Lagrangian submanifold in a symplectic manifold.
II.3.a Moser's method The next "lemma" contains all these results. Lemma 11.3.1. Let W be a 2n-dimensional manifold and let Q C W be a compact submanifold. Assume that wo and w1 are two closed 2-forms on W such that, at any point x of Q, w0 and w1 are equal and non degenerate on Then there exists open neighborhoods V0 and V1 of Q and a diffeomorphism
such that
= IdQ and
= w0.
11.3.
Thbu)ar neighborhoods of Lagrangian submanifolds
55
First proof. The vector bundle NQ retracts on its zero section. The inclusion As j*1wiI = V0 thus induces an isomorphism f —' Q
j
and the cohomology classes of that their difference is an exact form.
are equal in
Second proof. We explicitly construct a 1-form the dilatation of factor t in the fibers cOt'
0
that is a primitive of r. Consider
V0
VO
which means
tE1O 1
This is a diffeomorphism (onto its image) for t > 0 and we have cOo(Vo) = Q, Consider its = Idv0 and c°dQ = IdQ. The form i- = — w0 is a 2-form on restriction to Vo. It is identically zero along Q by assumption. We have
Consider now the (time depending) radial vector field dilatation) on Vo. This is the vector field defined by
Is=t'
=
It is defined only for t > 0, in the same way that
is a diffeomorphism only for
t > 0. In a very concrete way, the vector field is
+ v) = For all t, consider also the 1-form
defined by
= Notice that, if y is in Q, one has
cot(y) = y and
thus
(tangent to the
=
is zero along Q. For t> 0, one has
(Y) = (txer)x+tv
t > 0,
=
+
0
II. In symplectic and Calabi-Yau manifolds
56
and consequently
dat
+ tx,dr) = =
d
(tx,r) * r).
Eventually, we get
dat = for t >
0
and thus also for all t E T
= T —0 =
— WO*T
(0,
=
1]. Now
f
=
f
=
da
atdt. We has thus proved that, in a neighbourhood of Q, writing a = da is an exact form (with a identically zero on Q).
—
=
0
To finish the proof of Lemma 11.3.1, we use the actual method of Moser. We consider the path of symplectic forms
=
= w0 + tdcr.
w0 + t(w1 —
For t = 0, this is the non degenerate form Also, along Q, this is the very same form w0. Restricting again Vo if necessary (using compactness again) one can assume that is non degenerate on V0 for all t E [0,11. Let Yt be the vector
field defined by
= (the existence and uniqueness of degenerate). Let be its flow:
—a
are consequences of the fact that Wt is non
We have
=
(d(u) + +
=0 by definition of
Hence
=
=
wo
and eventually
=
0
11.3.
Tubular neighborhoods of Lagrangian subrnanifolds
57
Remark 11.3.5. In a general symplectic manifold W, the proof is identical to the one given here; what we need is the notion of a normal bundle, that is, of orthogonality in TW, and a way to replace the mapping (x, v) '—f x + v. One uses a Riemannian that replaces x + v is metric on W and its exponential mapping: the point (at time 0) with tangent the point reached at time 1 by a geodesic2 starting from x vector v. The most direct application of Lemma 11.3.1 is the Darboux theorem. This is a symplectic form on W and w0 is the is the case where Q is a point Xo, symplectic form induced on
Theorem 11.3.6 (Darboux theorem). Let x be a point of a manifold W endowed with a symplectic form w. There exists local coordinates
(xi,...,xn,y1 centered at x in which w =
yn)
A dx2.
W defines, using a diffeomorphism from on Proof. The form induced by a neighbourhood of 0 in Tro W onto a neighbourhood of xo in W, a symplectic on a neighbourhood of xo. Lemma 11.3.1 gives a diffeomorphism from a = wo. By definition neighbourhood of into itself, that fixes x0 and satisfies form
of w0, there exists local coordinates centered at x0 in which it can be written
o
II.3.b
Tubular neighborhoods
The next application is a theorem of Weinstein that describes the tubular neighborhoods of the Lagrangian submanifolds. Theorem 11.3.7 (Weinstein [34J). Let (W,w) be a symplectic manifold and let L C W be a compact Lagrangian submanifold. There exists a neighbourhood of the zero section in T* L, a neighbourhood V0 of L in W and a diffeomorphism Vo such that N0 —d,\
and PIL=Id
Proof. Let us check that we can apply Lemma 11.3.1. The submanifold Q is the is the restriction of w. The form w1 Lagraiigian submanifold L and the form is the symplectic form of T*L. We are going to compare them in T*L. As in the previous proof, let us assume firstly that W = C". Let be the composed mapping T*L
NL
2To extend the geodesics, we also need an assumption on the completeness of the metric, or on the manifold W.
II. In symplectic and Calabi-Yau manifolds
58
where
•
is the isomorphism between cotangent and tangent spaces given by restricted to L: the Eudlidean structure of '—p
a(u) = • J is the multiplication by i. Recall (see Lemma 1.2.1) that L is Lagrangian if and only if TL1- = JTL. a neighbourhood of the zero section in T*L, mapped onto a suitable Call is a diffeomorphism onto its image. We want to compare, so that To apply Lemma 11.3.1, we = --dA and w0 = C T*L, the two forms in have to check that they coincide along the zero section. Let (x, 0) e L c N0. We have T(xo)
Recall that
there is an exact
TL.
(T*L)
sequence a)IT
T(x,a) (T*L)
0 which splits along
the zero section s,
0
using
'T(so) (T*L), and that the kernel Ker T(X,Q)ir
is
For
v,
along
the
zero
section.
canonically identified with TL. Compute then /3 E TL, we have and w
((v,a),(w,/3)) = = =
+
(v +
+ —
= /3(v) — But
we have seen
(in
((v,
The forms the lemma.
Exercise 11.1) that
(w, /3))
=
dy3 A
dxi) ((v,
(w, /3)) =
—
/3(v).
and —dA coincide along the zero section, therefore we can apply 1i
In the general situation where W is a symplectic manifold, we need a Riemannian metric and an analogue of J. We use an "almost complex structure" J calibrated by w, namely an endomorphism J of the tangent bundle TW such that J2 = — Id and
(X,Y)u
w(X,JY)
is a Riemannian metric. Such structures exist and form a contractible set. See for instance [5, 24j. Notice that this notion is a globalization of the linear notion,
mentioned in §1.6.b.
11.3.
II.3.c
59
Thbular neighborhoods of Lagrangian submanifolds
"Moduli space" of Lagrangian submanifolds
We consider now, for a given manifold L, the space of Lagrangian immersions
f:L We call it a "space" because this set is actually a topological space, a fact which allows to consider immersions that are "close" to a given immersion. We use the Whitney e'-topology. The
Let V and W be two manifolds. The €1-topology is a topology on the space of e'TW) over V x W, the fiber maps from V to W. Consider the vector bundle The total space is usually called at (x, y) of which is the vector space £(TXV, J' (V, W) rather than L(TV, TW). Every map f e C1 (V, W) defines a mapping
j1f: V
J1(V,W)
x'
If U is an open subset of J1 (V, W), denote
V(u)=—{fEe1(v,w)Ij'feu}. The C1-topology is the topology for which the V(U) are a basis. It is said that a map f is "C'-close" to if it is close to fo for the C1-topology. Diffeomorphism group
f
The group of diffeomorphisms of L acts on this space by p I = o We want to consider Lagrangian immersions only up to this action: we do not want to take into account the way the manifold L is "parametrized". Moduli space
We consider the space of Lagrangian e'-ininiersions from L to W up to the action of the diffeomorphism group. The quotient space is called the "moduli space" of Lagrangian immersions from L to W and denoted 41(L). The next theorem describes the Lagrangian immersions that are close to a fixed Lagrangian embedding of L into W. Theorem 11.3.8. Let L be a compact and connected manifold. A neighbourhood of a Lagrangian embedding L
in the space 41(L) can be identified with a neighbourhood of 0 in the vector space of closed 1-forms of class C1 on L.
II. In symplectic and Calabi-Yau manifolds
60
W be a W be a Lagrangian embedding and I : L Proof. Let fo : L f is close to fo for the "e°Lagrangian immersion close to topology3", we can consider that everything lies in a neighbourhood of L. Thanks
to the tubular neighbourhood theorem (here Theorem 11.3.7) we can assume that everything takes place in a neighbourhood of the zero section in T*L. The map f is T*L (this is what ía has become e1-close to the inclusion of the zero section L when we have identified the neighbourhood of ía (L) in W with a neighbourhood of the zero section in T*L). Thus the composition of f with the projection of the cotangent is a e'-mapping L L, close to the identity. Recall the next lemma, which is a consequence of the inverse function theorem.
Lemma 11.3.9. Let L be a compact and connected manifold. Let I be a L —÷ L that is e' -close to the identity. Then f is a diffeomorphism.
-map D
According to this lemma, the composition is a diffeomorphism g of L. Com-
posing with g', we get an embedding
,T*L which is still
to the zero section... but now the composition L
T*L
is the identity. Thus is a section, that is, a 1-form on L, and is closed because the embedding is Lagrangian. Conversely, all the closed 1-forms that are close to the zero section define Lagrangian embeddings close to 0 Remark 11.3.10. One should have noticed that the section L —p T*L defined by a 1-form is a C1-mapping if and only if the form is a e1-form. The e'-topology thus defines the structure of a topological vector space on the space of 1-forms. In § 11.6 below, we will need a Banach space structure. Remark 11.3.11. The vector space we have obtained is infinite dimensional. It can be considered as a neighbourhood of ía in the "manifold" of deformations of fo, or as its tangent space at fo.
11.4
Calabi-Yau manifolds
We want now to describe, in a way analogous to what we have done in § 11.3.b, the moduli space of special Lagrangian submanifolds. In order to apply Theorem 11.3.7 (special Lagrangian submanifolds are, firstly, Lagrangian submanifolds) we need a compactness assumption on the Lagrangian submanifold. Unfortunately, as we have seen it in § I.5.b, the special Lagrangian submanifolds of C'2 are never compact. We thus need to consider more general manifolds, in which it is possible to define special Lagrangian submanifolds. These are the "Calabi-Yau" manifolds. 3The e°-topology, defined similarly to the e1-topology, is simply the compact open topology.
11.4.
61
Calabi-Yau manifolds
which The point is to define a structure that globalizes the structures on have allowed us to speak of special Lagrangian submanifolds. Recall that, in addition to the R-bilinear alternated form w, we have used the form Il = dz1 A• A of the complex determinant. We will use here the best adapted definition of a Calabi-Yau manifold, the
point is not to spend time on the Calabi-Yau manifold itself but rather on its special Lagrangian submanifolds. For more information on Calabi-Yau manifolds, see [33, 81 and the references they contain.
II.4.a Definition of the Calabi-Yau manifolds Our manifolds should be complex and endowed with a symplectic form w and a that is nowhere zero (this is sometimes called a type-(m, 0) holomorphic form holomorphic volume form). Consider thus a manifold M, on which are given
• a complex structure J (multiplication by i), • a closed non degenerate type (1, 1)-form w (the Kähler form)
• a Riemannian metric g(X, Y) =
iY),
• a Hermitian metric h(X, Y) = g(X, Y) — iw(X, Y), • a trivialization of the "canonical" bundle morphic form which is nowhere zero.
namely a type-(n, 0) holo-
We still need a relation between the forms w and Notice that both forms and A are of type (n, n) and both do not vanish on M, in particular, both arc volume forms. We thus have = A
f on M. The additional compatibility condition is that f should be constant. Let us look at the case of C". We have w"
Writing
dy =
—
and dx =
and noticing that
dyAdx =
—
(dz +d2) =
+
II. In symplectic and Calabi-Yau manifolds
62
we can also write
=
A d21
A
A
gives
A
We thus have
-
We will use the same normalization formula to define a Calabi-Yau manifold in general.
Definition 1L4.1. A complex manifold M is said to be a Calabi-Yau manifold if it is Kähler, has a trivialized canonical bundle, and if the Kàhler form w and the type-(n, 0) form trivializing the bundle AflT*M are related by 'n Remark
11.4.2. Recall that it is possible to express the fact that the form w is
Kähler by saying that the complex structure is "parallel" with respect to the LeviSimilarly, it is Civitá connection associated with the metric it defines with possible to express the compatibility condition for by saying that it is parallel with respect to the same connection. Remark 11.4.3. In general, it is required that the Kähler metric be complete, in other words that it is possible to extend geodesics. This is equivalent to requiring that the manifold be complete (in the sense of metric spaces).
II.4.b Yau's theorem Consider a (complex algebraic) projective smooth manifold M of complex dimension n. Assume that all the HP'°(M) are zero for 1 < p < n — 1 and that the canonical bundle = KM is trivialized by a type-(n, 0) form Notice that M is Kähler, call the Kähler form Rescaling w if necessary, we get
f
JM
•z
[
JM
A hard theorem of Yau [35] asserts that there exists a unique Kähler form on M such that e and which, together with f1, gives M the structure = of a Calabi-Yau manifold.
II.4.c
Examples of Calabi-Yau manifolds
Of course
is a Calabi-Yau manifold.
11.4.
63
Calabi-Yau manifolds
Affine quadrics
We have defined in § 11.1 a symplectic form on the unit sphere S2 C R3 by the formula
= det(x,X,X').
Similarly, the formula
Z, Z') defines a "holomorphic. symplectic" form of the complex quadric
+
+
Q=
=
i}
In "differential" terms,
=
z1dz2 A dz3 + z2dz3 A dz1 + z3dz1 A dz2.
0, z1 and z2 are coordinates and, using the
On the open subset of Q where z3 relation
z1dz1 + z2dz2 + z3dz3 = 0,
we can write
=
!dzi
A dz2,
Z3
so that A
=
A dz2 A 1z3
A d22.
I
Modifying the restriction to Q of the standard Kähler form of C3, let us construct a Kähler form w on Q such that wA W =
1
A
—
Call h the restriction to Q of the function z12. We look for w of the form
w= for some function f. A straightforward computation (see also [31]) shows that
f(h) =
works.
The quadric Q, equipped with and w is (thus) a Calabi-Yau manifold. Recall that Q is diffeomorphic to the tangent bundle TS2 by Q
X+iYu
TS2
x iivii2
In this way, what we have got is the structure of a Calabi-Yau manifold on the
tangent (or cotangent) bundle of the sphere S2. It is possible (but a little more complicated) to do the same for the cotangent bundles of all the spheres S" and more generally for those of all "rank-I symmetric spaces" (see [31]).
II. In symplectic and Calabi-Yau manifolds
64
Remark 11.4.4. Recall that we have identified C2 with the skew field H of quaternions (in § I.5.a). Similarly, the surface Q has the structure of a "quaternionic" or "hyperkähler" manifold. Call I the complex structure defined on Q by that of C3 (this is the multiplication by i) and notice that the symmetric bilinear form that is an equation for Define an operator on Q is still non degenerate when restricted to 21 = the tangent space by the fact that is the unique vector in that is orthogonal to Z for the complex bilinear form and such that
=
11Z112.
This is an almost complex structure since
detc(z, thus
=
— Id.
UZH2
This is an isometry since
IIJZII2 = detc(z, JZ, Moreover,
—Z) =
J2(Z)) = detc(z, JZ, —Z) =
IZH2.
J "anti-commutes" with I:
detc(z,IZ,JIZ) =
111Z112 = 11Z112 on the one hand idetc(z, Z, JIZ) by linearity.
We thus have
detc(z, Z, JIZ) =
—i
11Z112
= — detc(z, Z, IJZ)
so that JI = —IJ. Hence I, J and IJ form a quaternionic structure on Q. we
thus have
On Q,
• the Kähler form w, • the complex structure I defined by multiplication by i in C3, • the associated Riema.nnian metric g, so that w(X, IY) = g(X, Y), • the "holomorphic symplectic form" fi,
• the complex structure J defined in such a way that associated with the same metric g.
be a J-Kähler form,
It is said that Q is hyperkähler. See Exercise 11.7 for a kind of converse statement. Let us give now a few examples of compact Calabi-Yau manifolds.
11.4.
65
Calabi-Yau manifolds
Elliptic curves
The quotient M of C by a lattice A is an elliptic curve. The two forms 1
w = -dz A
and
2z
= dz
give it the structure of a dimension-i Calabi-Yau manifold. One can, more genby a lattice. It is time for a remark: no other erally, perform the quotient of "explicit" example of compact Calabi-Yau manifold is known. In all the known examples, the existence of the Kähler metric with all the desired properties is obtained as a consequence of the Yau theorem 114.b). Hypersurfaces Recall
that complex elliptic curves can be considered as degree-3 curves in P2 (C),
thanks to the Weierstrass p-function. They are thus the n = 1 case in the next theorem.
(C) is a dimension-n Calabi- Yau
Theorem 11.4.5. A degree-d manifold if and only if d = n + 2.
Proof. The condition on the degree is necessary, as we show it now by the computation of the first Chern classes. We want that the bundle M be trivializable, we must thus have ci(T*M) = —c1(TM) = 0. Calling j the inclusion of M in pn+1(C), we have
+ j*c(0(d)) = j*ci(Tpn+l(C)) since the normal bundle of M in the hyperplane section in H2
is 0(d). Denoting by t the dual class to (C))
,
we have
(n+2_d)j*t=0 so that d = n + 2. Assume conversely that d = n + 2. Let us construct explicitly a holomorphic n-form on M. Let F be a degree.-(n + 2) homogeneous polynomial that describes the hypersurface M. Every point of M lies in an affine chart 0 of (C). In affine coordinates zk = there is an index j such that
z3
since M is smooth. The formula "
OF
II. In syrnplectic and Calabi- Yau manifolds
66
defines a homogeneous holomorphic n-form on M that is nowhere zero. This is a consequence of the theorem of Yau 11.4.b) that there is, indeed, in the same cohomology class as the standard Kähler form w, another Kähler form + iOi9ço giving a Calabi-Yau structure on M. 0 Remark 11.4.6. The form the n + 1-form on
above is defined as "Poincaré residue4" starting from with poies along M defined by
(
in
the affine chart
1)
0.
Remark 11.4.7. Calabi-Yau manifolds of dimension 2 are hyperkahler. The proof of this fact is the subject of Exercises 11.6 and 11.7.
II.4.d Special Lagrangian submanifolds An immersion f : V
M from a manifold of real dimension n into a CalabiYau manifold M of complex dimension n is said special Lagrangian if it satisfies = = 0 and = 0. As in the case of the form is then a volume form.
11.5
Special Lagrangians in real Calabi-Yau manifolds
1L5.a
Real manifolds
A complex analytic manifold is real if it is endowed with a "real structure", that is, with an anti-bolomorphic involution S: an involution such that, for any holomorphic function f over an open subset U of M, 10 S is a holomorphic function. For example, on the algebraic submanifolds of P" (C) described by real polynomial equations, the complex conjugation is an anti-holomorphic involution. These manifolds are thus real manifolds. In particular, the projective space pN (C) itself is a real manifold. The real part, or set of real points of a real manifold is, by definition, the set of fixed points of S. For example, the real part of the real manifold PN(C) is p N (R). Notice that there exists respectable real manifolds that have no real point at all, as is, for example, the "Euclidean quadric" N+I
in 4See (151 p. 147.
11.5.
Special Lagrangians in real Calabi-Yau manifolds
67
Proposition 11.5.1. The real part of a real manifold of complex dimension n, if it is non empty, is a siibmanifold all connected components of which have dimension n.
Proof. The connected components of the set of fixed points of the action of a finite group (here the order-2 group generated by S) are always submanifolds. The
tangent space at x to such a component is the subspace of fixed points of the R-linear involution a = The fact that S is a real structure implies that foa is a complex linear form for any complex linear form f on the tangent space at x. We have to check that the fixed subspace of a has dimension n. To do this, we simply that the eigensubspaces associated with the eigenvalues 1 and —1 are isomorphic. Indeed, if or(X) = X, then for any complex linear form 1' we have
f o a(iX) =
if o a(X) = if(X) = f(—iX).
For any complex linear form f, we thus have
f(a(iX)) = f(—iX) so that cr(iX) = —iX. Hence, there are "as many" eigenvectors for the eigenvalue —1 than there are for the eigenvalue 1.
U
II.5.b Real Calabi-Yau manifolds A Calabi-Yau manifold is real if it is both a Calabi-Yau manifold and a real manifold, with a couple of compatibility conditions S*w = —w and
(similarly to what happens in forms Il and w).
=
with the complex conjugation and the two usual
• The affine quadric = 1 of C3, endowed with the complex conjugation of coordinates is a real manifold. It is also clear that this is a real Calabi-Yau manifold. Its real part is simply the unit sphere S2 C R3. If we consider Q as the tangent bundle to S2, notice that the complex conjugation is the multiplication by —1 on the fibers and that the real part is the zero section.
Examples 11.5.2.
• A real hypersurface of degree n+2 in P'"1(C) is a real Calabi-Yan manifold. S* This is checked by computing S the involution induced by the real structure (complex conjugation) of and w as in the proof of Theorem 11.4.5.
IL In syxnplectic and Calabi-Yau manifolds
68
II.5.c The example of elliptic curves Let us come back to the example of F = C/A where A is a lattice that we assume here to have the form
A={m+nrlm,nE Z} for some fixed r such that 0 Re(r) c 1 et Im('r) > 0. To define a real structure on C/A from the complex conjugation in C, it is necessary that A be invariant, that is, that = m + n'r for some m, it e Z. Considering the real and imaginary parts of r, it is seen that = 2Re(r), thus Re(r) = or 0.
/17 ReQr)
=
Re(r) = 0
Figure 11.1: Real elliptic curves
In the second case, the real part of F has two connected components, but in the first case, it has only one, as can be seen solving the equation 2
= z + m + n-i-
in both cases. These components are depicted in bold on Figure 11.1.
Notice (although this is a trivial remark) that the lines that are parallel to the x axis constitute a real foliation of C/A by circles (dimension-i tori) that are special Lagrangian submanifolds of F, represented by dotted lines on Figure 11.1. The space of these special Lagrangian submanifolds is parametrized by the axis generated by r or rather by its image in F, a circle. We shall see more generally in § 11.6 that the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold is, in the neighbourhood of a submanifold V, a manifold whose dimension is the first Betti number of V (here V is a circle and its first Betti number is 1).
II.5.d Special Lagrangians in real Calabi-Yau manifolds Assume now that M is a real Calabi-Yau manifold. We know that
=
—w
and S*Q =
69
Special Lagrangians in real Calabi-Yau manifolds
11.5.
Assume now that the real part MR is not empty. Call j the inclusion of Mft into M. We have So j = j and in particular j*(_w) = = (Soj)*w hence
=
0.
Similarly
= thus
f
13 = 0.
=
(S
=
=
We have proved:
Proposition 11.5.3. Let M be a real Calabi- Yau manifold. The real part of M, if it 0 is not empty, i,s a special Lagrangian submanifold of M. Let us describe now a few examples of this situation. The affine quadric
The sphere is a special Lagrangian submanifold of the affine quadric Q E C3. In other words, with the Calabi-Yau structure on TS2 defined in § II.4.c , the zero section is a special Lagrangian submanifold. In the next examples, we consider a smooth hypersurface defined by a real with its real Calabi-Yau homogeneous polynomial of degree n + 2 in structure. Effiptic curves
The n = 1 case, that of plane cubics, is isomorphic to the example of quotients of C by lattices II.5.c). The real part of a plane cubic has zero, one or two connected components (see Figure 11.1). All eomponents are (topologically) circles. Cubics are foliated by special Lagrangian circles, drawn in dotted lines on Figure 11.1. Degree-4 surfaces
Consider now real algebraic surfaces of degree 4 in P3(C) (the real part of this subject has been investigated and explained in [211). Here is an example from [9]. Consider the real polynomial
\_4
4
4
4
that describes a smooth surface M which has a non empty real part MR. This real
part is
= {(xo,x1,x2,x3)
R4
—
{0}
I
+ 4 = 4 + 4}/(s
Normalize the non zero vectors of R4 by the choice, in each real line through zero, of one of the two vectors such that +4
+4+4 = 2.
II. In symplectic and Calabi-Yau manifolds
70
Then MR is the quotient
{(xo,xi),(x2,x3)ER.2
It is clear that the curve C described in R2 by the equation x4 + y4 =
1
is
diffeomorphic to a circle (radially). Eventually
MR = (Cx C)/((u,v)
(—u
—
v))
is diffeomorphic to a torus. We have thus found a special Lagrangian torus in the Calabi-Yau surface M.
Moduli space of special Lagrangian submanifolds
11.6
We want now an analogue of Theorem 11.3.8, more precisely a description of a neighbourhood of a given special Lagrangian subma.nifold in the space of all special Lagrangian submanifolds.
Theorem 11.6.1 (McLean [25]). Let V be a compact manifold. The modiili space of the special Lagrangian embeddings of V in the Calabi- Yau manifold M is a manifold of finite dimension b1 (V) = dim H' (V; R). Its tangent space at a given point is isomorphic to the vector space of harmonic 1-forms on V. As usual, it is understood that the empty set is a manifold of any dimension — this is not an existence theorem.
Remark 11.6.2. There are two main differences between this statement and Theorem 11.3.8. The first one is that the moduli space here has finite dimension. The second one is that the condition "to be special Lagrangian" is no longer linear, so that this is indeed the tangent space that is identified to a space of differential forms.
Example 11.6.3. Let us come back to the example of the Calabi-Yau structure on TS2 described in §II.4.c. We have said in §II.5.d that the zero section is a special Lagrangian submanifold. As there are no non zero harmonic 1-forms on S2, the theorem of McLean asserts that the zero section is "rigid", that is, it cannot be deformed. In the moduli space of special Lagrangian submanifolds, this is an isolated point. Proof of Theorem 11.6.1. Using the tubular neighbourhood theorem (here in § II.3.b), replace M by a tubular neighbourhood of the submanifold V that is isomorphic to a neighbourhood of the zero section in the normal bundle of V (as are all tubular neighborhoods) and to a neighbourhood of the zero section in the cotangent T*V. We will use the structures induced by those of M on this neighbourhood, keeping their names, for example
= + i/3.
11.6.
71
Moduli space of special Lagrangian submanifolds
We have said in § H3.c that the space of Lagrangian submani folds can be identified with a neighbourhood of 0 in the space Z' (V) of the closed 1-forms on V. The special Lagrangian submanifolds are described, in this space, by the equation = 0 where
F:Z'(V) =
is the mapping defined by
Although these spaces are infinite dimen-
sional, the strategy of the proof is to show that F is submersive at 0. It is thus better to restrict, as much as possible, its target space. Notice first: Lemma 11.6.4. The image of F is contained in the subspace
of exact
n-forms on V.
T*V is the Proof of Lemma 11.6.4. If is the zero form, the mapping ij: V inclusion of the zero section, a special Lagrangian, thus F(0) = 0. Given a form joining it to the zero form... it is possible to consider the path (segment) and giving a homotopy from the section to the zero section. The cohomology class of the closed form does not depend on t, thus it is identically zero, and that means, indeed, that is an exact form. We thus consider F as a mapping
F: Z'(V) and compute its differential at 0. Lemma 11.6.5. The differential of F at 0 is the mapping
(dF)o(ij) = the Hodge star operators associated with the metric defined by the Cal abi- Yau structure on the special Lagrangian V. where * denotes
Proof of Lemma 11.6.5. To compute (dF)0(,7), one chooses a path of forms
whose tangent vector at 0 is the form ii. Let
be
a 1-form on V and X be
the vector field that corresponds to it via the metric on V, that is, the vector field such that g(X,.) = Let Y = JX be the vector field normal to V. This is the vector corresponding to under the isomorphism NV V. The vector field Y is only defined along V, we extend it (arbitrarily) in a vector field Y on the tubular neighbourhood under consideration. Call the flow of Y, so that is a diffeomorphism defined for t small enough. The restriction cot of to V is an embedding of V into NV (one pushes V using Wt)• For t = 0, this is the zero section. Hence for t small enough, this is still a section of NV. We have, for all x in V, d 5See Exercise 11.5,
= Y(x).
IL In symplectic and Calabi-Yau manifolds
72
Under the identification NV T*V, the section of NV corresponds to a section of T* V which is a path of forms, whose tangent vector at 0 is the form Consider now the (n, 0)-form Il, still on our neighbourhood of the zero section in NV. We have = applying Cartan formula together with the fact that Il is closed. For the embedding NV, we thus have V
=
= since 11 is C-linear. We then have
=
= Im(id(txIl)) = Red(cxQ) = The (n. — 1)-form
We still have to convince ourselves that txa =
is
the
unique form satisfying
=
A
But, as the (n + 1)-form is also zero and we have for any 1-form
A (txci) =
A ci
is zero, its interior product by X
=
by definition of X and of the metric g on the space of 1-forms.
Notice that this implies in particular that the differential dF0 is onto: if
0 is
an (ri — 1)-form, * a). du = To end the proof of the theorem, we need to precise what kind of implicit function theorem we use to go from "differential is surjective" to "inverse image is a submanifold". The simplest here is to use the standard implicit function theorem
for Banach spaces (see, for example, [10]). We need to endow the spaces of forms Z' (V) and with structures of Banach spaces. Let us precise the regularity of the forms we use. We consider forms of class in Z' (V) and of class in The Holder norm used here on forms is deduced from the usual Holder norm on functions: recall that (U) is the space of functions of class e" on the open set U of all the derivatives (of order 5 k) of which have a finite Holder norm huh6 (for EJO, 1]), with
hulL = sup
u(x)
—
lix — YhI
+ sup iu(x)I. xEU
11.7.
Towards mirror symmetry?
73
The implicit function theorem gives the fact that F-' (0) is a submanifold in a neighbourhood of 0, whose tangent space at 0 is the kernel (V) of It is important here that this kernel has finite dimension. The isomorphism between Hj,R(V) and the space (V) of harmonic 1-forms is the contents in degree 1 of the Hodge theorem, see [15). 0 Remark 11.6.6. The vector space to so that
(V) is isomorphic to the vector space dual
Hj,R(V) e has a natural symplectic structure (see Exercise 1.1), here A
=
— o' A
The space of harmonic 1-forms is a Lagrangian subspace, by
W(V)
—b (o
I
,
(this is the graph of the mapping *, which is symmetric with respect to the metric... see Exercise 1.8). If V W is a special Lagrangian submanifold, call the moduli space in a neighbourhood of jo• We thus have a Lagrangian subspace
H' "i"
T
•7()
DR V
and it is possible to "integrate" it in a Lagrangian embedding (see [19))
Iji
rV..2) See
L7fl1
also [11] for a description of all these structures by symplectic reduction.
11.7
Towards mirror symmetry?
The "mirror conjecture" asserts the existence, for any Calabi-Yau manifold M, of another Calabi-Yau manifold M* of the same dimension, related with M in a way we briefly describe now, sending the readers to [33] for missing detail. Call the space of isomorphism classes of
• a complex structure
deforming the complex structure J of M
• a "complexified Kähler class" on (M, J,), namely a cohomology class of the form + if3, for some Kähler class (for E and some element Z).
II. In symplectic and Calabi-Yau manifolds
74
Notice that, locally, +i/3 varies in an open subset of H2(M; C), so that the is, locally, a product. The manifold M and its "mirror" partner M* space should be related by an isomorphism of the moduli spaces
that exchanges the factors of this local decomposition as a product. Using in an essential way the symplectic structure of the loop space of M and techniques that go far beyond the level of these notes, Givental has proved the conjecture in [14], following a series of previous papers, the references of which can be found in [14] and [33]. Special Lagrangian submanifolds have been a few years ago the central object of another approach to mirror symmetry, more speculative and having given so far very few results — but a very beautiful approach indeed, that I intend to describe very briefly here.
II.7.a Fibrations in special Lagrangian submanifolds We are no more interested in a single special Lagrangian submanifold but in a whole family. More precisely, we consider a compact Calabi-Yau manifold M and a differential mapping
p:M
to a manifold B, whose general fibers are special Lagrangian submanifolds. The
dimension of B, as that of the fibers of p, must be n. It is not required that p be everywhere regular. Some of the fibers may be singular. The other ones, who correspond to regular values of p, are called general fibers. We know (see § L6.c and [4]) that in any proper Lagrangian fibration, the
general fibers are unions of tori, so this must be the case here. The first Betti number of a torus of dimension ii is precisely n, so that it can be expected that B "looks like" the moduli space of special Lagrangian submanifolds. So, let b E B be a regular value of p and let V C p'(b) be a connected component of the fiber (b). If X E TbB is a tangent vector, there exists a unique vector field Y normal to V in M and such that, for all x in V,
= X. To this field Y corresponds a harmonic 1-form on V, as in the proof of the theorem of McLean (here Theorem 11.6.1). As B has dimension n, starting from n independent vectors X1,. . , in TbB, one constructs n fields Y1,. . , that are normal to V and linearly independent at each point of V. Dually, we thus have n harmonic 1-forms ifl,. , that form a basis of (V) and are linearly independent at each point of V. In order that such a fibration p M —* B exists in a neighbourhood of a special Lagrangian tors V C M, it is necessary that, for the metric induced by the .
.
- .
11.7.
75
Towards mirror symmetry?
Calabi-Yau structure on V, there exists a basis of 5-C' (V) consisting of forms that are independent at each point of V. it is time to mention that (except in dimension 1) there is no known example having all the properties mentioned here.
• Notice first that, abstractly, a basis of harmonic 1-forms that are independent having this at each point exists on the flat torus, the basis dx,,. . , property. The metrics that are close enough to the flat metric thus have the same property. .
• We have seen in II.5.c that the situation of a Calabi-Yau manifold foliated by special Lagrangians submanifolds occurs in dimension 1.
• In dimension 2, on a special Lagrangian torus, one always has a basis of harmonic 1-forms as expected. We have seen that a special Lagrangian submanifold in dimension 2 is simply a complex curve (for a different complex structure). Assuming the submanifold is a torus, it must be an elliptic curve and it has a nowhere vanishing holomorphic form. Actually, the real and imaginary part of this form are harmonic forms on V and they are independent at every point.
II.7.b Mirror symmetry The Strominger, Yau and Zaslow approach to mirror symmetry [32] is to associate, to a Calabi-Yau manifold M endowed with a fibration in special Lagrangian tori
(assuming it exists), another Calabi-Yan manifold M*. The latter should be the "extended" moduli space of special Lagrangian submanifolds of M equipped with a fiat unitary line bundle. Call, as above, the moduli space of special Lagrangian submanifolds in the neighbourhood of V. Locally, the extended moduli space is M* =
x
H'(V;R/Z).
Its tangent space at a point m is TmM* = H'(V;R)@ H'(V; R)
H'(V;R)Ø C.
Thus, M* has a natural almost complex structure, it is even Kähler:
Theorem 11.7.1 (Hitchin [19]). The complex structure on M* is integrable, the metric of H' (V; R) defines a Kãhler metric on M*. We have seen (Remark 11.6.6 above) that
is a Lagrangian submanifold
of (V) e (V), a symplectic vector space endowed by the metric of an almost complex structure (see Exercise 1.12). It can be shown (see [19]) that M* is a Calabi-Yau manifold if is... a special Lagrangian submanifold in this complex vector space. See [11, 20].
II. In syrnplectic and Calabi-Yau manifolds
76
Exercises Exercise
IL 1.
Check that the Liouville form A of the cotangent T* V satisfies (w,/3)) = cx(w)
— 8(v)
(see Exercise 1.1).
Exercise 11.2. Let
L —b
L be a diffeomorphism. Prove that the formula —
defines a diffeomorphism of T*L into itself. Determine 4)*A and prove that 4' preserves the symplectic from.
Exercise 11.3. Let w be a non degenerate 2-form on a manifold W. Define the Hamiltonian vector fields and Poisson brackets as above (this does not use the fact that w is closed). Express Z)
when
X, Y et Z are tangent vectors to W at x that are the values at x of the
Haxniltonian vector fields of three functions f, g and h. Prove that w is a closed form if and only if the Poisson bracket it defines satisfies the Jacobi identity. 11.4. Assume X and Y are two "locally Hamiltonian" vector fields on a symplectic manifold, namely that txw et tyw are closed forms. Prove that their
Exercise
Lie bracket [X, Y) is a globally Haniiltonian vector field, namely that exact form.
is an
Exercise 11.5 (The Hedge star operator). Let V be an n-dimensional oriented manifold endowed with a Riemannian metric g and let a be the Riemannian volume form. Check that the formula g(u1 defines an metric on APT* V... and
that the map
*:APT*V
A
(*v) = g(u, v)a
for all u E APT* V defines, indeed, an operator, the Hodge star operator, which is an isometry. Check that
** = (_1)P(fl1)
77
Exercises
Exercise 11.6 (Multilinear algebra in R4). Consider the vector space R4, with its Euclidean structure g(X,Y) = (X,Y) azid canonical basis (el,e2,e3,e4), and the vector space A = A2(R4)*
of alternated bilinear forms on R4. (1) What is the dimension of A? Check that A is isomorphic to the vector space of skew-symmetric endomorphisms of R4.
(2) Endow A with the Euclidean structure (, ) induced by that of R4, namely <4) is orthonormal. Define (for 1
(where det, namely the determinant, is the generator of A4 (R4 )* such that e2 A e3 A e4) = 1. A e) and the eigenspaces Check that * is an involution. Determine the det(e1 A of *.
(3) Call A÷ the subspace of forms that are invariant by * (they are called "selfdual" forms). To any in A÷, associate as in (1) a skew-symmetric endomorphism R4
Prove that
=
— Id
if and only if (cr,
R4. = 1.
Exercise 11.7 (Calabi- Yau surfaces). Let M be a Calabi-Yau surface with Kähler form w and holomorphic 2-form
(1) There exists a local basis on M in which
of the vector space of holomorphic forms 1
= (see
[15]). Prove that, on the open set where =
and c°2 are defined, one has
A W2
for some constant A.
(2) Check that = Deduce that the real and imaginary /3 parts of self-dual in the sense that *ci = ci and = /3.
are
(3) Prove that the formula ci(X, JY) = g(X, Y) defines a skew-symmetric endomorphism J of the tangent bundle TM and
that
JX E (X. IX)1 = (C X)1.
II. In symplectic and Calabi-Yau manifolds
78
(4) Prove that J2 = — Id, so that J is an almost6 complex structure on M, and that J is an isometry for g. (5) Prove that M is endowed with a hyperkähler structure, namely with three isometries I, J and K that are almost complex structures and anti-commute and with three non degenerate 2-forms that are Kähler for the metric g and respectively for each of the complex structures I, J and K. Exercise 11.8. Using the notation of Exercise 11.7, prove that the special Lagrangian submanifolds of the Calabi-Yau manifold M are the complex curves for the complex structure .1. Exercise 11.9. In this exercise, W denotes a complex analytic manifold7 of complex dimension 2, endowed with the structure of a Calabi-Yau manifold, with the Kähler the holomorphic volume form and the metric -y. Assume moreover that form
HbR(W) =0. Consider a vector field X on W, assume that it is not identically zero, and = 0 and that it preserves w and (1, namely that it satisfies the relations = 0. Assume moreover that the 1-form LXI1 is holomorphic. The metric 'y and the vector field X are assumed to be complete.
(1) Prove that X is the Hamiltonian vector field of a function H: W —i R. is preserved by X and that there exists a holomorphic func= df. Let x E W be a point such that C such that spanned is the complex line in 0. Prove that the kernel of
(2) Prove that Lx
tion f : W by
(3) Assume that L is a Lagrangian submanifold of W that is preserved by X (this for all x L). Check that the connected components means that
of L are contained in the level sets H1(a) of the Hamiltonian H. (4) Call g and h respectively the real and imaginary part of f. Assume now that L is a special Lagrangian submanifold. Prove that h is locally constant on L.
(5) Let a R be a regular value of H and let Q = H1 (a) be the corresponding for level set in W. Fix a point x in Q. Prove that the orthogonal of and that the complex linear form is a complex line the metric in Deduce that the two real linear forms dh(x) and is non zero on dH(x) are independent. Prove that, for all b E R, L = Q fl h1(b) is a dimension-2 submanifold of W and that it is special Lagrangian.
(6) Prove that g IL has no critical point. 61t is not very hard to prove that this is a genuine complex structure, namely that M is a complex manifold for some structure such that the multiplication by i is J. 7This exercise (slightly) generalizes the construction given in § L5.e and in particular that of Exercise 1.19.
Exercises
79
(7) Assume that the Hamiltonian vector field X is periodic. Prove that the connected components of L are diffeomorphic to S' x B..
Bibliography [1] V. I. ARNOLD — "A characteristic class entering in quantization conditions",
Funct. Anal. Appl. 1 (1965). [2]
,
Mathematical methods in classical mechanics, Springer, 1978.
, "Lagrange and Legendre cobordisms I and II", Funct. Anal. AppI. 14 (1980), p. 167—177 et 252—260.
[3] —
[4] V. I. ARNOLD et A. B. GIVENTAL - "Symplectic geometry", Dynamical systems, Encyclopedia of Math. Sci., Springer (1985). [5] M. AUDIN — "Symplectic and almost complex manifolds", in /7], p. 41—74. [6]
, Les systèmes hamiltoniens et leur intégrabilité, Cours Spécialisés, 8, Société Mathématique de France & EDP Sciences, 2001.
[7] M. AUDIN et J. LAFONTAINE (éds.) — Holomorphic curves in sympleetic ge-
ometry, Progress in Math., Birkhäuser, 1994. [8]
J. BERTIN, J.-P. DEMAILLY, L. ILLUSrE et C. PETERS - Introduction a la théorie de Hodge, Panoramas et Syntheses, 3, Société Mathématique de France, 1996.
[9] R. BRYANT
"Some
examples of special Lagrangian tori", preprint (1998).
[101 H. CARTAN — Calcul différentiel, Hermann, Paris, 1967.
[11] S. DONALDSON — "Moment maps and diffeomorphisms", Asian Journal of
Math. (2000), a paraItre. [12] D. B. FuKs — "Maslov-Arnold characteristic classcs", Soviet Math. DokI. 9 (1968), p. 96—99. [13] A. B. GIVENTAL — "Lagrangian imbeddings of surfaces and the open Whitney umbrella", Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, p. 35—41, 96.
[14] A. B. GIVENTAL — "Equivariant Gromov-Witten invariants", Internat. Math. Res. Notices 13 (1996), p. 613—663.
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[15] P. A. GRIFFITHS et J. HARRIS — Principles of algebraic geometry, Wiley, 1978.
[16] M. GROMOV -. "Pseudo-holomorphic curves in symplectic manifolds", Invent. Math. 82 (1985), p. 307—347. [17] M. GROMOV — Partial differential relations, Springer, Berlin, 1986.
[18] R. HARVEY et H. B. LAWSON, JR. — "Calibrated geometries", Acta Math. 148 (1982), p. 47—157.
(19] N. J. HITCBIN - "The moduli space of special Lagrangian submanifolds", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 3-4, p. 503—515 (1998), Dedicated to Ennio De Giorgi. [201 —, "Lectures on special Lagrangian submanifolds", preprint (1999). [21] V. M. KHARLAMOV - "On the classification of nonsingular surfaces of degree
4 in RP3 with respect to rigid isotopies", Ftinktsional. Anal. i Prilozhen. 18 (1984), no. 1, p. 49—56.
[22] J. LAFONTAINE — Introduction aux variétés différentielles, Presses trniversi-
taires de Grenoble, 1996. [23] J. A. LEES — "On the classification of Lagrange immersions", Duke Math. J. 43 (1976), no. 2, p. 217—224.
[241 D. MCDUFF et D. SALAMON — Introduction to symplectic topology, The Clarendon Press Oxford University Press, New York, 1995, Oxford Science Publications. [25) R. C. MCLEAN — "Deformations of calibrated submanifolds", Comm. Anal. Geo7n. 6 (1998), no. 4, p. 705—747. [26] J. MILN0R — Morse theory, Princeton University Press, 1963. [27] K. MOHNKE — preprint (2001).
[28] J. MOSER — "On the volume elements on a manifold", Trans. Amer. Math. Soc. 120 (1965), p. 286—294.
[29] H. ROSENBERG et D. HOFFMAN — Surfaces minimales et solutions de problèmes variationnels, Société Mathématique de France, Paris, 1993. [30] A. CANNAS DA SILVA — Lectures on symplectic geometry, Lecture Notes in
Mathematics, Springer, 2001. [31] M. STENZEL — "Ricci-flat metrics on the complexification of a compact rank
one symmetric space", Manuscripta Math. 80 (1993), p. 151—163.
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[32] A. STROMINGER, S. T. YAu et E. ZASLOW - "Mirror symmetry is Tduality", Nuclear Phys. B 479 (1996), p. 243—259. [33] C.
VoIsIN — Symétrie miroir, Panoramas et Syntheses,
2,
Société
Mathématique de France, 1996, English translation: Mirror symmetry, SMF/AMS Texts and Monographs 1, 1999. [34] A. WEINSTEIN — Lectures on symplectic manifolds, CBMS Regional Confer-
ence Series in Mathematics, 29, Amer. Math. Soc., 1977.
[35) S. T. YAU - "On the Ricci curvature of a compact Kähler manifold and the complex Monge—Ampère equation I", Comm. Pure and Appi. Math. 31 (1978), p. 339—411.
Part B
Symplectic Toric Manifolds Ana Cannas da Silva
Foreword The goal of this text is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties) from the syrnplectic viewpoint. The study of toric
manifolds has many different entrances and has been scoring a wide spectrum of applications. For symplectic geometers, they provide examples of extremely symmetric and completely integrable hamiltonian spaces. In order to distinguish the algebraic from the symplectic approach, we call algebraic toric manifolds to the smooth toric varieties in algebraic geometry, and say symplectic toric manifolds when studying their symplectic properties. Native to algebraic geometry, the theory of toric varieties has been around for about thirty years. It was introduced by Demazure in [16] who used toric varieties
for classifying some algebraic subgroups. Since 1970 many nice surveys of the theory of toric varieties have appeared (see, for instance, [14, 21, 28, 41]). Algebraic geometers and combinatorialists have found fruitful applications of toric varieties to the geometry of convex polytopes, resolutions of singularities, compactifications of locally symmetric spaces, critical points of analytic functions, etc. For the last ten years, toric geometry became an important tool in physics in connection with mirror symmetry [13] where research has been intensive. In this text we emphasize the geometry of the moment map whose image, the so-called moment polytope, determines the symplectic toric manifolds. The notion
of a moment map associated to a group action generalizes that of a hamiltonian function associated to a vector field. Either of these notions formalizes the Noether principle, which states that to every symmetry (such as a group action) in a mechanical system, there corresponds a conserved quantity. The concept of a moment map was introduced by Souriau [45] under the french name application moment; besides the more standard english translation to moment map, the alternative momentum map is also used. Moment maps have been asserting themselves as a main tool to study problems in geometry and topology when there is a suitable symmetry, as illustrated in the book by Gelfand, Kapranov and Zelevinsky (22]. The material in some sections of the second part of these notes borrows largely from that excellent text, where details are given and where the discussion continues in exciting new directions. This course splits into two parts: the first concentrates on the symplectic viewpoint, and the second focuses on the algebraic viewpoint with links to symplectic geometry. Each of the two parts has a similar structure: introduction, classification, polytopes. So the lectures seem periodic though the languages for each half are noticeably different.
After introducing, in the first lecture, basic notions related to symplectic tone manifolds, in Lecture 2 we state their classification, and prove the existence part of the classification theorem by using the technique of symplectic reduction. Lecture 3 discusses moment polytopes, namely how to read from the polytope some topological properties of the corresponding symplectic toric manifolds, as well as how are some changes on the polytopes translated into changes of the cor-
88
Foreword
responding symplectic toric manifolds. Lecture 4 introduces toric manifolds from the algebro-geometric perspective, after reviewing definitions and notation in the theory of algebraic varieties. Lecture 5 describes the classification of toric varieties using the language of spectra and fans. Finally, Lecture 6 deals with polytopes now from the algebraic point of view, studying some geometric properties of toric varieties which polytopes encode. Lectures 3 and 6 underline the moment map potential. Geometry of manifolds at the level of a first-year graduate course is the basic prerequisite for this course. Some familiarity with symplectic geometry is useful to read these notes faster, though most of the needed definitions and results are stated here. Scattered through the text, there are exercises designed to complement the exposition or extend the reader's understanding. Throughout, the symbol denotes n-(complex-)dimensional complex projective space.
Chapter I
Symplectic Viewpoint Symplectic Toric Manifolds
1.1
In order to define symplectic toric manifolds, we begin by introducing the basic objects in symplectic/hamiltonian geometry/mechanics which lead to their consideration. Our discussion centers around moment maps.
Symplectic Manifolds
1.1.1
Definition 1.1.1. A symplectic form on a manifold M is a closed 2-form on M which is nondegenerate at every point of M. A symplectic manifold is a pair where M is a manifold and w is a symplectic form on M. (M, By linear algebra, a symplectic manifold is necessarily even-dimensional. Examples.
1. Let M =
with linear coordinates x1 symplectic form on is
..
,
The standard
= >dxk A dyk.
2. Let M =
with linear coordinates z1,.
=
. .
,
The form
dZk A d2k
a symplectic form on In fact, this form equals that of the previous example under the identification zk = Xk + is
1. Symplectic Viewpoint
90
Tangent vectors to regarded as the set of unit vectors in S2 at p may then be identified with vectors orthogonal to p. The standard
3. Let M =
syniplectic form on S2 is the form induced by the inner and exterior products:
:= (p,u x v)
,
for u,v E
=
This form is closed because it is of top degree; it is nondegenerate because (p, u x v) 0 when u 0 and we take, for instance, v = u x p.
Exercise 1
Check that, in cylindrical coordinates away from the poles (0 < 0 < 2ir and —1
The natural notion of equivalence in the symplectic category is expressed by a symplectomorphism: Definition 1.1.2. Let
and (M2,w2) be 2n-dimensional symplectic maniis a symplectomorphism
folds, and let cc: M1 —p M2 be a diffeomorphism. Then if (p*W2 =
The theorem (see, for instance, [12] or Theorem 1.3.1 for the case where the group is trivial) states that any symplectic manifold , w) is locally symplectomorphic to w0). In other words, the prototype of a local piece of a 2n-dimensional symplectic manifold is , we). Hence, this theorem provides the local classification of symplectic manifolds in terms of a unique invariant: the dimension.
Let (M, w) be a symplectic manifold of dimension 2n. A Darboux chart for
wIu=>dxkAdyk. By the Darboux theorem, there exists a Darboux chart centered at each point of
a symplectic manifold.
1.1.2
Ilamiltonian Vector Fields
Let (M, w) be a symplectic manifold. DefinItion 1.1.3. A vector field X on M is symplectic if the contraction Zxw is closed. A vector field X on M is hamiltonian if the contraction ZXW is exact.
91
Li. Symplectic Toric Manifolds
Locally on every contractible open set, every symplectic vector field is hamiltonian. If the first de Rham cohotnology group is trivial, then globally every syinmeasures the obstrucplectic vector field is hamiltonian; in general, tion for symplectic vector fields to be hamiltonian.
Note that the flow of a symplectic vector field X preserves the symplectic form:
=
d ZXW +zx
dw = 0.
dosed
= dH for some smooth function If a vector field X is hamiltonian with H: M R, then the flow of X a'so preserves the function H:
£xH =
=
Therefore, each integral curve {pt (x) set of H:
H(x) =
zxw = 0.
R} of X must be contained in a level
t
=
,
Vt.
Definition 1.1.4. A hainiltonian function for a hamiltonian vector field X on M is a smooth function H : M —i R such that zxw dH.
By nondegeneracy of w, any function H C°°(M) is a hamiltonian funcdH can be tion for some hamiltonian vector field because the equation ZXW always solved for a smooth vector field X. A hamiltonian vector field X defines a hamiltonian function up to a locally constant function. Examples.
1. On the standard syinplectic 2-sphere (S2. dO A dh), the vector field X = is hamiltonian with hamiltonian function given by the height function:
ix(dOAdh)= dli. The motion generated by this vector field is rotation about the vertical axis, which of course preserves both area and height.
2. On the symplectic 2-torus (T2,d01 A dO2), the vector fields X1 = X2 = are symplectic but not harniltonian.
1.1.3 Let
and
Integrable Systems
denote a hamiltonian vector field on a symplectic manifold (M,w) with hamiltonian function H E C°° (M).
92
1. Symplectic Viewpoint
Definition 1.1.5. The Poisson bracket of two functions tion
g E C°° (M) is the func-
{f,g} :=w(Xf,X9) We have X{f,9} = —[X1,X9] because XW(Xf,X9) = [Xg,Xf], where the Lie bracket of vector fields.'
iS
Exercise 2 Check that the Poisson bracket {•, .} is a Lie bracket and satisfies the Leibnlz rule:
Vf,g,h€C°°(M)
{f,gh}={f,g}h+g{f,h}
Therefore, if (M,w) is a symplectic manifold, then (C°°(M), {., .}) is a Poisson algebra, that is a Lie algebra with an associative product for which the Leibniz rule holds. Furthermore, we have an anti-homomorphism of Lie algebras
C°°(M) —f
X,,.
H
Definition 1.1.6. A hamiltonian system is a triple (M, w, H), where (M, ti.,) is a symplectic manifold and H E C°°(M) is a function, called the hamiltonian function. Exercise 3 Show that {f, H} = 0 if and only if f Is constant along integral curves of
A function I as in the previous exercise is called an Integral of motion (or a first integral or a constant of motion) for the hamiltonian system (M, w, H). In general, hamiltonian systems do not admit integrals of motion which are independent of the hamiltonian function. Functions f,,. . , on M are said to , be independent if their differentials (df, are linearly independent at . all points p in some open dense subset of M. Loosely speaking, a hamiltonian system is (completely) integrable if it has as many commuting integrals of motion as possible. Commutativity is with respect to the Poisson bracket. Notice that, if f,... . , are commuting integrals of motion for a hamiltonian system (M, w, H), then, at each p E M, their hamiltonian vector fields generate an isotropic subspace .
.
of
X13) =
f3 } = 0
If fi,... fn are independent at p, then, by symplectic linear algebra, n can be at ,
most half the dimension of M. 'The Lie bracket of two vector fields is their commutator, where they are regarded as firstorder differential operators. A bilinear function {., .} : x C°°(M) —. C°°(M) is a Lie bracket if it is antisymmetric, i.e., {f,g} = —{g,f}, Vf,g C°°(M), and satisfies the Jacobi identity:
If,{g,h}}+{g,{h,f}}+{h,{f,g}}=0
Vf,g,hE C°°(M)
1.1.
Symplectic Toric Manifolds
93
Definition 1.1.7. A hamiltonian system (M, w, H) is (completely) integrable if it which dim M independent integrals of motion, fi = H, 12,. admits n are pairwise in involution with respect to the Poisson bracket, i.e., = 0, .
.
for all i,j. Example. A hamiltonian system (M, u, H) where M is 4-dimensional is integrable if there is an integral of motion independent of H (the commutativity condition is automatically satisfied). For interesting examples of integrable systems, see [7]. Let (M, H) be an integrable system of dimension 2n with integrals of motion Ii H, 12,..., fr,. Let CE W' be a regular value off fe). The corresponding level set, f '(c), is a lagrangian submanifold. A submanifold Y of a 2n-dimensional symplectic manifold (M, is lagrangian if it is n-dimensional and if iw — 0 where i: Y M is the inclusion map. Exercise 4 By following the flows, show that if the hamilton Ian vector fields Xf1,..., Xf,r,
are complete on the level f'(c) (i.e., if their flows are defined for all time), the connected components of f1 (c) are homogeneous spaces for R", i.e., are of the form x Tk for some 0
Any compact component of f '(c) must hence be a torus. These compact components, when they exist, are called Liouville tori. Theorem 1.1.8. (Arnold-Liouville [2]) Let (M, w, H) be an integruble system of dimension 2n with integrals of motion Ii = H, f2,. . , fr,. Let c be a regular value off (11,. . , f,,). The corresponding level I '(c) is a lagrangian submanifold of M. .
.
(a) If the flows of X11,.. , starting at a point p E (c) are complete, then the connected component of 1—1(c) containing p is a homogeneous space for Namely, there is an affine structure on that component with coordinates .
p,,, known as angle coordinates, in which the flows of the vector fields
are linear.
,
(6) There are coordinates known as action coordinates, complementary to the angle coordinates such that the 's are integrals of motion and (pj,. . . form a Darboux chart. .
.
.
,
.
Therefore, the dynamics of an integrable system is extremely simple and the system has an explicit solution in action-angle coordinates. The proof of part (a) — the easy, yet interesting, part — of the Arriold-Liouville theorem is sketched above. For the proof of part (b), see [2, 17]. Geometrically, part (a) of the Arnold-Liouville theorem says that, in a neighborhood of the value c, the map I M R" collecting the given integrals of
I. Symplectic Viewpoint
94
motion is a lagrangian flbration, i.e., it is locally trivial and its fibers are lagrangian submanifolds. The coordinates along the fibers are the angle coordithe nates.2 Part (b) of the theorem guarantees the existence of coordinates on action coordinates, which (Poisson) commute among themselves and which satisfy with respect to the angle coordinates. Notice that, in general, the ac} = .. , tion coordinates are not the given integrals of motion because . . . , do not form a Darboux chart.
Hamiltonian Actions
1.1.4
Definition 1.1.9. An action of a Lie group U on a manifold M is a group homomorphism
C —+ Diff(M) 9
Diff(M) is the group of diffeomorphisms of M. The evaluation map associDiff(M) is ated with an action G where
ev,1,: MxG —p M (p,g)
The action
is smooth if
1I)9(p)
is a smooth map.
We will always assume that an action is smooth. Example. Complete vector fields3 on a manifold M are in one-to-one correspondence with actions of R on M. The diffeomorphism M M associated to t R is the time-t map exp tX defined by the flow of the vector field X.
Let (M,w) be a symplectic manifold, and G a Lie group with an action Diff(M).
U
Definition 1.1.10. The action is a symplectic action if it is by symplectomorphisms, i.e., C —p Sympl(M,w) C Diff(M), where Sympl(M,
is the group of symplectomorphisms of (M, u.i).
Examples.
1. On the symplectic 2-sphere (S2, dO A dh) in cylindrical coordinates, the oneparameter group of diffeomorphisms given by rotation around the vertical axis, h) = (0 + t, h) (t IR) is a symplectic action of the group S1 R/(2ir), as it preserves the area form dO A dh. 2The name angle coordinates is used even if the fibers are not tori. 3A vector field is complete if its integral curves through each point exist for all time.
1.1.
95
Symplectic Tone Manifolds
2. On the symplectic 2-torus (T2, dO1 A do2), the one-parameter groups of dif+ t, ,02) = feomorphisms given by rotation around each circle, similarly defined, are symplectic actions of S1. (t E IR) and
Let (M, w) be a symplectic manifold, C a Lie group with Diff(M), and g the Lie algebra of G with dual vector space g5.
action
:C
Definition 1.1.11. The action i,L' is a hamiltonian action if there erists a map
pM satisfying the following two conditions:
• For each XE g, letpX M along X, and let subgroup {exptX
(p(p),X), be the component ofp be the vector field on M generated by the one-parameter E R} c C. Then = Zx#W
i.e., the function
is a hamiltonian function for the vector field X#.
• The map p is equivariant with respect to the given action the coadjoint action Ads of G on
of C on M and
forallg€C. The vector (M, w, C, p) is then called a hamiltonian C-space and p is called a moment map. Exercise 5
Check that complete symplectic vector fields on M are in one-to-one correspondence with symplectic actions of R on M, and that, similarly, complete hamiltonian vector fields on M are in one-to-one correspondence with hamiltonian actions of R on M.
Examples. Consider the previous set of two examples (as well as in the examples of Section 1.1.2). The first — regarding S2 — is an example of a hamiltonian action of S1 with moment map given by the height function, under a suitable identification of the dual of the Lie algebra of S' with IR. The second example - regarding T2 is not hamiltonian since the one-forms dO1 and dO2 are not exact. Exercise 6 Let C be any compact Lie group and H a closed subgroup of G, with g and —p the respective Lie algebras. The projection i is the map dual to the inclusion i: lj '—* g. Suppose that (M, w, G, is a hamiltonian G-space. Show :
that the restriction of the C-action to H is hamiltonian with moment map 0
4: M —
1. Symplectic Viewpoint
96
Exercise 7
Suppose that a Lie group G acts in a hamiltonian way on two symplectic The product M, —p , w3), j = 1,2, with moment maps manifolds manifold M1 x M2 has a natural product symplectic structure given by the sum of the pull-backs of the symplectic forms on each factor, via the two projections. Prove that the diagonal action of C on M1 x M2 is harniltonian with moment given by map M1 x M2
€M3
,
From now on, we concentrate on actions of a torus G =
= lRm/Zm.
Hamiltonian Torus Actions
1.1.5
The coadjoint action is trivial on a torus (a product of circles S1 x x S'). Hence, if G = 1" is an n-dimensional torus with Lie algebra and its dual both identified and g* 1R11, a moment map for an action of G on with euclidean space, g (M,w) is simply a map p: M —+ satisfying:
• For each basis vector X2 of IR'2, the function is a hamiltonian function for xr and is invariant under the action of the torus.
If p.: M is a moment map for a torus action, then clearly any of its translations p + c (c is also a moment map for that action. Reciprocally, any two moment maps for a given hamiltonian torus action differ by a constant. Example. On (C,wo = 1} by rotations
A di), consider the action of the circle S' ,
where k
Z is fixed. The action
51
:
{t E C
tES' Diff(C) is hamiltonian with moment
= = r dr A dO, p(rezO) = kr2 and the vector field on C corresponding to the generator 1 of g is
This can be easily checked in polar coordinates, since —
Exercise 8 Let = {(t1,... on
e
:
=
1,
for allj} be a torus acting diagonally
by
(tj (zj, . = ta" zn) where k1,. , k,Z are fixed. Check that this action is hamiltonian with moment map iz : C" —+ R" given by .
. ,
.
. . ,
-
=
(
+ constant
1.1.
97
Symplectic Toric Manifolds
Theorem 1.1.12. (Atiyah [3], Guillemin-Sternberg [25]) Let (M, w) be a compact Tm connected .symplectic manifold, and let Tm be an rn-torus. Suppose that M —i Rm. Then: Sympl(M,w) is a hamiltonian action with moment map
(a) the levels of p are connected; (b) the image of p is convex;
(c) the image of p is the convex hull of the images of the fixed points of the action.
The image p(M) of the moment map is called the moment polytope. A proof of Theorem 1.1.12 can be found in [36).
An action of a group G on a manifold M is called effective if it is injective as a map G Diff(M), i.e., each group element g e moves at least one point, that is, flPEMGP = {e}, where = {g e C g p = p} is the stabilizer of p. I
Exercise 9
Suppose that acts linearly on Let be the E weights appearing in the corresponding weight space decomposition (further discussed in Section 11.3.2), that is,
where, for
= (Ar,...
e m).v=e i>.2
Tm acts on the complex line VA(k) by
3v, Vk=1 n. (a) Show that, if the action is effective, then m n and the weights A(12) are part ofaZ-basis of Zm. A(1) (b) Show that, if the action is symplectic (hence, hamiltonian), then the (e It1
3
weight spaces VA(k) are symplectic subspaces.
(c) Show that, if the action is hamiltonian, then a moment map is given by
=
(k)II2 ( + constant
is the standard norm° and v where II weight space decomposition. Cf. Exercise 8.
+
... +
is the
(d) Conclude that, if T'1 acts on way, then any moment map
in a linear, effective and hamiltonian is a submersion, i.e., each differential —. R" (v E C't) is surjective.
=
that the standard inner product satisfies (v, w) = wO(v, iv) where and = In particular, the standard norm is invariant for
a symplectic complex-linear action.
The following two results use the crucial fact that any effective action Tm Diff(M) has orbits of dimension m; a proof may be found in [10].
I. Symplectic Viewpoint
98
Corollary 1.1.13. Under the conditions of the convexity theorem, if the Tm-action is effective, then there must be at least m + 1 fixed points.
Proof. At a point p of an rn-dimensional orbit the moment map is a submersion, i.e., (dpi ),,... (dpi,,),, are linearly independent. Hence, p(p) is an interior point of p(M), and p(M) is a nondegenerate convex polytope. Any nondegenerate convex polytope in Rm must have at least m +1 vertices. The vertices of p(M) are images o of fixed points. ,
Theorem 1.1.14. Let (M,w,
p) be a hamiltortian Tm-space. If the Tm-action
is effective, then dimM > 2m.
Proof. Since the moment map is constant on an orbit 0, for p
0 the exterior
derivative :
maps
to 0. Thus
= is the symplectic orthogonal of This shows that orbits 0 of a hamiltonian torus action are always isotropic submanifolds of M. In particular, by symplectic linear algebra we have that dim 0 dim M. Now consider an rn-dimensional orbit. 0 where
1.1.6
Symplectic Toric Manifolds
Definition 1.1.15. A symplectic toric manifold is a compact connected symplectic manifold (M, equipped with an effective hamiltonian action of a torus T of dimension equal to half the dimension of the manifold,
dimT= and with a choice of a corresponding moment map p. Exercise 10
Show that an effective hamiltonian action of a torus F' on a 2n-dimensionai symplectic manifold gives rise to an integrable system. Hint: The coordinates of the moment map are commuting integrals of motion.
DefinitIon 1.1.16. Two symplectic toric manifolds, (Mi, w,, p,), i = 1,2, are equivalent if there exists an isomorphism A : T1 T2 and a A-equivariant symplectomorphism M1 M2 such that = P2 °
_________________ 1.1.
Symplectic Toric Manifolds
99
Equivalent symplectic tone manifolds are often undistinguished. Examples of symplectic toric manifolds. 1.
The circle 51 acts on the 2-sphere (52, Wstandard = dO A dh) by rotations ett . (0, h) = (0 + t, h)
with moment map J.L = h equal to the height function and moment polytope [—1,1].
Equivalently, the circle S' acts on P1 = C2 by e" . [20 zij = [z0 form WFS =
0/
with the Fubini-Study This is hamiltonian with and moment polytope —
:
moment map p[zo zj]
2. Let (P2, wFs) be 2-(complex-)dimensional complex projective space equipped with the Fubini-Study form defined in Section 1.2.3. The T2-action on P2 by [z0 : z1 z2j = [zo : zi z2] has moment map 1z112
2
\Iz012 + IziI2 + Iz2I2
(0,0)
(0,
2012 + 12,12 + 1z2I2
1. Symplectic Viewpoint
100
The fixed points get mapped as
[1:0:0] [0:1:0]
'—i
(0,0)
[0:0:1]
Not ice that the stabilizer of a preimage of the edges is S', while the action
is free at preimages of interior points of the moment polytope. ExorcIse 11
Compute a moment polytope for the T3-action on P3 as = [zQ
:
e'03z3]
Exercise 12
Compute a moment polytope for the T2-action on (e'9,
(Izo
Zi], fwo
w1])
=
({zo
etOzjl,
x Ipi
as :
101
Classification
1.2.
1.2
Classification
Recall that a 2n-dimensional symplectic toric manifold is a compact connected w) equipped with an effective harniltonian action of an symplectic manifold and with a corresponding moment map p.: M In this lecture n-torus we describe the classification of equivalence classes of symplectic toric manifolds
by their moment polytopes p(M). Symplectic reduction is the quotienting technique which we use for the construction of a symplectic toric manifold out of an appropriate polytope, thus proving the existence part in the classification theorem. 1.2.1
Deizant's Theorem
We now define the class of polytopes4 which arise in the classification of symplectic toric manifolds.
Definition 1.2.1. A Delzant polytope
in
is a polytope satisfying:
• simplicity, i.e., there are n edges meeting at each vertex;
• rationality, i.e., the edges meeting at the vertex p are rational in the sense that each edge is of the form p + tu2, t 0, where smoothness, i.e., for each vertex, the corresponding u1,. . ,u,, can be chosen to be aZ-basis of Z'2. .
Examples of Delzant polytopes hi R2:
Ni_
The dotted vertical line in the trapezoidal example is there just to stress that it is a
picture of a rectangle plus an isosceles triangle. For "taller" triangles, smoothness would be violated. "Wider" triangles (with integral slope) may still be Deizant.
The family of the Delzant trapezoids of this type, starting with the rectangle, correspond, under the Delzant construction, to
surfaces; see Lecture 3. <0>
4A polytope in R" is the convex hull of a finite number of points in A convex polyhedron is a subset of R" which is the intersection of a finite number of affine half-spaces. Hence, polytopes coincide with bounded convex polyhedra.
1. Symplectic Viewpoint
102
Examples of polytopes which are not Deizant:
The picture on the left fails the smoothness condition, since the triangle is not
isosceles, whereas the one on the right fails the simplicity condition.
Deizant's theorem classifies (equivalence classes of) symplectic toric manifolds in terms of the combinatorial data encoded by a Delzant polytope. Theorem 1.2.2. (Delzant [15]) Tone manifolds are classified by Delzant polytopes. More specifically, the bijective correspondence between these two sets is given by the moment map: {
{ Deizant polytopes}
tonic manifolds}
p(M)
In Section 1.2.5, we describe the construction which proves the (easier) existence part, or surjectivity, in Deizant's theorem. In order to prepare that, we will next give an algebraic description of Deizant polytopes.
be a Delzant polytope in
Let
and with d facets.6 Let
1,. . . , d, be the primitive7 outward-pointing normal vectors to the facets of as an intersection of halfspaces Then we can describe
i=
i=1,...,d}
forsome.\2eR.
Example. For the picture on the next page, we have
= {x E (R2)* I (x,(—1,0))
<0, (x,(0,—1))
0, (x,(1,1)) 1)
with its dual via the euclidean inner product, it may be more clear 5Although we identify to see in (R") for Delzant's construction. 6A face of a polytope is a set of the form F = Pfl{x E R" I 1(x) = c} where c E R satisfies f(x) c, Vx P. A facet of an n-dimensional polytope is an (n — 1)and f dimensional face. 7A lattice vector v E is primitive if it cannot be written as v = ku with U E Z", k E Z and
IkI > 1; for instance, (1,1), (4,3), (1,0) are primitive, but (2,2), (4,6) are not.
1.2.
103
Classification
V3 —
V2 = (—1,
v1
1.2.2
Orbit Spaces
Let
C —i
=(0,—lj
Diff(M) be any action.
Definition 1.2.3. The orbit of C through p E M is (p) g E G}. The stabilizer (or isotropy) of p E M is {g e C p}. = J
Exercise 13
If q is in the orbit of p, then Gq and
are conjugate subgroups.
Definition 1.2.4. We say that the action of G on M is:
• transitive if there is just one orbit, • free if all stabilizers are trivial {e}, • locally free if all stabilizers are discrete. Let
be the orbit equivalence relation; for p, q E M, p
q
p and q are on the same orbit.
I. Symplectic Viewpoint
104
is called the orbit space. Let
The space of orbits M/G := M/
ir:M—*M/G p '—p
orbit through p
be the point-orbit projection. We equip M/G with the weakest topology for which ir is continuous, i.e., U C M/G is open if and only if 1r '(U) is open in M. This is called the quotient topology. This topology can be "bad." For instance: Example. Let G = C\{0) act on M = by A
= multiplication by A.
'—p
The orbits are the punctured complex lines (through non-zero vectors z E Ca), plus one "unstable" orbit through 0, which has a single point. The orbit space is
M/G = pn_l U {point}. The quotient topology restricts to the usual topology on The only open set containing {point} in the quotient topology is the full space, hence the topology in M/G is not Hausdorif. However, it suffices to remove 0 from to obtain a Hausdorif orbit space:
r—'. Then there is also a compact (yet not complex) description of the orbit space by taking only unit vectors under the action of the circle subgroup:
=
1.2.3
(cn\{o})/(c\{o})
=
Symplectic Reduction
Let = form on
= >dxkAdyk = >rkdrkAd9k be the standard symplectic Consider the following S'-action on t E S1 '—p
The action
= multiplication by t.
is hamiltonian with moment map
—i R
,a: z
'—p
—
+ constant
since
=
X#
—
ZX#W =
a a a —-l-——+...+-—-and 00, 002
=
1.2.
105
Classification
is the unit sphere. The
then (0) = If we choose the constant to be orbit space of the zero level of the moment map is
=
= pn_l 1,
Moreover, this construction induces a symplectic form on instance of the following major theorem.
as a particular
C, ji) be a hamil'(0) '—i M be the inclusion
Theorem 1.2.5. (Marsden-Wernstein [34], Meyer [37)) Let (M,
tonian C-space for a compact Lie group C. Let i map. Assume that G acts freely on 1r'(O). Then (a) the orbit space Mred =
(b) ir : jr'(O)
:
p' (0)/C is a manifold,
Mreci is a principal C-bundle, and
(c) there is a symplectic form Wred Ofl Mred satisfying
=
IT*Wred.
For a proof of Theorem 1.2.5, see for instance [12]. Definition 1.2.6. The pair (Mred, Wred) is called the symplectic reduction of (M, w)
with respect to C and p (or the reduced space, or the symplectic quotient, or the Marsden-Weinstein-Meyer quotient, etc.).
Remark. When M is Kähler, i.e., has a compatible complex structure, and the action of C preserves the complex structure, then the symplectic reduction has a natural Kähler structure. Example. Consider the S'-action on w0) which, under the usual identification of with corresponds to multiplication by eit. This action is hamiltonian with a moment map p —f R given by
p '(O)/S' =
with the so-called Fubini-Study symplectic form Exercise 14 Recall that
11"
S2 as real 2-dimensional manifolds. Check that WFS
where
=
dO
=
A dh is the standard area form on S2.
equipped
1. Symplectic Viewpoint
106
1.2.4
Extensions of Syinpiectic Reduction
We consider three basic extensions of the procedure of symplectic reduction.
1. Reduction for product groups. Let C1 and 02 be compact connected Lie groups and let C = x 02. Then Suppose that (M,w, C, ii) is a hamiltonian 02 and o C-space with moment map
ii: M Write ii = (111, v2) where
: M —*
for i =
1,2.
The fact that jI is
equivariant implies that a'1 is invariant under 02 and 112 is invariant under Now reduce (M, w) with respect to the C1-action. Let Z1
= "I'(O)
Assume that 01 acts freely on Z1. Let M1 = Z1/G1 be the reduced space and let w1 be the corresponding reduced symplectic form. The action of 02 on Z1 commutes with the G1-action. Since G2 preserves w, it follows that G2 acts symplectically on (M1 , Wi). Since preserves also preserves 112 t1 where t1 Z1 '—i M is inclusion. Thus 112 0 is constant on fibers of Z1 M1. We conclude that there exists a smooth map
Exercise 15 Show that: (a) the map is a moment map for the action of C2 on (b) if C acts freely on v'(O,O), then G2 acts freely on is
a natural
and and there
This technique of performing reduction with respect to one factor of a prod-
uct group at a time is called reduction in stages. It may be extended to reduction by a normal subgroup H C C and by the corresponding quotient group C/H. 2. Reduction at other levels. Suppose that a compact Lie group C acts on a symplectic manifold (M,w) in a hamiltonian way with moment map p : M Let To e reduce at the level of p, we need to be preserved by C, or else take the C-orbit of (c), or equivalently take the inverse image (Of) of the coadjoint orbit through or else take the quotient by the maximal subgroup
1.2.
107
Classification
of C which preserves (c). Of course the level 0 is always preserved. Also, when C is a torus, any level is preserved and reduction at for the moment map is equivalent to reduction at 0 for a shifted moment map 4: M —
3.
Orbifold singularities. Roughly speaking, orbifolds (introduced by Satake in [42]) are singular manifor some finite group folds where each singularity is locally modeled on 1' c GL(m; IR). For the precise definition, let Ml be a Hausdorif topological space satisfying the second axiom of countability.
Definition 1.2.7. An orbifold chart on Ml is a triple (V, F, connected open subset of some euclidean space
where V is a
F is a finite group which
acts linearly on V so that the set of points where the action is not free has codimension at least two, and V Ml is a F-invariant map inducing a homeomorphism from V/F onto its image U C IMI. An orbifold atlas A for MI is a collection of orbifold charts on Ml such that: the collection of images U forms a basis of open sets in IMI, and the charts are compatible in the sense that, whenever two charts and satisfy Ui ç U2, there exists an injective
F1 —+ F2
and a A-equivari ant
o such that = Two orbifold atlases are equivalent if their union is still an atlas. An rn-dimensional orbifold M is a Hausdorff topological space IMI satisfying the second axiom of countability, plus an equivalence class of orbifold atlases on IMI.
open embedding
: V1 —* V2
Notice that we do not require the action of each group F to be effective. Given a point p on an orbifold M, let (V, 1', be an orbifold chart for a neighborhood U of p. The orbifold structure group of p, Fr,, is (the isomorphism class of) the isotropy group of a pre-image of p under We may always choose an orbifold chart (V, F, such that is a single point (which is fixed by F). In this case F F',, and we say that is a structure chart for p. An ordinary manifold is a special case of orbifold where each group F is the identity group. Quotients of manifolds by locally free actions of Lie groups are orbifolds. In fact, any orbifold M has a presentation of this form obtained as follows. Given a structure chart (V, F, for p E M with image U, the
orbifold tangent space at p is the quotient of the tangent space to V at '(p) by the induced action of F: :=
The collection of the orbifold tangent spaces at all p, builds up the orbifold tangent bundle TM, which has a natural structure of smooth manifold outside the zero section. The general linear group GL(m; R) acts locally freely
I. Symplectic Viewpoint
and M (TM — 0)/GL(m; R). Choosing a riemannian metric and taking the orthonorrnal frame bundle, O(TM), we present M as O(TM)/O(m).
on TM \
0,
be an n-torus acting on a symplectic manifold (M, w) Example. Let C = For any E in a hamiltonian way with moment map ,i M Suppose that is a regular is preserved by the the level be the is a submanifold of codimension n. Let Then value of stabilizer of p, and g' its Lie algebra. Note that regular
dp7'
is surjective at all p
the stabilizers on
are
finite.
the orbit space By the slice theorem (see, for instance, [6, 12]), near is modeled by S/C7', where S is a G7'-invariant disk in is an orbifold. through p and transverse to 0,. Hence, Example. Consider the S1-action on C2 by eth (zi, z2) = some fixed integer k 2. This is hainiltonian with moment map
ii:
C2
—.+
R
+
(zj,z2) Any
< 0 is a regular value and
stabilizer of (zi, z2)
e'8z2) for
1z212)
a 3-dimensional ellipsoid. The 0, and is
is
is {1} if z2
is called a teardrop orbifold or 0. The reduced space conehead; it has one cone (also known as a dunce cap) singularity with cone that is, a point with orbifold structure group Zk. angle
if z2 =
Example. Let S1 act on C2 by etO . (zj, Z2) = (eikOzi, eieoz2) for some integers
k, t 2. Suppose that k and £ are relatively prime. Then (z1, 0) (0, z2)
(zi,z2) The quotient larities, one with angle
has stabilizer Zk has stabilizer has stabilizer {1}
(for (for z2
(for z1,z2
0) 0) 0)
a football orbifold. It has two cone singuand another with
8By Sard's theorem, the singular values of
form a set of measure zero.
1.2.
109
Classification
Example. More generally, the reduced spaces of 5' acting on Ctm by • (z,,. .
—
.
. .
are called weighted (or twisted) projective spaces. The differential-geometric notions of vector fields, differential forms, exterior
differentiation, group actions, etc., extend naturally to orbifolds by gluing corresponding local r-invariant or F-equivariant objects. In particular, a symplectic orbifold is a pair (M, w) where M is an orbifold and w is a closed 2-form on M which is nondegenerate at every point of M. Definition 1.2.8. A syinpiectic tone orbifold is a compact connected symplec-
tic orbifold (M, u) equipped with an effective hamiltonian action of a torus T of dimension equal to half the dimension of the orbifold,
dimT= and with a choice of a corresponding moment map ji. Symplectic toric orbifolds have been classified by Lerman and Tolman (33] in a theorem which generalizes Delzant's theorem: a symplectic toric orb-
ifold is determined by its moment polytope plus a positive integer label attached to each of the polytope facets. The polytopes which occur in the Lerman-Tohnan classification are more general than the Delzant polytopes iii the sense that only simplicity and rationality are required; the edge vecIn the case where the need only form a rational basis of tors u1,. .. integer labels are all equal to 1, the failure of the polytope smoothness accounts for all orbifold singularities. Throughout the rest of these notes, we concentrate on the manifold case.
1.2.5
Deizant's Construction
Following [15, 24], we prove the existence part (or surjectivity) in Delzant's theorem, by using symplectic reduction to associate to an n-dimensional Delzant polytope A a symplectic tone manifold Ttm, Let A be a Delzant polytope with d facets. Let v2 E 7U', i = 1,..., d, be the primitive outward-pointing normal vectors to the facets. For some A
E
Let e1 =(1,0,...,O),...,ed =(0,...,O,1) be the standard basis ir:
lRd
,.
—+ v,.
Consider
1.
110
Sympiectic Viewpoint
Lemma 1.2.9. The map it is onto and maps Zd onto
for the The set {vi,.. . , Vd} spans form a basis following reason. At a vertex p, the edge vectors u1,. . , which, by a change of basis if necessary, we may assume is the standard for basis. Then the corresponding primitive normal vectors to the facets meeting at p are symmetric (in the sense of multiplication by —1) to the ui's, hence form a
Proof. The set {ei,... , ed} is a basis of
.
0
basis of Z12. Therefore, it induces a surjective map, still called it, between tori:
II
II
Ta
The kernel N of it is a (d — n)-dimensional Lie subgroup of Td with inclusion i: N '—÷ Ti'. Let a be the Lie algebra of N. The exact sequence of tori Td
N
1
—f
1
induces an exact sequence of Lie algebras
with dual exact sequence (Rd)* _!1+ n*
0 —4
0.
Now consider with symplectic form w0 = hamiltonian action of Td given by
(eitl,.
. .
The moment map is qS : Cd ,Zd)
- (z1,.
..
,
*
,Zd)
= (eitlz1,.
and standard
. .
defined by
=
+
..
constant
we choose the constant to be (A1,. . , Ad). By Exercise 6, the subtorus on Cd in a hamiltonian way with moment map
where acts
dzk A d2k,
.
o
Let Z = (i5
o
Claim 1. The set Z is We postpone the
:
Cd
-
be the zero-level set. compact and N acts
freely on Z.
proof of this claim until further down.
N
111
Classification
L2.
is a regular value of
Since i' is surjective, 0 E
o
Hence,
Z is a
of (real) dimension 2d — (d — n) = d + n. The orbit compact submanifold of = Z/N is a compact manifold of (real) dimension dim Z — dim N = space is a principal N-bundle (d + n) (d — n) = 2n. The point-orbit map p: Z over Consider the diagram
z where j: Z
Cd
is inclusion. The Marsden-Weinstein-Meyer theorem guarantees the existence of a symplectic form wA on satisfying '—f
= j*wo Z is connected, the compact symplectic 2n-dimensional manifold (Ms, is also connected. Since
Proof of Claim 1. The set Z is clearly closed, hence in order to show that it is compact it suffices (by the Heine-Borel theorem) to show that Z is bounded. Let be the image of by irk. We will show that = Lemma 1.2.10. Let y E
Then:
ye
y is in the image of Z by
Proof.
The value y is in the image of Z by conditions hold:
if and only if both of the following
1. y is in the image of 2. i*y=0.
Using the expression for conditions are equivalent to:
and the third exact sequence, we
see
1. (y,ej)A2fori=1,...,d; 2. y =
for some x
Suppose that the second condition holds, so that y = (y,e2)
A2,Vi
(x). Then
(lr*(x),ej) < A2,Vi
xEA.
that these
I. Symplectic Viewpoint
112
Thus, y E q5(Z)
yE
= A'.
0
= A', Since we have that A' is compact, that is a proper map and that we conclude that Z must be bounded, and hence compact. It remains to show that N acts freely on Z. be the set of indices for the n Pick a vertex p of A, and let I = {i1,. . = 7r* (p). Then p is characterized facets meeting at p. Pick z E Z such that A, where i ranges in I: by n equations (p, = (p,v,) =
A,
(p,lr(ej)) =
A,
= (çb(z),e2) =
A,
i-th coordinate of cb(z) is equal to A,
zi=o. those z's are points whose coordinates in the set I are zero, and whose other coordinates are nonzero. Without loss of generality, we may assume that I {1,... ,n}. The stabilizer of z is Hence,
= {(t1,.
E
.
As the restriction ir : —p IR" maps the vectors er,. .. , v1,. . , v,, of Z" (respectively), at the level of groups, ir bijective. Since N = ker(ir Td T"), we conclude that N fl
to a Z-basis T" must be = {e}, i.e., N, = {e}. Hence all N-stabilizers at points mapping to vertices are trivial. But this was the worst case, since other stabilizers N,' (z' E Z) are contained in sta.
:
bilizers for points z which map to vertices. This concludes the proof of Claim 1.
0 Given a Delzant polytope A, we have constructed a symplectic manifold (Ms, where = Z/N is a compact 2n-dimensional manifold and is the reduced symplectic form.
Claim 2. The manifold having image
is a hamiltonian T"-space with a moment map = A. Proof of Claim 2. Let z be such that 4(z) = lr* (p) where p is a vertex of A, as in the proof of Claim 1. Let o• : T" be the inverse for the earlier bijection (Td), T". Since we have found a section, i.e., a right inverse for ir, in the exact sequence ' " d 1
—iT"
1,
the exact sequence splits, i.e., becomes like a sequence for a product, as we obtain an isomorphism
(i,o) : N x T"
Td.
113
L2. Classification
factor (or, more rigorously, The action of the quotient MA = Z/N.
C Td) descends to the
It remains to show that the ¶"7-action on MA is hamiltonian with appropriate moment map.
Consider the diagram
z
Cd ..±4 (lRd)*
11*
(lRfl)*
MA
where the last horizontal map is simply projection onto the second factor. Since the composition of the horizontal maps is constant along N-orbits, it descends to a map MA —p
which satisfies
/AAOp=O.*OcbOj.
By Exercise 15 on reduction for product groups, this is a moment map for the action of TTh on (MA, WA). Finally, the image of is: liA(MA) = (ILA op)(Z) = (cc
because cL(Z) = and We conclude that (MA, WA,
o4oj)(Z) =
=
= (ir o = id. is the required toric manifold correspond-
ingto.&
0
Exercise 16 Let be an n-dimensional Delzant polytope, and let (MA, WA, T", be the associated symplectic toric manifold. Show that maps the fixed points of bijectively onto the vertices of
By the remark in Section 1.2.3, Delzant's construction yields a natural Kahier structure on each symplectie toric manifold.
1.2.6
Idea Behind Deizant's Construction
The space Rd is universal in the sense that any n-dimensional polytope with d facets can be obtained by intersecting the negative orthant with an affine plane A. Given to construct A first write as:
i=l,...,d}.
I. Symplectic Viewpoint
114
Define
ir: Rd —*
Rd.
lr*:
with dual map
u—+
Then lr* — A: R't —p W' is an affine map, where A =(Al,...,Ad). Let A be the
image of ir' —
A.
Then A is an n-dimensional affine space.
=
Lemma 1.2.11. We have the equality (ir —
fl A.
Proof. Let x e R8. Then —
A)(x) E
—
A,e2) —
O,Vi O,Vi
A2,Vi
0
is the image of the moment map for fl A. Now We conclude that the standard hamiltonian action of T" on Cd —+
(z1,. .,Zd)
Rd
'—4
.
.,1Zd12)
Facts.
• The set q5'(A) c
Cd is a compact submanifold. Let i Cd q51(A) denote inclusion. Then is a closed 2-form which is degenerate. Its kernel is an integrable distribution. The corresponding foliation is called the null foliation. :
• The null foliation of
is
a principal fibration, so we take the quotient:
N 'I
:= Let
be the reduced symplectic form.
• The (non-effective) action of = Nx on p1(A) has a "moment map" with image = (By "moment map" we mean a map satisfying the usual definition even though the closed 2-form is not symplectic.)
I. Symplectic Viewpoint
116
Exercise 18 Follow through the details of Delzant's construction for the case of
(n =
1,d
=
2).
V,
V2 = v, A, = 0 and A2 = a.
= to, a) C
Let v(= I) be the standard basis vector in R. Then
is
described by
(z,—v)O where vi =
and
V
—ti
The projection R2
R
—f-.
'—b
—v
'—i
V
has kernel equal to the span of (e, + e2), so that N is the diagonal subgroup of T2 = S' x S'. The exact sequences become 1
o
—+
—b
N
—L
T2
t
'—i
(t,t) (ti,t2)
n
-L '—.
o
—. R
*
—p
x
s' —i
—.
tj-'t2
—
R
1R2
(x,x) (xi,x2) 2 (R)
'—+
X2Xl
—i
n
0
(—x,x)
(x,,x2)
X1
The action of the diagonal subgroup N =
+X2
ES'
x
S'}
= has moment map
= with zero-level set (z
0
= {(z,,z2) E C2 :
1z112
Hence, the reduced space is a projective space; (j*
=
1?'
+
2aJ
on C2,
1.2.
Classification Exercise 19 Consider the standard (S' )3-action on P3: :
: z2 : za]
Izo
:
&°'zi
:
Exhibit explicitly the subsets of P3 for which the stabilizer under this action is {1), S', (S')2 and (51)3. Show that the images of these subsets under the
moment map are the interior, the facets, the edges and the vertices, respectively.
Exercise 20 What would be the classification of symplectic toric manifolds if, instead of the equivalence relation defined in Section 1.1.6, one considered to be equivalent T1, i = 1,2, related by an isomorphism A : T1 —. T2 and a those (Me, A-equivariant symplectomorphism Mj —. M2 such that:
(a) the maps
and
l.L2
o
are equal up to a constant?
(b)
Exercise 21
(a) Classify all 2-dimensional Delzant polytopes with 3 vertices, i.e., triangles, up to translation, change of scale and the action of SL(2; Z). hUnt: By a linear transformation in SL(2; Z), we can make one of the angles in the polytope into a square angle. How are the lengths of the two edges forming that angle related? (b) Classify all 2-dimensional Deizant polytopes with 4 vertices, up to trans-
lation and the action of SL(2; Z). Hint: By a linear transformation in SL(2; Z), we can make one of the angles in
the polytope into a square angle. Check that automatically another angle also becomes 900. (c) What are all the 4-dimensional symplectic toric manifolds that have four
fixed points?
ExercIse 22 Let be the n-simplex in spanned by the origin and the standard basis vectors (1,0 0),. . . , (0 0, 1). Show that the corresponding symplectic toric manifold is projective space, MA =
Exercise 23 Which 2n-dimensionai toric manifolds have exactly n + 1 fixed points?
I. Symplectic Viewpoint
118
1.3
Moment Polytopes
The general theme behind this and Lecture 6 is how to understand a toric manifold from its polytope. After reviewing the basics of Morse theory following [39], we compute the homology of symplectic toric manifolds using Morse theory; an appropriate Morse function is provided by a moment map with respect to a suitable circle subgroup. We go on to describe elementary surgery constructions based on symplectic reduction, which hold in the category of symplectic tone manifolds.
1.3.1
Equivariant Darboux Theorem
The following two theorems describe standard neighborhoods of fixed points. Their proofs rely on the equivariant version of the Moser trick and may be found in [27]. Theorem 1.3.1. (Equivariant Darboux) Let (M, w) be a 2n-dimensional symplectic manifold equipped with a symplectic action of a compact Lie group G, and let
q be a fixed point. Then there exists a C-invariant chart (U,xi,.. . centered at q and G-equivariant with respect to a linear action of G on that ,
,
such
wIu=EdxkAdyk. A suitable linear action on Ofl
is equivalent to the induced action of C
TqM. In particular, if C is a torus, this linear action is characterized by the
weights occuring in the representation of C on TqM. Now any symplectic action is locally hamiltonian. In order to prepare the computation of the Betti numbers of a symplectic toric manifold by using a moment map as a Morse function, we next
specify the local picture for a moment map near a fixed point of a hamiltonian torus action. Theorem 1.3.2. Let w, Tm, ji) be a hamiltonian Tm-space, where q is a fixed point. Then there exists a chart (U, x1,... , x,,, centered at q and weights , E Zm such that
(a) and (b)
=p(q) — This theorem guarantees the existence of a Darboux chart centered at any fixed point where the moment map looks like the moment map for a linear action
119
1.3. Moment Polytopes
In other words, the real analogue of the model in Exercise 9 is a general local picture near a fixed point of a hamiltonian torus action. on
Exercise 24
Show that for a symplectic toric manifold the weights A1 1),••
,
form a
Z-basis of Zm.
As a consequence of Theorem 1.3.2 and of the previous exercise, the bijection claimed in Delzant's theorem is well-defined. Indeed, each vertex of a moment polytope satisfies the simplicity, rationality and smoothness conditions which characterize Deizant polytopes. Another consequence of Theorem 1.3.2 is that a moment map for a symplectic toric manifold yields a lot of Morse functions, as we will next explore.
1.3.2
Morse Theory
Let M be an rn-dimensional manifold. A smooth function f: M R is a Morse function on M if all of its critical points are nondegenerate.9 R is the maximal dimension The index of a bilinear function H: Rm x where H is negative definite. The nullity of H is the dimension of such that H(v, w) = 0 its nullspace, that is, the subspace consisting of all v e for all w C lRtm. Hence, a critical point q of f : M —' R is nondegenerate if and x R has nullity equal to zero. only if the hessian Hq : Let q be a nondegenerate critical point for f: M IR. The index off at q is the index of the hessian Hq : lRtm x lRtm —÷ R. This is well-defined, i.e., the index is independent of the choice of local coordinates. Moreover, the Morse lemma states that there is a coordinate chart (U,xi,. . , Xm) centered at q such that of a subspace of
.
= f(q)
—
(x1)2
—...
—
(XA)2 + (XA+1)2
+... + (x)2
where A is the index of f at q. In particular, nondegenerate critical points are necessarily isolated.
Let f be a Morse function on M. For a C R, let Ma = f'(—oo,a] = (pc Mj 1(p) S a}. 9A point q E M is a critical point off if matrix
= 0. A critical point is nondegenerate if the hessian
( 02f )q
is
nonsingular, where the x, 's are local coordinates near q. (The condition that the hessian
matrix is nonsingular is independent of the choice of coordinates.) The hessian matrix defines a symmetric bilinear function Hq : R given by inner product
/ 821 \
w)
/q and also called the hessian of f at q relative to the local coordinates the hessian is in fact the expression in coordinates of a natural bilinear form on the tangent space at q.
I. Symplectic Viewpoint
120
Theorem 1.3.3. (Morse [40], Milnor (39])
[a, b], consisting of all p E M with (a) Let a < b and suppose that the set a f(p) b, is compact, and contains no critical points of f. Then
is a deformation retract of Mi', so Furthermore, diffeomorphic to M'-' is a homotopy equivalence. that the inclusion map
(b) Let q be a nondegenerate critical point with index A and f(q) = c. Suppose that f'[c — c, c + €] is compact, and contains no critical point of f other has than q, for some e > 0. Then, for all sufficiently small e, the set with a A-cell attached. the homotopy type of is compact, then the manifold M has the homotopy type of a CW-complex with one cell of dimension A for each critical point of index A.
(c) If each set
A k-cell is simply a k-dimensional disk Dk, and it gets attached along its
boundary S'_'. Morse's original treatment did not include part (c) of Theorem 1.3.3. Instead, his main results were phrased in terms of inequalities. Let dim Hk(M) be the k-th Betti number of M. Let M be a compact manifold and f a Morse function on M. Let CA be the number of critical points of f bk(M)
with index A.
Theorem 1.3.4. (Morse inequalities [40]) (a) bA(M)
=
,
and
(c) A perfect Morse function is a Morse function for which the inequalities in the previous statement are equalities.
Corollary 1.3.5. If all critical points of a Morse function f have even index, then f is a perfect Morse function.
Homology of Symplectic Toric Manifolds
1.3.3
Let (M, w, p) be a 2n-dimensional symplectic toric manifold. Choose a suitably generic direction in by picking a vector X whose components are independent
over Q. This condition ensures that: • the one-dimensional subgroup, TX C T't, generated by the vector X in
is
dense
121
1.3. Moment Polytopes
• X is not parallel to the facets of the moment polytope
:= j.t(M), and
have different projections along X.
• the vertices of Exercise 25
Check that the fixed points for the r-action are exactly the fixed points of the action restricted to TX, that is, are the zeros of the vector field, X# on M corresponding to the
R
projection
M
:= (p, X) : M —' R be the projection of along X. By definition Let of moment map, pX is a hamiltonian function for the vector field X# generated by X. We conclude from the previous exercise that the critical points of pX are precisely the fixed points of the Ta-action. then there exists a By Theorem 1.3.2, if q is a fixed point for the ,A") E such at q and weights centered chart (U,x1,. .. that . .
. .
=
=
—
Since the components of X are independent over Q, all coefficients (A(k), X) are nonzero, so q is a nondegenerate critical point of Moreover, the index of q is twice the number of labels k such that are precisely X) <0. But the the edge vectors u, which satisfy Deizant's conditions. Therefore, geometrically, the index of q can be read from the moment polytope by taking twice the number of edges whose inward-pointing edge vectors at point up relative to is a perfect X, that is, whose inner product with X is positive. In particular, Morse function. By applying Corollary 1.3.5 we conclude that:
Theorem 1.3.6. Let X E have components independent over Q. The degree-2k homology group of the symplectic toric manifold (Al, w, T, has dimension equal to the number of vertices of the moment polytope where there are exactly k (primitive inward-pointing) edge vectors which point up relative to the projection along the X. All odd-degree homology groups of M are zero.
I. Symplectic Viewpoint
122
Of course, by Poincaré duality (or by taking —x instead of X), the words "point up" may be replaced by "point down". ExercIse 26 Let (M, T, p) be a symplectic toric manifold. What is the Euler characteristic of M?
1.3.4
Symplectic Blow-Up
Let L be the tautological line bundle over
1,
that is,
L={([p1,z)IpECTh\{O}, z=ApforsomeAEC} given by ([p], z) i-4 [p]. The fiber of L over the point represented by that point. is the complex line in
with projection to [p] E
at the origin is the total space of the bundle L. DefinItion 1.3.7. The blow-up of The corresponding blow-down map is the map /3: L —* C" defined by j3( [ps, z) = z.
Notice that the total space of L may be decomposed as the disjoint union of two sets,
E:={([pj,O) IPEC"\{O}} and
S:={([p],z)IpeC"\{O} , z=Ap for some
The set E is called the exceptional divisor; it is ditfeomorphic to P"1 and gets mapped to the origin by /3. On the other hand, the restriction of /3 to the complementary set S is a diffeomorphism onto C" \ {O}. Hence, we may regard L as by smoothly replacing the origin by a copy of being obtained from There are actions of the unitary group U(n) on all of these sets induced by the standard linear action on C", and the map /3 is U(n)-equivariant. Definition 1.3.8. A blow-up symplectic form on L is a U(n) -invariant symplectic form t. such that the difference w — fl*wo is compactly supported, where w0 = A is the standard symplectic form on C". Two blow-up symplectic forms are called equivalent if one is the pullback of the other by a U(n)-equivariant diffeomorphism of L. Guillemin and Sternberg [26J have shown that two blow-up symplectic forms are equivalent if and only if they
have equal restrictions to the exceptional divisor E c L. Let 1" (e > 0) be the set of all blow-up symplectic forms on L whose restricis CWFs, where is the Fubim-Study tion to the exceptional divisor E described in Lecture 2. An c-blow-up of C" at the origin is a pair form on (L,w) with w E
1.3. Moment Polytopes
123
Let (M, w) be a 2n-dimensional symplectic manifold. It is a consequence of z1, - . . , the Darboux theorem that, for each point q M, there exists a chart where centered at q and with image in =
It is shown in [26] that, for c small enough, we can perform an c-blow-up of M at q modeled on at the origin, without changing the symplectic structure outside of a small neighborhood of q. The resulting manifold is theii called an c-blow-up of M at q. Example. Let
C) be the P' -bundle over obtained by projectivizing the direct sum of the tautological line bundle L with a trivial complex line bundle. Consider the map —p
13:
([p], [Ap : w])
'—p
r
[Ap :
where lAp w] on the right represents a line in forgetting that, for 1, that line sits in the 2-complex-dimensional subapace C C [p] e Notice that (3 maps the exceptional divisor :
each
E:={([p],[0:... :0:
r,
to the point [0: ... 0: 1] :
whereas
(3
is
a diffeomorphism on the complement ,
... :0: 1]}
Therefore, we may regard l?(L
C) as being obtained from by smoothly reThe space !F(L C) is the blow-up of at the point [0: ... 0: 1], and (3 is the corresponding blow-down map. The manifold P(L C) for n = 2 is known as the first liirzebruch surface. placing the point [0
:
... 0: 1] by a copy of :
:
Exercise 27 Write a definition for blow-up of a .symplectic manifold along a complex subby considering the of the normal bundle to the submanifold.
Symplectic blow-up is due to Gromov according to the fIrst printed exposition of this operation in [35].
1. Symplectic Viewpoint
124
1.3.5
Blow-Up of Toric Manifolds
Suppose that a compact Lie group acts on a symplectic manifold (M, w) in a hamiltonian way, and that q E M is a fixed point for the G-action. Then, by Theorem 1.3.1, there exists a Darboux chart (U, zr,.. . G-equivariant with respect to a linear action of C on of M relative to this chart, for e sufficiently small.
,
centered at q which is Consider an c-blow-up
Exercise 28 Check that C acts on the blow-up in a hamiltonian way. Describe the moment map.
Let A be an n-dimensional Deizant polytope, and let (MA, be the associated symplectic toric manifold. The c-blow-up of (MA, WA) at a fixed
point of the r'-action is a new symplectic toric manifold. What is the moment polytope corresponding to this new symplectic tone manifold? Let q be a fixed point of the Vt-action on (MA,WA), and let p = be the corresponding vertex of A. (Cf. Exercise 16.) Let u1,. . , be the primitive (inward-pointing) edge vectors at p, so that the rays p + tu,, t 0, form the edges of A at p. .
Theorem 1.3.9. The c-blow-up of (MA, WA) at a fixed point q is the symplectic toric
manifold associated to the polytope by the n vertices
p+cUi,
obtained from A by replacing the vertex p
i=1,...,n.
In other words, the moment polytope for the blow-up of (MA ,WA) at q is obtained from A by chopping off the corner corresponding to q, thus substituting the original set of vertices by the same set with the vertex corresponding to q replaced by exactly n new vertices: p
Proof. Exercise: Check that the new polytope is Delzant. We may view the eblow-up of (MA,WA) as being obtained from MA by smoothly replacing q by , cwFS). Compute the restriction of the moment map to this set. Recall Exercise 22.
0
125
1.3. Moment Polytopes
Example. The moment polytope for the standard T2-action on (IF'2, WF$) is a right If we blow-up 1P2 at [0 : 0 : 1] we obtain a symplectic toric isosceles triangle manifold associated to the trapezoid below.
/9
ExercIse 29 Cheek that this manifold is the first Hirzebruch surface, defined in Section 1.3.4.
Example. The following moment polytope corresponds to a toric manifold obtained by blowing-up IF'2 at the three fixed points:
Exercise 30
The toric 4-manifold corresponding to the polygon with vertices (0,0), (n + 1,0), (0, 1) and (1, 1), for n a nonnegative integer, is called a Hirzebruch surface.
The manifold 1to is just a product of two spheres, whereas by the previous example i-li is a blow-up of p2 at a point (a) Construct the manifold 1-ti, by symplectic reduction of C4 with respect
to an action of(s')2. (b) Exhibit as a p1-bundle over P'.
L Symplectic Viewpoint
126
1.3.6
Syniplectic Cutting
Let (M, w) be a symplectic manifold where S' acts in a hamiltonian way, p: S' —' Diff(M), with moment map : M IL Suppose that: 0, and • M has a unique nondegenerate minimum at q where
• for c sufficiently small, S1 acts freely on the level set
Then the action of S' on
Let C be equipped with the symplectic form —idz A the product ,
is hanultonian with moment map
Observe
that S1 acts freely on the c-level of
= =
for c small enough:
{(p,z)eMxCIp(p)—1z12=e}
{(p,O)EMxCIp(p)=c} U {(p,z)EMxCHzJ2=p(p)—e>0}.
The reduced space is hence
,f'(e)/S1
U
{p E M /A(p) > c} I
The open submanifold of M given by {p E M I p(p) > c} embeds as an open dense submanifold into Exercise 31 Show that the reduced space
is the
of M at g.
This global description of blow-up for hamiltonian S'-spaces is due to Lerman [31], as a particular instance of his cutting technique. Symplectic cutting is the application of symplectic reduction to the product of a hamiltonian S'-space with the standard C as above, in a way that the reduced space for the original hamiltonian S1-space embeds symplectically as a codimension 2 submanifold in a symplectic manifold. As it is a local construction, the cutting operation may be more generally performed at a local minimum (or maximum) of the moment map p. There is a remaining S'-action on the cut space := '(c)/S' induced by
r: S' —i Diff(M xC),
rt(p,z) = (pt(p),z) In fact, r is a hamiltonian S'-action on M x C which commutes with descends to an action
S1 —*
thus
1.3. Moment Poiytopes Exercise
127
32
Show that
is hamiltoman by describing a moment map.
Loosely speaking, the cutting technique provides a hamiltonian way to close > e}, by using the reduced space at level e, We may similarly close (p e M jz(p) <e}. The resulting hamiltonian S1-spaces are called cut spaces, and denoted and Of course, if another
the open manifold (p e M
I
group G acts on M in a hamiltonian way which commutes with the S'-action, then the cut spaces are also hamiltonian G-spaces.
Chapter II
Algebraic Viewpoint 11.1
Toric Varieties
The goal of this lecture is to explain toric manifolds as a special class of projective varieties. The first five sections contain a crash course on notions and basic facts about algebraic varieties, mostly in order to fix notation. The combinatorial flavor of tone varieties is postponed until Lecture 5.
11.1.1
Affine Varieties
Let C[zj,. .. , Zn] be the algebra of polynomials in the n complex coordinate functions on Throughout this lecture, we consider the Zariski topology on a (Zariski) closed set in is a set of common zeros of a finite number of polynomials from C[zi,... , Zn]; naturally, the complement of a Zariski closed set is called a (Zariski) open set. The fact that any infinite intersection of closed sets is indeed a closed set follows from the stabilization property for sets given as zero sets of polynomials: any decreasing sequence of such sets X1 j X2 ... stabilizes, i.e., there exists an integer r such that X,. = = .... This is a restatement of Hilbert's basis theorem which says that any ideal in C[zi,. . , is finitely generated; see, for instance, [50, §IV-1]. Notice that any nonempty open set is dense (i.e., its closure is the full space), hence the Zariski topology is not Hausdorif. .
Definition 11.1.1. An afline variety is a nonempty closed set in a C".
The Zariski topology on an affine variety X C C" is the topology on X which declares to be (Zariski) open (respectively, closed) every set which is the intersection of X with an open (resp., closed) subset of C". An affine variety X is irreducible if it cannot be written as the union of two proper closed subsets. On an irreducible affine variety, any nonempty open subset is dense.
II. Algebraic Viewpoint
130
Example. The zero locus in C2 of the polynomial Z2 is not irreducible; its irreducible components are given by the lines z1 = 0 and Z2 0. Exercise 33
Let X C Ctm be the affine variety defined as the zero locus of polynomials Pi.• .. ,Pr E C(zj,. . , zn]. Show that X is irreducible if and only if the ideal generated by pi,. . . ,i-'r is prnne (an ideal I C CEzj,..., Zn) prime if whenever u,VECIZI .
Let X
C C72 be an
affine variety presented as the zero locus of polynomials
Pi,".,Pr Definition 11.1.2. A regular function on X is a function X —# C which is the restriction to X of a polynomial function in Ctm. The ring of regular functions on X is denoted C[X]. Exercise 34
Show that the ring Clxi generated by pi Pr
Let X C
is
isomorphic to Cizi,.
.
.
,
where I is the ideal
C Ctm be affine varieties.
Definition 11.1.3. A regular map from X to X' is a map cm. restriction of a polynomial map
: X —+
X' which is the
Exercise 35
Show that a map w: X —. X' is regular if and only if it pulls back regular functions on X' to regular functions on X.
Definition 11.1.4. An isomorphism from X to X' is a regular map X —' X' which is invertible by a regular map. The symbol indicates an isomorphism. The group
of isomorphisms X X is denoted by Isom(X). The affine varieties X and X' are isomorphic when there eDists an isomorphism between them. Exercise 36 Show that X and X' are isomorphic if and only if the associated rings of regular functions, C[X) and C[X'], are isomorphic.
Example. Consider the variety X = = 1 ,i 1,... ,n} C for n = 1, this is the (complex) hyperbola with {zi = O} and {z2 0] as asymptotes. On X the functions z1,. , E C[X1 are invertible by regular functions: = . .
11.1.
131
Toric Varieties
Hence the ring of regular functions on X is the ring of Laurent polynomials in n variables: The projection C2" —# C" onto the first vi components maps X isomorphically
onto the n-dimensional algebraic torus
=C"\ hyperplanes z1 =0 ,i= 1,...n.
(C*)hl := (C\{0})"
Hence we have
X
(Zl,...,zn,zn+l,...,Z2n)
(zi,zj',. ..
(Cs)"
*—
(z1,...
This shows that the torus (C* )fl is an affine variety. Note that the coordinates whereas they were not are invertible by regular functions on invertible by regular functions on C". - .
. ,
11.1.2
Rational Maps on Afline Varieties
If X C C" and X' C Ctm are affine varieties, the symbol X --÷ X' denotes a map defined on some open subset of X. Example. If Pl,P2
C[zi,.
. .
then the rational function defined on the set
Let X C C" be an irreducible affine variety. Definition 11.1.5. A rational function on X is a map X --÷ C which is the restriction of a rational function on C" whose denominator does not vanish identically on X. The ring of rational functions on X is denoted Ox-
Let C(X) denote the field of fractions of C[X}, and let Ix be the ideal in Ox formed by the rational functions on C" whose numerator vanishes identically on X. Exercise 37 Show that the field C(X)
is
isomorphic to the quotient Ox/Ix
An irreducible affine variety X is normal if its ring of regular functions C[XJ is integrally closed in its field of fractions, that is, for any I E C(X), if f satisfies an equation of the form
fm+gifm_l+...+9rn0 with coefficients g,
C[X], then f E C[X]. Any smooth variety is normal and the set of singular points of a normal
variety has codimension at least 2 [44].
II. Algebraic Viewpoint
132
Examples.
1. On the curve X C C2 defined by y2 = x2 + x3, the rational function t = C(X) is integral over C[X] since t2 — 1 — x = 0, but t C[X], hence X is not normal.
2. The cone X C C3 given by x2 + y2 =
z2
is normal [44], though it has a
singular point at the origin C Ctm be slIme varieties.
Let X C
Definition 11.1.6. A rational map from X to X' is a map w: X --. X' which is the restriction of a rational map C'1 --÷ Ctm whose denominator does not vanish identically on X. 38
X - -. X' is rational if and only if it pulls back rational Show that a map functions on X' to rational functions on X.
Definition 11.1.7. A birational equivalence from X to X' is a rational map X --s X' which is invertible by a rational map. The affine varieties X and X' are birationally equivalent when there exists a birational equivalence between them. ExercIse 39
Show that X and X' are birationally equivalent if and only if the associated fields of rational functions, C(X) and C(X'), are isomorphic.
Any is the zero set of one polynomial in C[zj,.. . , A hypersurface in affine variety is birationally equivalent to a hypersurface of some space cm; for a proof see, for example, [44].
11.1.3
Projective Varieties
vanishes at a point two : ... e P'1 We say that a polynomial p C(zo,.. , A e C*. Notice that this condition implies that each if p(Awo,... , homogeneous component of p vanishes. We consider the Zariski topology on a (Zariski) closed set in P" is a set of common zeros of a finite number of polynomials from C[zo, . . . ,z,,]; as usual, the complement of a Zariski closed set is called a (Zariski) open set. By considering homogeneous components, we may assume that each of those polynomials is homogeneous. A hypersurface in is the zero set of one (reduced) homogeneous polynomial in C[zo,. .. , \ {0}, the degree of which is called the degree of the .
11.1.
133
Toric Varieties
hypersurface'. A hypersurface of degree 2,3,4,... is traditionally when it is called a conic), a cubic, a quartic, etc.2 (except in
called
a quadric
Definition 11.1.8. A projective variety is a nonempty closed subset of some projective space
is the topology on X The Zariski topology on a projective variety X C which declares to be (Zariski) open (respectively, closed) every set which is the intersection of X with an open (resp., closed) subset of r. A projective variety X is irreducible if it cannot be written as the union of two proper closed subsets. Examples. 1.
The product of two projective spaces is a projective variety. This can be seen via the Segre embedding
S:
P" x
[z®wI.
([zI,[w])
The homogeneous coordinates of an image point [z ® to] are
yjj:=zjw,, i=O,...,n,j=O,...,m. The image of S
is
the set cut out by the system of equations
Ii,k_—O,...,n
YijYkt = ykjYit
S
is a nonempty closed subset of jpnrn+fl+rn In x x thus particular, 5(1?' x IF') is the subset of points [yoo 110, : 111° y,,] E determined by the single quadratic equation = YoiYio
YooYii hence
S(P' x p1) is a nondegenerate quadric in
2. The set of lines in IPhl through the point [0: ... 0: 1] E IF" (or, similarly, through any other point), is naturally identified with Jl" via :
1
IF"—' [to]
{ lines in 1?" through [0: ... : 0: 1]} L1,,,1
:= {[Awo: ...
:
:
y]
E C2 \ {(0,0)}}
'A polynomial is reduced if each of its irreducible factors has multiplicity 1. For some applications it is convenient to allow for nonreduced polynomials and then consider that some components of the hypersurface have multiplicities. 2A hypersurface of degree 1 is called a line when in P2, a plane when in in higher projective spaces.
and a hyperplane
II. Algebraic Viewpoint
134
The blow-up ofr at [0: ... :0:
1],
:
... :0: 1]), is the subset of
defined by the incidence relation
x
B(lr, [0: ... 0: 1]) := {([w], [z]) :
pn._l x
I
[z] E
defined by the
x
This can be translated as the closed subset of system of equations
+n I,
as a projective variety in x IF" The Segre embedding exhibits hence B(P'2, [0: ... : 0: 1]) is a projective variety.
Since the point [0 B(P'2, [0
:
...
:
0
:
...
:
0
:
1] belongs to any line through it, the set
1]) decomposes as the disjoint union of the so-called
:
exceptional divisor,
E:= {([w],[0: ... :0:1])
11w]
with
S:= {([w),[w:y])
\{[0: ... :0:
, yE C)
[w]
1])
Let X C r be a projective variety. C which, locally near Definition 11.1.9. A regular function on X is a function X each point x E X, may be written as a quotient of two homogeneous polynomials
of the same degree such that the denominator does not vanish at x. The ring of regular functions on X is denoted C[X]. Any regular function on P'2 is constant. More generally, it can be shown that CLX] = C whenever X is an irreducible projective variety [44]. Therefore, the ring
C[X] will not give much information.
Let X C P'2 and X' C projective space
standard charts (k = 0.. Vk := {[z0
IPtm
be projective varieties. Recall that complex
comes equipped with n + 1 affine neigbborhoods given by the
.
. .
, n):
z,,]
I
0}
[zo:...:z] '2
C'2 i—i
'
'
'
X —i X' such that X there exists a neighborhood U of x and an affine neighborhood V of ço(x) for which ço(U) C V and cc : U V is given by m regular functions.
Definition 11.1.10. A regular map from X to X' is a map
for each x
11.1.
135
Toric Varieties Exercise 40
Check that the regularity condition at a point z
X is independent of the
choice of the afline neighborhood containing f(x).
pN where Example. A Veronese embedding of degree d is a map V N= is mapped to the point with homogeneous ... — 1 and [z0 :
coordinates given by the various monomials + ... + being nonnegative integers such that
the exponents A0,. ..
. . .
,
= d.
Exercise 41 Check that the set of all homogeneous polynomials of degree d in n+ 1 variables z,, forms a vector space of dimension 20
In particular, if n = 1 and d = 2, the Veronese embedding is simply
V: [zo : z1]
'—÷
: z0z1 : z?]
Exercise 42
Check that V is a regular map.
The Veronese embedding of degree d allows to translate the study of some into the case of hyperplanes problems concerning hypersurfaces of degree d in inlPN.
X Definition 11.1.11. An isoinorphism from X to X' is a regular map indicates an isomorphism. The which is invertible by a regular map. The symbol group of isomorphisms X —÷ X is denoted by Isom(X). The projective varieties X and X' are isomorphic when there erists an isomorphism between them. 11.1.4
Rational Maps on Projective Varieties
are projective varieties, the symbol X --. X' still denotes and X' c If X C a map defined on some open subset of X.
Let X C
be an irreducible projective variety.
Definition 11.1.12. A rational function on X is a function - - s C whose restriction to each affine neighborhood is a rational function on whose numerator and denominator have the same degree and whose denominator does not vanish identically on X. The ring of rational functions on X is denoted Ox. Let C(X) denote the field of fractions of CIX], and let Ix be the ideal in Ox formed by the rational functions on whose numerator vanishes identically on X.
II. Algebraic Viewpoint
136 ExercIse 43 Show
Let
that the field C(X)
Xc
ffsfl
and X' c
is
Ptm
isomorphic to the quotient Cxlix.
be projective varieties.
is a map X --+ Ptm which Definition 11.1.13. A rational map from X to by m + 1 rational functions on X. A is given in homogeneous coordinates for rational map from X to X' is the restriction to X' of a rational map : X ---. is regular and ço(U) C X'. such that there is an open set U C X where Definition 11.1.14. A birational equivalence from X to X' is a rational map X -- + X' which is invertible by a rational map. The projective varieties X and X' are birationally equivalent when there exists a birational equivalence between them. Exercise 44
that X and X' are birationally equivalent if and only if the associated fields of rational functions, C(X) and C(X'), are isomorphic. Show
11.1.5
Quasiprojective Varieties
The notions before make sense in the broader class of quasiprojective varieties, which encompasses both affine and projective varieties. Definition 11.1.15. A quasiprojective variety is a nonempty open subset of a projective variety. Zariski topology, regular function, regular map, isomorphism, rational function, rational map and birational equivalence are defined for quasiprojective varieties analogously to projective varieties. For instance, a (Zariski) closed set of a quasiprojective variety is the intersection of the variety with a closed subset of projective space. From now on, the term variety (without specifying affine or projective) refers to a quasiprojective variety. It is a fact that two irreducible varieties X and X' are birationally equivalent if and only if they contain isomorphic open subsets U C X and U' C X' [44]. Example. The tautological line bundle L defined in Section 1.3.4 is a quasiprojective variety (yet not affine, nor projective). In fact, the inclusion of in as the open set of points (zo: ...: Zn] with 0 induces an inclusion of the blow-up of at the origin as an open subset of the projective variety [0: ... : 0: 1]); cf. Section 11.1.3.
The blow-down map /3 of quasiprojective varieties.
L —p
is a birational equivalence in the category
Blowing-up at a point is a local operation which extends to any quasiprojective variety modeled on the blow-up of at the origin or of at [0: ... : 0: 1].
11.1.
137
Toric Varieties Exercise 45
C), where P(L C) was discussed in [0: ... 0: 1]) Check that Section 1.3.4, and that the blow-down map f3: P(L C) —. 1?" is a birational equivalence in the category of projective varieties. :
An irreducible projective variety X
is
normal if every point has a normal
affine neighborhood.
A normalization of an irreducible variety X is an irreducible normal variety X, together with a regular map v : X —, X which is a finite birational equivalence. X' between varieties is finite, if any point x E X' has an affine X A map '(V) is afline and the restriction neighborhood V such that the preimage U := V is a finite map, that is every point has a finite number of preimages. U presents Any variety is a finite union of irreducible varieties [44]. If X = 2(3 for all i j, then X as a finite union of irreducible closed subsets and X2 the Xe's are called irreducible components of X.
One can define a normalization of an arbitrary variety X as a disjoint union of normalizations for each of the irreducible components of X.
Toric Varieties
11.1.6
is a 2n-dimensional Lie group The n-complex-dimensional algebraic torus under multiplication of complex numbers. The weight lattice of (C* is the lattice is a group homomorphism A character (or a Laurent monomial) of —'
There is a bijective correspondence between weights and characters of (C5
A =(A1,...,Afl)
E
An action of a torus (C5
A: w = (tv1,.
C5 . . ,
:=
...
on a variety X is a group homomorphism
Isom(X).
Definition 11.1.16. A toric variety is an irred'ucible variety3 X equipped with an action of an algebraic torus having an open dense orbit. Definition H.1.17. Two toric varieties are equivalent if there exists an equivariant isomorphism between them. Toric varieties are easy to construct, as the following examples illustrate. 3Toric varieties are usually required to be normal and normality is always the case for smooth varieties, which interest us mostly. However, our discussion does not require normality and is shortened without this assumption.
II. Algebraic Viewpoint
138
Examples.
1. Let A = {A('),. an action of w
[21
:
Associated to A, there is
be a finite subset of on the projective space
. .
...
:
Zk] =
[W
21
:
...
defined by: Zk]
:
,
* for w E (C
'1
Let XA be the closure of the (Ct)'1-orbit through [1:...: 1]. Then XA is a tone variety.
2. Let A = {A('),... ,A(k)} be a finite subset of
The action of
on the
vector space Ck associated to A is defined by:
forwe (C *
zl,...,W
w-(zl,...,Zk)=(W
through (1,..
Let YA be the closure of the tone variety.
,
1).
Then YA is a
3. Suppose that we have an action of a torus (C*)fl on some variety V. Let v E V. The closure of the (C* )'1-orbit through v is a toric variety. The important example XA has the following particular instances. Examples.
1. For a fixed positive integer d, let A1
= =
{A = (A0,.. {A(°>,.
.
,
A,2) e
> 0 all i , Ao + ... +
I
= d}
(which corresponds to the set of all Laurent monomials in n variables of
degree d containing no negative powers). Then
XA, = closure of
where V :
:
:
...
w E (C*)n+l} = V(P'2)
:
pN is the Veronese embedding of degree d defined in
Section 11.1.3.
2. Similarly, for a fixed positive integer d, let A2
=
P"
{A(°),. .. ,A(")}
As before XA2 = V(IP'1)
P'1. For
n
2 and d = 3, the set A2 is:
11.1.
139
Toric Varieties
I I •
II. Algebraic Viewpoint
140
11.2
Classification
Recall that a toric variety is an irreducible quasiprojective variety equipped with
an action of an algebraic torus having an open dense orbit. In this lecture we begin by reviewing the language of spectra used for classifying affine toric varieties. Arbitrary normal toric varieties are classified by combinatorial objects called fans.
11.2.1
Spectra
Let A be a finitely generated C-algebra without zero divisors. An ideal I in A is prime if
u,vEAanduvEl An ideal I in A is maximal if I is I itself.
uEIorvEI.
A and the only proper ideal in A containing I
Exercise 46
Regard A simply as a commutative ring with unity. Show that the ideal I is prime if and onJy if the quotient ring A/I is an integral domain (i.e., A/I has no zero divisors), and that the ideal I is maximal if and only if A/I is a field.
Exercise 47
Check that every maximal ideal is prime. Give an example of a prime ideal which Is not maximal.
Exercise 48
Let I be an ideal in A, and let p: A —. A/I be the surjective ring homoinorphism given by taking an element to its coset in the quotient ring A/I. Check that there exists a bijective correspondence between the ideals J of A which contain I, and the ideals J of A/I, given by J = (J).
p'
The spectrum of the algebra A is the set Spec A
{ prime ideals in A }
equipped with the Zariski topology, which declares to be closed a subset of Spec A consisting of all prime ideals containing some subset of A. The maximal spectrum of A is the set SPecm A
{ maximal ideals in A }
equipped with the Zariski topology, which declares to be closed a subset of consisting of all maximal ideals containing some subset of A.
A
11.2.
141
Classification
Examples.
Associated to the point E x there is a maximal ideal I,, C A consisting of all polynomials which vanish at x, that is, is the ideal generated by the monomials z1 — x1,. , z,-, —
1. Let A = C[zi,.. ,z,,] and let x = (x1,.
.
.
By Hubert's Nullstellensatz (see, for instance, [5] or [20]), any maximal ideal for some x C". Moreover, the corresponof C[z1,. .. , z,,j is of the form dence
C" c[c"I is
a homeomorphism for the Zariski topology, where we stress that the poly-
nomial ring C[zj,... , z,,J is the ring of regular functions on Exercise 49 Show that there exists a bijective correspondence between Spec CEz1,.. and the set of irreducible subvarieties in C".
flint: Any prime ideal in C[zi,..
,
z,,]
is finitely generated.
2. Let A = .,z,,,z;'] and let x = (x],. to the point x there is the maximal ideal in A: .
r
._
/
—1
,
—1
.
,x,,) E (C*)fl. Associated —1
X1
—.1
— yr,...,) is the full algebra A since it contains (z1 —xi)zj' +xi(zj' —yj) E Hubert's
By observing that the ideal (z1 — x1, when Yl
Nullstellensatz implies that any maximal ideal of C[zi, the form Moreover, the correspondence
.
, z,,,
z; 1] is of
C[Zi,Zi',...,Zn,Z'] is a homeomorphism for the Zariski topology, where Cf(C*)fJ is the ring of regular functions on cf. Section 11.1.1. Exercise 50
that Spec CIa1, Show
there
exists a
a
bijective correspondence between of irreducible subvarieties in
and the set
Let X be an affine variety in C" defined by polynomials p1,... ,Pr from C[zi,. ,z,,]. Let I = (pi,. . ,Pr) be the ideal generated by those polynomials. . .
.
II.
142
Then
Algebraic Viewpoint
X= = {
ideals
13
c C[zi,..
,
1
= { maximal ideals Jc = { maximal ideals in C[zi,. = { maximal ideals in C[X]
1
ç
}
J} .
.
,
z,,]/I
}
}
SPeCm
is a homeomorphism for the Zariski topol-
The correspondence X ogy.
More generally, if A is a finitely generated C-algebra without zero divisors, the set XA = SPeCm A is called an abstract affine variety. While maximal ideals in A play the role of points in XA, arbitrary prime ideals are thought of as irreducible subvarieties, by analogy with the ring of regular functions on an affine variety. The dimension of XA is defined to be
c
sup{3 chain
dimXA
nEZ
c ... C
of distinct prime ideals}
When A = C[X) is the ring of regular functions on an irreducible affine variety X, the dimension dim XA coincides with the (complex) dimension dim X since both are equal to chain X0 C X1 C nEZ
11.2.2
...
C
X,. = X of distinct irreducible subvarieties}
Toric Varieties Associated to Semigroups
Let S be a commutative semigroup. Its seinigroup algebra C[S] is the C-algebra generated as a complex vector space by the symbols z° with o E S and multiplication defined by the rule = In particular, a C-algebra.
generators
for S as a semigroup yield generators z° for C[SI as
Examples.
1. If S =
then
= C[zi,... ,
is the algebra of polynomials in n
variables.
2. If S =
then C[S] = C[zi,zj',...
polynomials in n variables.
is the algebra of Laurent
11.2.
143
Classification
Notice that in the previous two examples, the maximal spectrum of the semigroup algebras are toric varieties: and
SPeCm
SpeCm
In fact, we have the following general result: be a finitely generated Proposition 11.2.1. Let S ç mal spectrum SpeCm C[SJ is an affine tone varzety.
Then the maxi-
Proof. By shrinking the lattice if necessary, we may assume that S generates V2 as an abelian group, in which case Specm C[SI has dimension n. The inclusion of semigroups S ç V2 and hence of semigroup algebras C[S] C C[ZThJ gives an embedding of the torus (C*)fl
=
Specm
Specm
C[S].
Let 0 be the image of this embedding. The torus (C* )fl acts on C[S1 by
wz°:=vfz°, This action induces an action of (CS)fl on Spec1,, C[S]. By considering the dimen-
sion, the set 0
is an open orbit for this action. We conclude that its 0 closure 0 must be the full Spec,,, C[SI and hence this is a toric variety. Example. The complex curve inC2 with equation yk affine toric variety with given by t
(k = 1,2,3,...) is an
(x,y) =
It may be obtained as Spec,,, C[S] for the semigroup S =
generatedby{k,k+1,...,2k—1}.
\ {1,2,. . . ,k
—
1}
Exercise 51
Show that the variety in the previous example is not normal for k > 1, and that its normalization is the affine line X.
11.2.3
Classification of Affine Toric Varieties
Theorem 11.2.2. (Classification of alfine toric varieties) Any affine toric variety is eQuivalent to one of the form Spec,,., C(SJ for some finitely generated semigroup
S C Z" (n 0).
II. Algebraic Viewpoint
144
Proof. Let X be an affine toric variety for the torus (C*)rn. Let 0 be the open orbit for the (C* )m-action on X. Then 0 may be identified with the quotient of (C* by the stabilizer of some point in 0. Since this quotient is itself a (possibly acting on X. smaller dimensional) torus, we can regard 0 itself as a torus By irreducibility of X, we have an embedding C[X] C C[OJ = C[Z'2]. The subring C[X] c C[0] is (Ct)'2-invariant with respect to the induced actions of (C*)'2 on C[X] and on C[(C*)'2]. As a representation of an algebraic torus, the space C[XJ decomposes into one-dimensional weight spaces. The weight spaces are generated by monomials as C-algebras, hence the vector space C[X] itself is generated by Q monomials, i.e., it is a semigroup algebra. Example. Recall the construction of the affine toric variety YA from a finite set A= .. , C Z'2, described in Section 11.1.6. By the previous theorem, we must have YA Specm C[SJ for some finitely generated semigroup S C Z'2. It is not hard to see that S is the semigroup generated by A. In fact, the ring C[YA] of regular functions on YA is generated by the restrictions to YA of the coordinate functions on C'2. Since YA is the closure of {
(zAp>
. , I
z E (C*)'2}
the ring C[YAJ is generated by the monomials
,...,z
z
i.e., is the semigroup algebra of the semigroup in Z'2 generated by A.
Remark. The only smooth affine toric varieties are products of the form (C*)P x This follows from the classification of affine toric varieties (Theorem 11.2.2), the classification of normal toric varieties (Theorem 11.2.10) and the study of conditions for smoothness (Exercise 53). See the remark at the end of Section 11.3.4.
11.2.4
Fans
Definition 11.2.3. A (convex polyhedral) cone in R'2 is a set of the form
C = {aivi + . .. +
ER" I a1,... ,a,. 0}
for some finite set of vectors v1,.. , ii,. R'2, then called the generators of the cone C. The cone C is rational if it admzts a set of generators in Z'2. The cone C is smooth if it admits a set of generators which is part of some 7L-basis of Z'2. .
The dimension of a cone C is the dimension of the smallest R-subspace containing C (which is the vector space C + (— C)).
11.2.
145
Classification
Definition 11.2.4. The dual of a cone C C I
f(x) 2 0 Vx
E C}.
Farkas' theorem states that the dual of a rational cone is a rational cone [21]. From the theory of convex sets [20], it follows that (C* ) * = C. is a hyperplane of the form A supporting hyperplane for a cone C C
Hf:={xER'2If(x)=0}
forsomef€C*\{0}.
A face of a cone C c is either C itself (a nonproper face) or the intersection of C with any supporting hyperplane (proper faces). A face of a cone is itself a cone; indeed the face C fl H1 (with f E C*) is generated by those vectors in a set of generators for C such that 1(v2) = 0. Exercise 52
Show that a cone C has only finitely many faces and that any intersection of faces is also a face. Hint: The face C fl Hf is generated by those vectors v2 in a generating set for
C such that 1(v2) =
0.
If 0 is a face of C, then C is called strongly convex; this is the case precisely when C contains no one-dimensional R-subspaces, that is, when C fl (—C) = {0}.
If C is strongly convex, then its dual regardless of the dimension of C.
Let C be a rational cone in Lemma 11.2.5.
is n-dimensional (i.e., C*
and let C* be its dual,
+ (_C*) =
also
rational.
The intersection C* fl (V' ) * is a finitely generated semigroup.
Proof. Let v1,... ,v,. be generators of C* and let K = 0 r,v2 where r 2 0, so r2 = m2 + with a nonnegative integer and 0 t2 1. Then v = each v1 and t2v, in K fl
The finitely generated semigroup C* fl
)*
is
m2v2
+
with
0
denoted
Examples.
1. For the cone C given by the first octant in Z'2, the semigroup Sc consists of elements in (Z11)* with all coordinates nonnegative, and is generated by where 2.
= (0,...,0,1) E
e1
For the trivial strongly (ZTh)* is generated by
convex cone C
= {0} in
the semigroup
=
II. Algebraic Viewpoint
146
3. For the cone C C Z2 generated by e2 and el
C
e2
the
semigroup
Sc is
+
and
generated by
— e2,
• •
+
•
• • •
Sc
e1 — e2
4. For the cone C C generated
Z2
+
by
generated by e2 and and +
2e1 — e2,
• • • •
• e2
Corollary 11.2.6. For a rational cone C C toric variety. Proof. This follows
• • • • •
the semigroup Sc is
Sc
e
the affine variety
immediately from Proposition 11.2.1 and Lemma 11.2.5.
is a 0
Examples.
1. For the cone C given by the first quadrant in algebra is and the corresponding toric variety is
the associated semigroup
11.2.
147
Classification
the associated semigroup
2. For the trivial cone C = {O} in
C[Sc]C[zj,zi—1 and
algebra is
—1
the corresponding toric variety is the torus SpeCm CISc]
3. For the third cone in the previous list of examples, the associated semigroup algebra is
=
C[w1,w2]
and the corresponding toric variety is Specm C[Sc]
C2.
4. For the fourth cone in the previous list of examples, the associated semigroup algebra is
C[w1,w2,w3]/(wiw3 =
C[Sc] =
and the corresponding toric variety is the quadratic cone Spec,,, C[Sc]
{(w1,w2,w3)
C3
=
I
which has an orbifold Z2 singularity at the origin. Note that, for the cone C C 7L2 generated by e2 and e1 — 2e2, the associated semigroup algebra is C[Sc] = C[zi,
C[wi, W2)
which corresponds to a smooth toric variety.
If we have an inclusion of rational cones C C C', then (C')' C C', hence C[Sc'] is a subalgebra of C[ScJ. It follows that we get a map Spec,,, CISc] —p
Spec,,,
C[Sc'];
to see this, first notice that, since nontrivial C-algebra homomorphisms C[Sc] C are uniquely determined by their kernels which are exactly the maximal ideals of CISc], we obtain a bijection Specm CISc]
Horn
\ {O}
Thus, by restricting homomorphisms from C[ScJ to CISc'], we get the map.
asserted
II. Algebraic Viewpoint
148
Lemma 11.2.7. Let C and C' be rational cones. If C is a face of C', then the induced map Specm CISc]
Specm C[Sc'J is an open injection for the Zar'iski
topology.
In other words, if C is a face of C', then
C[Sc] is an open subset of
Specm CISc'].4
Proof. If C is a face of C', then (see, for instance, [211) there is v E Sci such that
Sc = Sc' + Zt(—v) for some o E Sc' thus each element of C[Sc] may be written in the form C[Sc'] is injective, since if two and n E The map Specm CISc] SPecm C-algebra homomorphisms C[Sc] —p C coincide on C[Sc'], then they also coincide on elements The map Specm CISc] _4 SPeem CISc'] misses exactly the maximal ideals in C[Sc'] containing the set {z'-'}, since any C-algebra hoC which does not vanish on Z° extends to a nontrivial momorphism C[Sc'] and if a C-algebra hohomomorphism h: CISc] C where h(z_") momorphism CISc'] —+ C vanishes on vanish identically since h(1) =
Definition 11.2.8. A fan in rational cones such that
then any extension h : C[Sc] —+ C must
= h(zt7)h(z_t')
0.
0
is a (nonempty) finite collection F of strongly convex
• every face of every cone C e F belongs to F, • the intersection of any two cones from F is a face of both of them. The fan F is smooth if all of its cones are smooth. The support of F is the union cones from F. The fan F is complete if is the whole space.
IFI of all
11.2.5
Toric Varieties Associated to Fans
Definition 11.2.9. The tone variety Xjrr associated to a fan F in is the result of gluing the affine toric varieties Xc := SPeCm C[ScJ (for all C E F) by identifying with the correponding Zariski open subset in whenever C is a face of C'.
Each affine chart has a natural torus action defined as in the proof of Proposition 11.2.1. Those actions are compatible under the identifications dictated
by the face relations; hence there is a well-defined torus action on the variety XF. Moreover, Xy contains indeed an open dense orbit of (Cs as the open set corresponding to the zero cone in F: by strong convexity the zero cone is a face of every other cone, thus producing an open subset of each other affine piece, and the dual of the zero cone is the full set so the corresponding algebra is C[zi, zj1,... , whose maximal spectrum is (C5 )fl 4This functorial property partly justifies working with cones and not their duals right away.
11.2.
Classification
149
The variety X, is normal since it is glued out of normal afilne varieties and normality is a local property. Exercise 53 Show that:
(a) The variety is compact if and only if the fan is complete. (b) The variety Xy is smooth if and only if every cone from is smooth.
As a consequence of the first part of the previous exercise, if F is a complete fan in then X, is a compactification of the torus (C*)n Examples.
1. Consider the fan F consisting of the three cones
C{O},
and
depicted below.
—e1
Each 1-dimensional cone represents the affine variety C:
C1
ei
•scI
C[Sci]rC[z}
•
C[S00] = C[z
—ci Co
1]
IL Algebraic Viewpoint
The gluing of these 1-dimensional charts is prescribed by the 0-dimensional cone C representing C:
. C[Sc] = C[z, z')
4 Sc
I
I In Xc1, the subset Xc corresponds to C := {z C Xc0, the subset Xc corresponds to C_1 := {z' C glue Xc1 to Xc0 along Xc by using the gluing map z
0}, whereas in 0}. We can thus producing C1, z
X, =
2.
Consider the fan F consisting of seven cones (three 2-dimensional cones, three 1-dimensional cones and one 0-dimenisonal cone C = {0}), as sketched below. The shaded areas represent 2-dimensional cones.5
C1
We will check that the toric variety X, is P2. In fact, each 2-dimensional cone corresponds to an affine chart C2:
5ff instead of the cones we considered their duals, the drawing would be messy with overlappings, hence the reason why we stick to this side of duality.
11.2.
151
Classification
SCL2
: : (z z ) =
*
Cl
C1
*
*
.
(z' 'z 12
/
A'IC.
f"f
1
=
1
,Z1
1
Z2
C0,2
:
The expressions in terms of homogeneous coordinates [w0 :
: w2] on these
three affine charts are written just to help keep track of the gluing maps below.6
The gluing of these aflIne charts along their intersections is prescribed by the 1-dimensional cones representing x C: 6The initial chosen identification {(I
determines the other two.
Z2J
(ZL,z2)
C2)
/
152
ii. Algebraic Viewpoint •
Sco
• • •
• • SI
S
. .
C[zi,z2,z2T'l
C[Sc11
Sc2
• •
S
•
=
•
x C2 whereas For instance, in Xc12, the subset Xc1 is represented by x We can glue Xc12 to in Xe,,,, the subset Xç1 corresponds to to obtain by using the gluing map (z1,z2) X001 along \ {[O: 1: o]}. 3.
= (0,.. .,O,1) be the standard Z-basis of Z", Let e1 = generated by the be the cone in and let eo := —. . —en. Let the set vectors efl,.. . es,,. Then .
is a complete fan (where we include the trivial cone C = {0}.) In particular, for n = 1,2 we get the fans encoding P' and p2, respectively.
11.2.
153
Classification Exercise 54
Check that the toric variety associated to the fan is projective space:
in the previous example
n
Exercise 55 Find the toric variety corresponding to the fan depicted below.
11.2.6
Classification of Normal Tone Varieties
Theorem 11.2.10. (Classification of normal tone varieties) Any normal toric where variety X is equivalent to a variety of the form XF for some fan F in n is the dimension of the torus acting on X. This fan is determined uniquely up to a transformation from GL(n; 7L). For a proof of Theorem 11.2.10, see for instance [41]. Thanks to this theorem, normal toric varieties are often defined in terms of fans.
has finitely many Then the variety Proposition 11.2.11. Let F be a fan in and there is a natural bijection between the (nonempty) orbits of the torus (C*
cones in F and the (C)'t-orbits in I
cones
I
frornFj>
} Oc
has dimension equal to the codimension of C. Moreover, for where the orbit cones C, C' E F we have
c c'. For a proof of Proposition 11.2.11, see again [41]. In the next lecture we state and prove the polytope analogue of this proposition.
II. Algebraic Viewpoint
154
11.3
Moment Polytopes
We have seen that normal toric varieties are classified by fans. Most interesting for our purposes are those fans dual to polytopes: a fan associated to a polytope defines an equivariantly projective toric variety. Moreover, a polytope encodes other geometric information such as a symplectic form and an equivariant complex line bundle. We will relate the dual languages of polytopes and fans, and review the link to the symplectic approach. 11.3.1
Equivariantly Projective Toric Varieties
Let X be a toric variety for a torus (CS )n. We say that X is equivariantly projective pk for some k and some action embedding X if there exists a of The first example of a toric Let A = {A('),. . , .A(k)} be a finite subset of variety in Section 11.1.6 is .
: ...
XA := closure of
:
wE
(C5 )'1-orbit through [1: ... : 1] for the action defined by the weights A(t), i = 1,. . , k. The variety XA is clearly equivariantly 1, i.e., the action of the torus on XA extends to the whole ambient embedded in The following theorem shows that, conversely, any equivariantly projective toric variety is equivalent to one of type XA for some finite set A C Z'1. .
Theorem 11.3.1. Let X be a tone variety which is (C5 )fl -equivariantly embedded containing X, and 1. Let Y be the minimal projective subspace in in let k — 1 be the (complex) dimension of V. Then there exists a subset A C X extending containing k elements and a (C5 isomorphism XA to an equivaniant projective isomorphism !Pk1
Proof. Any action of (C5 on IP'' by projective transformations can be lifted to a linear action on Ce. Any linear action of (C5)'2 on Ce is diagonalizable, so in suitable coordinates it is given by a collection of weights i = 1,. , £ E such that w E (C5 )fl acts as multiplication by . .
Pick a point z = [z1 : ... E X lying on the open orbit of the torus. Let A C be the collection of those for which z2 0. Then the minimal projective subspace of pe_1 containing X is :
155
11.3. Moment Polytopes
The dimension of Y is k —
1
where k = #A. We obtain a (CS
isomorphism by collecting the nonzero coordinates: pk-1
y
U
U
x.
x/t
0
Example. Not all projective toric varieties are equivariantly projective. For instance, the nodal7 cubic curve X C P2 given by the equation = is
—
a (not normal) toric variety as its smooth part,
is an open orbit for C5. For a reason why X does not admit an equivariant projective embedding see [22, p.l69J.
11.3.2
<0>
Weight Potytopes
acting linearly on a complex vector space V and consider Consider a torus the associated action on the projectivization IP(V). Let v be a nonzero vector in V, through [v]. By construction, be the closure in P(V) of the (C5 and let is a toric variety equivariantly embedded in P(V). By Theorem 11.3.1, the toric is equivalent to XA,,, where A,, is the finite subset of Z" prescribed as variety follows (adapting the proof of the previous theorem). of For a given a weight A E representation on V is the subapace
the A-weight space of the (C5)m-
VA={vEVIw.v=w"v, Vw€(C5)m}. The weight space decomposition of this representation is the isomorphism [11} V cf.
Exercise 9. Given a vector v E V, the component of v of weight A is the
component vA E VA of v in the weight space decomposition. Set
7The adjective nodal refers to having no singularities other than ordinary double points.
II. Algebraic Viewpoint
156 Exercise 56 Show that the toric variety
is equivalent to
Definition 11.3.2. The weight polytope P,., of the vector v V \ {O} is the convex described in the previous paragraph. hull in W2 of the set
is the weight polytope Example. Given a finite set A C its convex hull P in of the vector v (1,... , 1) for which the closure of the orbit through [ii] is XA. 11.3.3
Orbit Decomposition
indexorbit decomposition
Let A = {A('),
be
. . . ,
a finite subset of
The toric variety associated
to A is (cf. Section 11.1.6) pk—1
: ...
XA := closure of
wE
Proposition 11.3.3. Let P be the convex hull in of the set A above. Then there is a bijection between the (nonempty) faces of the polytope P and the in XA given by
J faces of j polytopeP f
F
J
X°(F)
where
The dimension of X°(F) is the dimension of F. Moreover, the closure X(F) of X°(F) is equivalent to the toric variety XAnF. If F and F' are two faces of P, then
F'cF.
X(F')cX(F)
Proof. For u = (it1,... ,uk)E Ck such that [uJ := [it1 E A we have that the
ut] E XA, given
of u is
(0,... ,0,u2,O,
.
.
. ,
0)
We first show that the correspondence is well-defined, i.e., that X°(F) is indeed an orbit. Of course, X°(F) is invariant as [it]
X°(F)
w [u]
X°(F).
157
11.3. Moment Polytopes
Besides, any two [u], [u'] E X°(F) are related: to show that there exists indeed and UA' = for all A E F, notice that uA = such that UA = to V)2 some W1,W2 E (C*)n, hence just take w = Wi for To prove injectivity, we need to show that
Without loss of generality, we assume that there exists A E F' such that A F. Therefore, if [u] E X°(F), then uA = 0, so that [U] X°(F'). To prove surjectivity, we need to show that any [u] e XA belongs to some X°(F). We introduce the notation [Vp] :=[vj :
...
:
Iv1=1
Vk] such that
=0
F;
[1:...: 1]. By definition of XA, any [u] E XA is
in particular we have that [Vp] of the form [u]
. [Vp]
=
for some analytic map of a punctured disk given by
f: {z E C* I 1(z) = (CiZal
(C*)fl
<e}
Z and where the underbraced dots E Ct, some exponents represent terms of higher degree. Given a = (a1,.. , R be E the linear function defined by inner product with a, fa(A) := A. Let for some constants
.
Fa := {A e P where ía
achieves
its minimum }
the set Fa is a face of P, called the supporting face of fa. For simplicity, suppose that Fa fl A = {A('), A(2)}; the general case is just harder for notation. Let e = (c1,.. Then .
[u]
= lim f(z). [Vp] :
=
... :
=[c +...:c +...:...] =
I.
[AA(2)
0
= C.[VFJEX°(Fa),
...: 0]
_________
II. Algebraic Viewpoint
158
where in the middle equality we have divided all homogeneous coordinates by and the underbraced dots represent terms with only positive = 0 powersofz.
11.3.4
Fans from Polytopes
R be a linear function. We denote be a polytope, and let f: R'2 Let P C is, the set of points in P where f that by supppf the supporting face of f in P,
achieves its minimum, as defined in the previous section.
The cone associated Definition 11.3.4. Let F be a face of a polytope P C to F is the closure of the subset CF,P C (Rfl)* consisting of all linear functions such that supppf = F. fE
Exercise 57 Show that Cp,p is a convex cone, and that the collection of cones CF.P for all faces of P forms a complete fan. Hint: Read the description in terms of the dual polytope, after the next defini-
tion, and translate P so that it contains the origin in its interior.
Definition 11.3.5. The fan of the polytope P for all faces F of P.
is the collection
of the
cones CF,P
Suppose that the polytope P contains the origin in its interior. The fan of the polytope P coincides with the fan spanned by the faces of the dual polytope: :=
{f
(Rfl)* I
f(v) —1 ,
Vv
E P}
that is, the collection of cones formed by the rays from the origin through the proper faces of plus the origin. For instance, the dual of a cube is an octahedron, so the fan of a cube has eight 3-dimensional triangular cones, together with all corresponding faces. Simple examples in R2 are: (—1,1)
(—1, —1)
(1,1)
(1, —1)
Ps
159
11.3. Moment Polytopes
(0,1)
(—1,1)
(3,—i)
(—1,—i)
(—i,—2)
If P is rational, then then .?p is smooth.
is rational and)
is
rational, and if P is smooth,
Example. Let P be the polytope in the following picture.
F6 ={(0,i)}
F2
F4 = {(i,0)}
F5
The fan of P is depicted below. The cone associated with the full polytope is the origin, whereas the cones associated with each of the facets F1, F2 and F3, are half-lines, and the cones associated with each of the vertices F4, F5 and F6 are two-dimensional (shaded regions). CF1 .P
Cp5 P CF4 ,P
P
CF6 P
CFa,P
II. Algebraic Viewpoint
160
As in Section 11.1.6, consider a torus (C*)fl acting linearly on a vector space
V and the associated action on the projectivization P(V). Let v be a nonzero through [v]. be the closure in IID(V) of the (C* vector in V, and let Then the toric variety
is equivalent to XA0, where A,, =
{)¼ E
0}.
Proposition 11.3.6. The fan of the toric variety 0,, eq'uals the fan of the weight = dim P,,. polytope Pv. In particular, dim For a proof of the previous proposition, see for instance [22, p.191] and [41]. Exercise 58
Check that, in a similar way, we can define a fan of a convex polyhedron, though in this case the fan may not be complete.
C[S1 be the aifine toric variety associated to the finitely genas an abelian where P is the group. The variety Specm C[S] is normal if and only if S = P fl convex hull of S in IR". When SPecm CESI is normal, its fan coincides with the fan of the convex polyhedron P.
Remark. Let
erated semigroup S ç Z'2. We may assume that S generates
11.3.5
Classes of Tone Varieties
Since all smooth varieties are normal, we restrict our attention to the universe of normal toric varieties, which are classified by fans. The affine ones correspond to fans consisting of the set of all faces of a single n-dimensional cone (see the remark at the end of the previous section). The compact ones correspond to complete fans. The equivariantly projective ones are necessarily of the form XA for some set of the form A = Z"ri P where P is a polytope.8 Since projective spaces are compact (or because fans of polytopes are complete), any equivariantly projective toric variety is compact. Relation between these classes:
• Not all equivariantly projective normal toric varieties are smooth. To see a nonsmooth (i.e., singular) one just take the fan of a simple rational nonsmooth polytope. For instance, the triangle on the next page fails the smoothness condition at the top vertex. 8When A C Zk is finite yet not of the form Z" fl P for some polytope P, the corresponding XA is not normal, and its normalization is XA', where A' = Z' fl P' and P' is the convex hull of A.
1L3. Moment Polytopes
161
• Not all compact normal toric varieties are equivariantly projective, though in (complex) dimensions 1 and 2 this is always the case. Equivalently, not all complete fans come from polytopes in the sense of Definition 11.3.5, though in dimensions 1 and 2 they do. There are plenty of complete fans in R3 which do not come from polytopes. For example, the collection of cones over the
subdivision below of the boundary of the tetrahedron is not associated to any polytope [21, 411.
• Of course, not all normal toric varieties are compact — any affine toric variety is not compact; more generally, any fan which is not complete corresponds to a noncompact toric variety.
11.3.6
Symplectic vs. Algebraic
A lattice polytope in is a polytope whose vertices belong to Z". Suppose that is an n-dimensional polytope which is both Delzant and lattice. As a Deizant polytope, it is the moment polytope of a symplectic toric manifold , by Delzant's construction.
On the other hand, consider the set of integral points in
II. Algebraic Viewpoint
162
where k := #A is the number of such points. The convex hull of A is obviously Then the associated variety XA is a toric variety for (C*)n. The variety XA is smooth and compact because the fan of the polytope is smooth and complete, Moreover, by and XA is connected because it is the closure of a definition XA is equivariantly embedded in pk_1
i : XA
to its real subgroup
and the restriction of the
= is
I
effective, because the action of (C*
was
1t21 = 1
for all i }
already effective.
Recall that projective spaces have canonical symplectic structures provided with the symby the Fubini-Study forms. For later convenience, we equip and WFS is a Since XA is a complex submanifold of plectic structure Kähler form, we obtain that the restriction WA
:= i*(—2wFS)
is nondegenerate, hence a symplectic form on XA. The structure WA
15
because WFS is
We will check that the Ta-action on XA is in fact hamiltonian by exhibiting a moment map.
The action of
on
(ti,...
by
(zr,.
. .
= (t
z1,.. . ,t
2k)
is hanjiltonian with moment map
p:Ck The action of S1 on
,
—2w0) by diagonal multiplication w
to E S'
i—' multiplication by to
is hamiltonian with moment map
The manifold
1,
is the symplectic reduction of (Ck, —2Wo) with respect
to the S'-action and the moment map
Since
the S'-action commutes with
163
11.3. Moment Polytopes
the action of action and
and preserves the moment map 1, descend to the quotient
we
conclude that the
Therefore, the T"-action on
by
(tj,..
: ... :
(zi
.
zk]
=
z1
... tAzk]
is hamiltonian with moment map k
p
.
p[z1
,
...
Zk]
k
j_-rl
The image of p is the convex hull of A, i.e., is the polytope the restriction of p is As the symplectic submanifold (XA, to XA produces a moment map for the restricted action. We claim that the image so that the two constructions (Deizant's and the toric variety) of PIXA is still yield equivalent symplectic toric manifolds. Since p(XA) is a Deizant polytope, it suffices to show that each vertex of E ZTh be such that the be a vertex, and let a = (aj,. is in p(XA). Let a• A, achieves its R, f0(A) restriction to of the linear function Consider the map of a punctured disk to the torus minimum in . .
:
f: {z
E
I
Izi
1(z) = (ZL1,. . .
By definition of XA, we have that :
...
E
XA.
Hence, by continuity of p, we conclude that
If =
12
E
p(XA)
The coincidence of the two constructions allows to see that a symplectic tone manifold is Kähler, because it inherits a compatible invariant complex structure from its equivariant embeddings into projective spaces.
Remark. Not all toric varieties admit symplectic forms. A compact normal toric variety admits a symplectic form if and only if its fan comes from some polytope. Changing the cohomology class of the symplectic form corresponds to changing the lengths of the edges of the polytope. The size of the faces of a polytope cannot be recovered from the fan, where only the combinatorics of the faces is encoded. Hence, the fan does not give the cohomology class of the symplectic form.
164
II. Algebraic Viewpoint Exercise 59 Find the toric variety corresponding to the fan depicted below.
Bibliography [11 Abraham, R., Marsden, J. E., Foundations of Mechanics, second edition, Addison-Wesley, Reading, 1978.
[2] Arnold, V., Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1978.
[3] Atiyah, M., Convexity and conunuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
[4] Atiyah, M., Bott, R., The moment map and equivariant cohomology, Topology 23 (1984), 1-28.
[5] Atiyah, M., Macdonald, I., Introduction to Commutative Algebra, AddisonWesley, Reading, 1969.
[6] Audin, M., The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics 93, Birkhäuser Verlag, Basel, 1991.
[7] Audin, M., Spinning Tops. A Course on Integrable Systems, Cambridge Studies in Advanced Mathematics 51, Cambridge University Press, Cambridge, 1996.
[8] Batyrev, V., Mirror symmetry and toric geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, 239-248. [9] Bott, R., Tn, L., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer-Verlag, New York-Berlin, 1982.
[10] Bredon, G., Introduction to Compact Transformation Groups, Pure and Applied Mathematics 46, Academic Press, New York-London, 1972. [11] Bröcker, T., torn Dieck, T., Representations of Compact Lie Groups, Graduate Texts in Mathematics 98, Springer-Verlag, New York, 1985.
[12] Cannas da Silva, A., Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag, Berlin, 2001.
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[15] Delzant, T., Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. [161 Dema.zure, M., Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Ecole Norm. Sup. (4) 3 (1970), 507-588.
[17] Duistermaat, J.J., On global action-angle coordinates, Comm. Pure Appi. Math. 33 (1980), 687-706.
[18] Duistermaat, J.J., Equivariant cohomology and stationary phase, Symplectic Geometry and Quantization (Sanda and Yokohama, 1993), edited by Maeda, Y., Omori, H. and Weinstein, A., 45-62, Contemp. Math. 179, Amer. Math. Soc., Providence, 1994.
[19] Duistermaat, J.J., Heckman, G., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268; Addendum, Invent. Math. 72 (1983), 153-158.
[20] Ewald, C., Combinatorial ConvexUy and Algebraic Geometry, Graduate Texts in Mathematics 168, Springer-Verlag, New York, 1996.
[21] Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, 1993. [22] Gelfand, I., Kapranov, M., Zelevinsky, A., Discriminants, Resultants,
and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser Boston, 1994. [23] Griftiths, P., Harris, J., Principles of Algebraic Geometry, Chapter 0, reprint of the 1978 original, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.
[24] Guillemin, V., Moment Maps and Combinatorial Invariants of Hamiltonian TTh-spaces, Progress in Mathematics 122, Birkhäuser, Boston, 1994. [25] Guillemin, V., Sternberg, S., Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513.
[26] Guillemin, V., Sternberg, S., Birational equivalence in the symplectic category, Invent. Math. 97 (1989), 485-522.
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[27] Guillemin, V., Sternberg, S., Symplectic Techniques in Physics, second edition, Cambridge University Press, Cambridge, 1990. [281 Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B., Toroidal Embeddings, I, Lecture Notes in Mathematics 339, Springer-Verlag, Berlin-New York, 1973.
[29] Kirwan, F., Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton, 1984.
[301 Kodaira, K., On the structure of compact complex analytic surfaces, I, Amer. J. Math. 86 (1964), 751-798.
[31] Lerman, E., Symplectic cuts, Math. Res. Lett. 2 (1995), 247-258. [32] Lerman, E., Meinrenken, E., Tolman, S., Woodward, C., Nonabelian convexity by symplectic cuts, Topology 37 (1998), 245-259.
[33] Lerman, E., Tolman, S., Hainiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201-4230. [34] Marsden, J., Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), 121-130. [35] McDuff, D., Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984), 267-277. [36] McDuff, D., Salamon, D., Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford University Press, New York, 1995. [37] Meyer, K., Symmetries and integrals in mechanics, Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), 259-272. Academic Press, New York, 1973.
[38] Mikhalkin, G., Amoebas of algebraic varieties, survey for the Real Algebraic and Analytic Geometry Congress, Rennes, June 11-15, 2001, preprint available at http: //xxx. lanl .gov/abs/mat.h.AG/0108225. [39] Milnor, J., Morse Theory, based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies 51, Princeton University Press, Princeton, 1963.
[40] Morse, M., The foundations of a theory in the calculus of variations in the large, Trans. Amer. Math. Soc. 30 (1928), 213-274. [41] Oda, T., Convex Bodies and Algebraic Geometry - An Introduction to the Theory of Toric Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 15, Springer-Verlag, Berlin, 1988.
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[42] Satake, I., On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359-363. Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.
I., Basic Algebraic Geometry, 1, Varieties in Projective Space, 2nd edition, translated from the 1988 russian edition and with notes by Miles Reid, Springer-Verlag, Berlin, 1994.
[44] Shafarevich,
[45] Souriau, J.-M. , Stracture des Systèmes Dynamiques, Maitrises de Mathematiques, Dunod, Paris 1970.
[46] Spivak, M., A Comprehensive Introduction to Differential Geometry, Vol. 1, second edition, Publish or Perish, Inc., Wilmington, 1979. [47] Sturmfels, B., Gröbner Bases and Convex Polytopes, University Lecture Series 8, Amer. Math. Soc., Providence, 1996. [48] Weinstein, A., Lectures on Symplectic Manifolds, Regional Conference Series in Mathematics 29, Amer. Math. Soc., Providence, 1977.
(49] Wells, R.O., Differential Analysis on Complex Manifolds, second edition, Graduate Texts in Mathematics 65, Springer-Verlag, New York-Berlin, 1980. [50] Zariski, 0., Samuel, P., Commutative Algebra, Vol. 1, with the cooperation of LS. Cohen, Graduate Texts in Mathematics 28, Springer-Verlag, New YorkHeidelberg-Berlin, 1975.
Index action coadjoint, 95 coordinates, 93 definition, 94 effective, 97 free, 103 hamiltonian, 94, 95 locally free, 103 smooth, 94 symplectic, 94 torus, 96 transitive, 103 action-angle coordinates, 93 afline neighborhood, 134 affine variety abstract, 142 definition, 129 algebraic torus, 131 angle coordinates, 93 Arnold Arnold-Liouville theorem, 93 Atiyah Atiyah-Guillemin-Sternberg theorem, 97 birational equivalence, 132, 136 blow-down map, 122 blow-up along a submanifold, 123 definition, 122, 123, 134 equivalence, 122 of toric manifold, 124 symplectic form, 122
character, 137
classification toric manifolds, 101, 102 tone varieties, 140, 143, 153 coadjoint action, 95 complete fan, 148 integrability, 93 vector field, 94 complex projective space, 104, 105, 116 component, 156 cone definition, 144 dimension, 144 dual, 145 face, 145, 159 generator, 144 rational, 144 smooth, 144 strongly convex, 145 conehead orbifold, 108 conic, 133 connectedness, 97 constant of motion, 92 convex polyhedron, 101 convexity, 97 cubic, 133 cut spaces, 127
Darboux chart, 90 equivariant Darboux theorem, 118 theorem, 90 degree, 132
Index
170
Deizant construction, 109, 113 examples of Delzant polytopes, 101
examples of non-Delzant polytopes, 102
polytope, 101, 117 theorem, 102, 109 dimension, 142, 144 dual cone, 145 dunce cap orbifold, 108 effective action, 97 equivalence toric manifold, 98 tone variety, 137 equivariant Darboux theorem, 118 moment map, 95 equivariantly projective, 155 evaluation map, 94 example 2-sphere, 105 complex projective space, 116 Deizant construction, 116 Deizant polytope, 101 fan, 160 Hirzebruch surface, 101, 123, 125 non-Deizant polytope, 102 quotient topology, 104 toric varieties, 149 weighted projective space, 109 exceptional divisor, 122
support, 148 toric variety, 148 finite map, 137 first integral, 92 form
Fubini-Study, 99, 105 symplectic, 89 free action, 103 Fubini-Study form, 99, 105 function hamiltonian, 95 rational, 131, 135 regular, 130, 134 G-space, 95 generator, 144 group of symplectomorphisms, 94 Guillemin Atiyah-Guillemin-Sternberg theorem, 97
hamiltonian action, 94, 95 function, 91, 92, 95 G-space, 95 moment map, 95 system, 92 vector field, 90 Hausdorif quotient, 104 hessian, 119 Hubert's Nullsteilensatz, 141 Hirzebruch surface, 101, 123, 125 homology of toric manifolds, 120 hypersurface, 132
face
cone, 145, 159 polytope, 102 facet, 102 fan complete, 148 definition, 144 examples, 160 polytope, 159 smooth, 148
ideal
maximal, 140 prime, 130, 140 independent, 92 index, 119 integrable completely, 93 system, 91, 98 integral of motion, 92
Index irreducible component, 137 variety, 129, 133 isomorphism, 130, 135 isotropic, 92 isotropy, 103
Kähler structure, 105 lagrangian fibration, submanifold, lattice polytope, 162 Laurent monomial, 137 Leibniz rule, 92 Lie bracket, 92 Liouville
Arnold-Liouville theorem, 93 torus, locally free action, 103 manifold symplectic, 89 with corners, 115 map rational, 131, 132, 135, 136 regular, 130, 134 Marsden-Weinstein-Meyer quotient, 105 theorem, 105 maximal ideal, 140 spectrum, 140 moment map definition, 95 equivariance, 95 hamiltonian G-space, 95 moment polytope, 97, 118, 155 Morse function, 119 inequalities, 120 lenuna, 119 theorem, 120
nodal, 156 nondegenerate critical point, 119, 121 normal variety, 131, 137 normalization, 137 null foliation, 114 orbifold atlas, 107 chart, 107 conehead, 108 definition, 107 dunce cap, 108 equivalence, 107 reduced space, 108 107
structure group, 107 symplectic, 109 tangent bundle, 107 tangent space, 107 teardrop, 108 tone, 109 orbit definition, 103 point-orbit projection, 104 space, 103, 104 unstable, 104 perfect Morse function, 120 plane, 133 point-orbit projection, 104 Poisson algebra, 92 Poisson bracket, 92 polytope definition, 101 Delzant, 101, 117 examples of Deizant polytopes, 101 examples of non-Deizant polytopes, 102 face, 102 facet, 102 fan, 159 lattice, 162 moment, 97, 118
Index
172
rational, 101 simple, 101 smooth, 101 weight, 156, 157 prime ideal, 130, 140 primitive vector, 102 projective variety, 132, 133 quadric, 133 quartic, 133 quasiprojective variety, 136 quotient Hausdorif, 104 Marsden-Weinstein-Meyer, 105 symplectic, 105 topology, 104
rational cone, 144 function, 131, 135 map, 131, 132, 135, 136 polytope, 101 reduced polynomial, 133 space, 105 reduction at other levels, 106 for product groups, 106 in stages, 106 orbifolds, 107 regular function, 130, 134 map, 130, 134
Segre embedding, 133 semigroup algebra, 142 simple polytope, 101 smooth cone, 144 fan, 148 polytope, 101 spectrum, 140 stabilizer, 103
Sternberg Atiyah-Guillemin-Sternberg theorem, 97 strongly convex, 145 support, 148 supporting face, 158 supporting hyperplane, 145 surgery, 118 symplectic action, 94 blow-up, 122 cutting, 126 form, 89 Fubini-Study form, 99, 105 manifold, 89 quotient, 105 reduction, 104-106 standard form on 89 standard form on 89 standard form on S2, 90 toric manifold, 89, 98 toric orbifold, 109 vector field, 90 symplectomorphism, 90 system hamiltonian, 92 integrable, 91, 98 tautological line bundle, 136 teardrop orbifold, 108 theorem Arnold-Liouville, 93 Atiyah-Guillemin-Sternberg, 97 convexity, 97 Darboux, 90 Delzant, 102, 109 equivariant Darboux, 118 Marsden-Weinstein-Meyer, 105 tone manifold classification, 101, 102 definition, 101 equivalent, 98 examples, 99 4—dimensional, 117
173
Index homology, 120 symplectic, 89, 98
toric variety affine, 143, 161 classes, 161 classification, 140, 143, 153 compact, 161 definition, 129, 137 equivalent, 137
equivariantly projective, 155 examples, 149 fan, 148 semigroups,
142
torus action, 96
algebraic, 131, 137 Liouville, 93 transitive action, 103 twisted projective space, 109
unstable orbit, 104 variety affine, 129 irreducible, 129, 133 normal, 137 projective, 132, 133 quasiprojective, 136 toric, 129, 137 vector field complete, 94 hamiltonian, 90 symplectic, 90 Veronese embedding, 135 weight
component, 156 lattice, 137 polytope, 156, 157 space,
156
space decomposition, 97, 156 weighted projective space, 109
Weinstein Marsden-Weinstein-Meyer quotient, 105 Marsden-Weinstein-Meyer theorem, 105
Zariski topology, 129, 132, 136, 140
Part C
Geodesic Flows and Contact Toric Manifolds Eugene Lerman
Foreword The main theme of these notes is the topological study of contact toric manifolds, a relatively new class of manifolds that I find very interesting. A motivation for studying these manifolds comes from completely integrable systems {fi,. , fn} is homogeneous of degree on punctured cotangent bundles where each function 1 (one can think of 's as symbols of first order pseudo-differential operators, but this is not essential). A punctured cotangent bundle is a symplectic cone whose base is naturally a contact manifold (this is explained in detail in Chapter 1.2). This observation leads to studying completely integrable systems on contact manifolds, whatever those are. The simplest (symplectic) completely integrable systems are the ones with global action-angle coordinates. The next simplest case is that of Hamiltonian torus actions. If the phase space is compact one ends up with (compact) symplectic toric manifolds. This is the theme of Ana Cannas's lectures delivered at the summer school. The corresponding case in the contact category is that of compact toric manifolds. We will use the excuse of studying completely integrable geodesic flows with homogeneous integrals to introduce various ideas essential for the classification of contact and symplectic tone manifolds. More specifically we will discuss in . .
these notes contact moment maps, slices for group actions, sheaves and cohomology, orbifolds and Morse theory on orbifolds.
Chapter I
From toric integrable geodesic flows to contact toric manifolds 1.1
Introduction
We start with an innocuous sounding problem. Problem Consider the cotangent bundle of the n-torus minus the zero section 0. That is, consider the manifold
M = {(q,p)
E
x
Ip
0}.
Suppose further that the torus G = T'1 = acts on M effectively and preserves the standard symplectic form w = A (i.e., the action is symplectic). Suppose further that the action of G commutes with dilations, i.e., the action p of R on M given by
pA(q,p) = (q,eAp).
Is the action of C necessarily free? Remark 1.1.
1. Recall that an action of the group C on a manifold M is effective if the only element of C that fixes all the points of M is the identity.
2. There is an "obvious" action of on M which has the above properties and is free it is the lift of left multiplication: (q,p) = (aq,p),
where a E
(q,p) E M. The issue is whether an arbitrary action of
which is effective, symplectic and commutes with dilations is necessarily free.
I. From toric integrable geodesic flows to contact toric manifolds
180
3. We will see later (Proposition 1.13) that an action of a Lie group G on a punctured cotangent bundle which is symplectic and commutes with dilations is necessarily Hamiltonian. That is, there is a moment map : M Recall the definition of the moment map for a symplectic action of a Lie group C on a symplectic manifold (M, w): for any vector X in the Lie algebra g of C d(4',X)
—W(XM,.),
g —p IR is the canonical pairing and XM is the vector field induced by the infinitesimal action of X: XM(x) (exptX) x. The problem is due to John Toth and Steve Zelditch [TZ]. The context is classical and quantum integrability of geodesic flows. Recall that for a manifold Q with a Riemannian metric g the corresponding geodesic flow is the flow on the cotangent bundle T*Q of the Hamiltonian vector field Xh of the function h = h9 E Coo(T*Q) which is the square root of the energy:
where
:
.
h9(q,p) = (g,(p,p))'12 for
all q E Q,
p
E TQ.
Here
denotes the inner product on the cotangent
bundle T*Q Q dual to the inner product g; is the so called dual metric. For an analyst the function h is the principal symbol of the square root of the Laplace operator defined by the metric g. A precise definition of will play no role in these notes. For a Riemannian geometer it's important that the integral curves of the vector field Xh project down to geodesics on the manifold Q. Toth and Zelditch were interested in the meaning of L°° boundedness of the L2-normalized eigenfunctions of They observed that the question is easier if is quantum completely integrable. Again, it will not be important to us as to what that means precisely. What will matter is that quantum integrability of implies (classical) homogeneous complete integrability of the geodesic flow. Namely, it implies that there exist functions fi = h,f2,. .. E Coo(T*Q 0), n = dimQ,' such that
1. the functions fr,.
,
ft-,
are functionally independent on an open dense set
2. the functions Poisson commute with each other: n;
0 for all 1
3. the functions ft are homogeneous of degree 1:
p;J, = for all
: T*Q 0 E IR, where (q,eAp), q E Q, p E TQ*Q.
pj(q,p) = 1TQ deleted.
0
—*
eAf2
T*Q
0
again denotes the dilation
denotes the punctured cotangent bundle of Q, that is, TQ with the zero section
181
1.1. Introduction
Exercise 1.2. Suppose I is a smooth function on the punctured cotangent bundle = e'f for all T*Q 0 of a manifold Q which is homogeneous of degree 1, i.e., Show that its Hamiltonian vector field Xf (relative to the standard dilations symplectic structure on T*Q) satisfies = X1 opA. Conclude that the flow of Xh commutes with dilations.
Given a completely integrable system, we know that locally around any generic point there exist action-angle variables. Toth and Zelditch observed that things are considerably simpler if the action-angle variables are global. Then there on T*QNO preserving the function = exists an effective action of a torus h and the symplectic form, and commuting with dilations PA. Toth and Zelditch proved (op. cit.): acts effectively onM = T*TflNO, Theorem 1.3. Suppose that the Lie group G = preserving the standard syrnplectic form and the function h9 (q, p) = (g (p, p) ) 1/2 for some metric g on T'2. Suppose further the action commutes with dilations. If the action of G is free then the metric g is flat, that is, g=
dq2 0 dq3
for some constants gjj (with gzj = The eigenfunctions of a flat metric Laplace operator on a torus are well understood.
Let us now go back to the problem. The answer to the question is yes [LS]:
acts effectively on the Theorem 1.4. Suppose the Lie group G = F' = punctured cotangent bundle M = T* N 0 preserving the standard symplectic form and commuting with dilations : M —' M, = (q,e"p). Then the action of G is free.
The main purpose of these notes is to explain why Theorem 1.4 is true. I will now try to motivate the proof and to put it in a broader context. As was
remarked previously (Remark 1.1(3)), the action of the torus C on M = T*Tfl N 0 is Hainiltonian. By the dimension count the manifold M together with its natural symplectic structure and the action of C is a symplectic toric manifold. Recall the definition. g*) where Definition 1.5. A symplectic toric manifold is a triple (M, w, : M M is a manifold, w is a symplectic form on M, and is a moment map for an effective Hamiltonian action of a torus G on (M, w) satisfying 2 dim G = dim M.
Compact symplectic toric manifolds are well understood thanks to a classification theorem of Delzant [D], which says that all such manifolds are classified by the images of the corresponding moment maps. Note that by the Atiyah-GuilleminSternberg convexity theorem [A, GS1], the images are convex rational polytopes.
182
1.
toric integrable geodesic flows to contact toric manifolds
Deizant proved that in the tone case the polytopes are simple2 and additionally satisfy certain integrality conditions. Finally any simple polytope satisfying the integrality conditions occurs as an image of the moment map for a compact symplectic toric manifold. Exercise 1.6. Show that a Hamiltonian torus action on a compact symplectic manifold is never free. Hint: any smooth function on a compact manifold has a critical a maximum and a minimum). point (in fact it has at least two
The manifold M = 0 we are interested in is not compact. Worse, is homogeneous we will see that the corresponding moment map 4) : M (Proposition 1.13):
= for all m E M, A This is a bit of bad news Morse theory is an essential ingredient in the proof of the Atiyah-Guillemin-Sternberg convexity theorem and hence of Delzant's classification. Morse functions on a noncompact manifold which are not bounded either above or below are in practice impossible to work with. On the other hand, because of homogeneity, the moment map descends to the quotient of M by the action of R. The quotient is diffeomorphic to the co-sphere
= 1} for some dual metric S*Tfl is odd-dimensional, it is not a symplectic manifold.3 bundle S*Tn := {(q,p) E
I
Since
It is also easy to see in an example that the map induced on S*Tfl by a homogeneous moment map on T*Tfl symplectic moment maps:
0 behaves rather strangely if one is used to
Example 1.7. Consider the standard action of G = T*Tfl, the lift of the left multiplication:
on the cotangent bundle
(q,p) = (a. ...4 The corresponding moment map 4) : T*G = C X is given by 4)(q,p) = p. Fix the standard metric on C and identify the dual of the Lie algebra 9* with 9* Then the co-sphere bundle S*G is x The map 4)' = 4)IS*G : S*C is also given by 4)'(q, p) = p. Note that for any nonzero vector X e g the function (4)', X) has exactly two critical manifolds even though the action of C on S*C is
freer
Compare this with the symplectic situation where critical points of components of moment maps are points with nontrivial isotropy groups. What are we dealing with? We are dealing with moment maps for group actions on contact manifolds. 2A polytope in an n-dimensional real vector space is simple if there are exactly n edges meeting at each vertex. Equivalently, all the supporting hyperplanes are in general position. Thus a cube and a tetrahedron are simple and an octahedron is not. 3The manifold S*Tn is contact. See next Chapter and the Appendix.
Li. Introduction
183
We finish the section by sketching a strategy for our proof of Theorem 1.4.
The effective action of the torus C on M = T*Tfl 0 descends to an effective of M by dilations. The action of C on B action on the quotient B = preserves a contact structure (see Definition 1.21 below) making (B, C) into a
compact connected contact toric C-manifold (c.c.c.t.m., see Definition 1.37). We
then study all c.c.c.t.m.'s with a non-free torus actions and argue that none of them can have the homotopy type of B = TTh x
I. From toric integrable geodesic flows to contact toric manifolds
184
1.2
Symplectic cones and contact manifolds
In this section we define symplectic cones, contact forms and contact structures. Given a symplectic cone we show how to construct the corresponding contact manifold, and conversely, given a contact manifold we construct the corresponding symplectic cone. Thus symplectic manifolds and contact manifolds are "the same thing." Next we show that a symplectic action of a Lie group on a symplectic cone induces a contact action on the corresponding contact manifold. This will give us tools to set up a proof of Theorem 1.4 as a study of contact toric manifolds. The material in this section is fairly well known. We now start by defining symplectic cones.
Definition 7.8. A symplectic manifold (M, w) is a symplectic cone if
• the manifold M is a principal JR bundle over some manifold B, called the base of the cone, and • the action of the real line JR expands the symplectic form exponentially. That is, = eAw, where denotes the diffeomorphism define by A E JR.
Definition 7.9. Recall that a map f : X Y between two topological spaces is proper if the preimage of a compact set under I is compact. An action of a Lie group G on a manifold M is proper if the map G x M (g,m) (g. m,m) E M x M is proper.
By a theorem of Palais [P] the quotient M/G of a manifold M by a free proper action of a Lie group G is a manifold and the orbit map M MfC makes M into a principal C bundle (see also Remark 1115 below). It follows that if a symplectic manifold (M, W) has a complete vector field X with the following two properties: 1. the action of JR induced by the flow of X is proper, and
2. the Lie derivative of the symplectic form w with respect to the vector field X is again w: Lxw = then (M,
is a symplectic cone relative to the induced action of JR. This gives us an equivalent definition of a symplectic cone.
DefinitIon 7.10. A symplectic cone is a triple (M, w, X) where M is a manifold, w is a symplectic form on M, X is a vector field on M generating a proper action of the reals JR such that LXW = W. Example 1.11. Let (V, WV) be a symplectic vector space. The manifold M V {O}
is a symplectic cone with the action of JR given by p,, (v) = e'v. Clearly eAwv. The base is a sphere.
—
1.2.
Symplectic cones and contact manifolds
185
Example 1.12. Let Q be a manifold. Denote the cotangent bundle of Q with the
zero section deleted by TtQ .. 0. There is a natural free action of the reals R on p) = (q, e'p). It expands the the manifold M := T*Q 0 given by dilations standard symplectic form on the cotangent bundle exponentially. Thus T*Q 0 is naturally a symplectic cone. The base is the co-sphere bundle S*Q. Proposition 1.13. Suppose (M, X) is a symplectic cone and suppose a Lie group G acts on M preserving the symplectic form w and the expanding vector field X. Then the action of G on the symplectic manifold (M, w) is Jiamiltonian. Moreover we may choose the moment map 'I': M to be homogeneous of degree 1, i.e.,
= for all A E R and m E M. Here is, the time A flow of X.
denotes the action of R generated by X, that
Proof. Note first that since Lxw = and since dw 0, it follows from Cartan's formula = t(X)dw + dt(X)w) that = w. Since the action of C preserves X and w, it preserves the contraction t(X)w. Therefore for any vector A in the Lie algebra g of C we have LAM (t(X)w) = 0, where AM as before denotes the vector field on M induced by A. Therefore 0=
+
=
+ t(AM)w,
and consequently L(AM)W
We conclude that the map 4: M
d(w(AM,X)). defined by
(4(m), A) = iim(X(m) , is
a moment map for the action of C on (M,w).
0
Exercise 1.14. Suppose a Lie group C acts on a manifold M preserving a 1-form /3. Define the /3-moment map M by (Wa(m),A) = for all A g and all m E M. Here as usual denotes the vector field induced by A on M. Show that is G-equivariant, that is, show that for any a E C and any
mEM
m) = map
GL(g*) denotes the coadjoint representation. Conclude that the Adt C defined in Proposition 1.13 is equivariant.
I. From toxic integrable geodesic flows to contact toric manifolds
186
Definition 1.15. A 1-form a on a manifold B is a contact form if the following two conditions hold: 1.
a vector subbundle of the tangent bundle TB. 2.
is a symplectic structure on the vector bundle
B (i.e.
is
nondegenerate). Remark 1.16. 1. If —p B is a symplectic vector bundle, then the dimension of its fibers is necessarily even. Hence if a manifold B has a contact form, then B is odd-dimensional.
2. A 1-form a on 2n + 1 dimensional manifold B is contact if and only if the form a A is never zero, i.e., it is a volume form. [Prove this). Example 1.17. The 1-form a = dz + xdy on R3 is a contact form: a A da = dz A dx A dy.
Example 1.18. Let B =
IR
x T2. Denote the coordinates by t, 01 and 02 respectively.
The 1-form a = cost dO1 + sin t dO2 is contact. [Check this.) Lemma 1.19. Suppose a is a contact form on a manifold B. Then for any positive function f on B the 1-form ía is also contact. Proof. Note first that since f is nowhere zero, kerfa = kera. Thus to show that fa is contact, it is enough to check that d(f a) is nondegenerate, where e = kera = kerfa. Nowd(fa) = 0. Therefore = fdak. But f is nowhere zero and is nondegenerate by assumption. Thus is nondegenerate.
U
Definition 1.20. We define the conformal class of a 1-form a on a manifold B to be the set [a] = {e"a I h E C°°(B)}, that is, the set of all 1-forms obtained from a by multiplying it by a positive function.
Thus if a 1-form a on a manifold B is contact, then its conformal class
consists of contact forms all defining the same subbundle e of the tangent bundle of B.
Definition 1.21. A (co-orientable) contact structure on a manifold B is a subbundle of the tangent bundle TB of the form = ker a for some contact form a. A co-orientation of a contact structure is a choice of a conformal class of contact forms defining the contact structure. Remark 1.22. More generally a contact structure on a manifold B is a subbundle of the tangent bundle TB such that for every point x E B there is a contact 1-form a defined in a neighborhood of x with ker a = There exist contact structures which are not co-orientabje. For such structures a one-form a with kera =
exists only locally. IN THESE NOTES WE WILL ONLY DEAL WITH CO-ORIENTABLE CONTACT STRUCTURES.
1.2.
187
Symplectic cones and contact manifolds
Exercise 1.23. Let /3 be a nowhere zero 1-form on a manifold B and let Let if —' B denote the annihilator of ij in T*B: the fiber of if at b vector space
= ker (3. B is the
=0). B. Show that any Show that (3 is a nowhere zero section of real line bundle if B is of the form = fj3 for some nowhere other nowhere zero section /3' of if zero function f on B. Conclude that a contact structure is co-orientable if and only if the punctured real line bundle 0 has two components (0 of course denotes the zero section). Show that a choice of co-orientation of is the same as a choice of a of the punctured bundle r 0. component
= kera2) be two co-oriented Definition L24. Let = kercii) and contact manifolds. A diffeomorphism B1 B2 is a contactomorphism if the to preserving the co-orientations. That is, differential dço maps = Ia1 for some positive function f.4 Definition 1.25. An action of a Lie group G on a manifold B preserves a contact structure and its co-orientation if for every element a E G the corresponding diffeomorphism aB : B —' B is a contactomorphism. We will also say that the action of G on (B, is a contact action.
Definition 1.26. Let a be a contact form on a manifold B. The Reeb vector field Y0 of a is the unique vector field satisfying t(Y0)da = 0 and a(Ya) = 1. Exercise 1.27. Why does the definition of the Reeb vector field makes sense?
Remark 1.28. The Reeb vector field depends strongly on the contact form. And it is not just its magnitude: if a is a contact form and Y0 is its Reeb vector field, then there is no reason for t(Y0)d(f a) = 0 where f is a nowhere zero function. Exercise 1.29. Compute the Reeb vector field of the contact form a = dz + x dy on R3.
Exercise 1.30. Compute the Reeb vector field of the contact form a of Example 1.18: a = cost do1 + sint dO2 on B = x T2. Symplectic cones and contact manifolds are intimately related:
Theorem 1.31. Suppose a compact connected Lie group G acts effectively on a symplectic cone (M, w, X) preserving the symplectic form w and the expanding vector field X. Then G induces an effective action on the base B of the cone making the projection w: M B G-equivariant. Moreover, the base B has a natural cooriented contact structure and the induced action of G on B preserves a contact form a defining In particular the action of C on (B, is contact. 4Since the lift = co-orientation" means that maps
T*Bi to
TB2 of
maps to (cf. Exercise 1.23).
"d4p
preserves the
I. From toric integrable geodesic flows to contact toric manifolds
188
We start the proof of Theorem 1.31 with an observation that the action of G on M descends to an effective action of C on the base B making the projection B G-equivariant. Next we prove: M
M
Proposition 1.32. Any principal R-btrndle R
B is trivial.
Proof. The proposition is true because the real line is contractible. Here is an
M is a (local) section of w : M elementary argument. Note first that if 8: B is a function, then s—f makes sense; it is again a (local) section Band f E of w: M B. To prove that a principal bundle is trivial it is enough to construct a global section. To this end choose an open cover {U0 } of B such that for each M. Choose a partition of unity r0, subordinate to the there is a section cover {U0}. Two sections of a principal R-bundle differ by real-valued function. Thus by abuse of notation on an intersection fl is a real-valued — function. Now define for each index :
=
—
—
s0).
Then on an intersection U0 fl —
= (sfl
—
—
\
—
—
J
/
\c3,.y
=
sp —
—
—
—
\
Therefore the collection of local sections B. Consequently the bundle is trivial.
=
0.
defines a global section of w : M —*
0
Thus any symplectic cone is of the form B x JR where B = M/IR is an odddimensional manifold. Lemma 1.33. Let (M, X) be a symplectic cone, let B be its base and let w: M B denote the projection. Pick a trivialization cc: B x JR —÷ M. Then ço*w = d(etcx)
where t is a coordinate on JR and a is a contact form on B. Conversely, if a is contact form on B then (B x IR, d(eta), is a symplectic cone. Proof. By Proposition 1.32 the principal R bundle M B is trivial. Choose a trivialization M B x JR. Under this identification the vector field X becomes Since dw = 13 = t(X)w.
and
=
Then t(X)/3 =
0
0
dt(X)w = w (c.f. proof of Proposition 1.13). Let and = dt(X)/3 + t(X)d/3 = 0+ e(X)w = (3.
1.2.
189
Symplectic cones and contact manifolds
Hence for any point (b, t) E B x R 13(b,t) = et/3(b,o).
on B. It remains (dcx)d is nowhere zero, is contact. For this it suffices to show that to show that where d = dim M — 1. Now w = d(eta) is syniplectic. Hence wd is nowhere = etddt A A = (et(dt A + Hence A vanishing. Now is nowhere vanishing. = d(eta) and let Conversely suppose is a contact 1-form on B. Let = d(eta+O) = w. = = X = fit. Then It remains to check that is nondegenerate. For any (6, t) E B x R, the tangent where Y0 space T(b,t)(B x R) decomposes as T(b,t)(B x R) = is contact is the Reeb vector field of (cf. Definition 1.26). Since JR is nondegenerate as well. (6) to nondegenerate. The restriction dt A = et(dt A + dc') is nondegenerate. This proves that (B x IR, d(eta), Hence fl is a symplectic cone.
Since t(X(b, O))/3(b,O) = 0 it follows that 13(b.o) = ab for a 1-form
Exercise 1.34. Let (M,w,X) be a symplectic cone, let B be its base and let w M B denote the IR-orbit map. Pick a global section s : B —* M of tv: M —' B and let = s*(t(X)w). Show that & is a contact form on B. Show that it is the same contact form that the proof of Lemma 1.33 would produce from the denotes the t) Here Pt trivialization w: B x JR —b M, :
MM
action of JR.
B give rise to Remark 1.35. Different choices of trivializations of w : M different contact forms on B. However, they all define the same contact structure on the base B. Intrinsically can be defined as follows: for a point 6 E B,
= dtvm(ker(t(X)w)rn)
for any m E
(1.1)
It is not hard to check that is well-defined. First note that JR acts transitively = Second observe that for any A JR we have on the fiber eA(i(X)w), and hence = (ker(t(X)w)PA(,,,). Here again Pt M M denotes the action of JR. It follows that the action of C on B induced by an action of C on the symplectic cone w : M —* B preserves the contact structure defined by (Li). Since C is connected and since the identity map preserves the co-orientation of all the other elements of C also preserve the co-orientation.
It remains to show that there is a G-invariant 1-form with ker = By Lemma 1.33 a choice of a trivialization of tv: M —' B gives us a 1-form on B with ker = but this form need not be G-invariant. It is only C-invariant if the trivialization is G-equivariant. Therefore we proceed as follows. Lemma 1.36. Suppose a compact Lie group C acts on a manifold B preserving a (co-oriented) contact structure = ker & for some 1-form Then there exists a C-invariant 1-form & with ker& =
190
1.
toric integrable geodesic flows to contact tone manifolds
B —p B is a contactomorphism. Hence (aB)*c = for some positive function fa depending smoothly on a. Define a new contact form & to be the of & over the action
Proof. For every a E C the corresponding diffeomorphism aB
ofC: ab
f
da
da
Ir
= (J fa(b) da)/
= 0 = J0 for all b E B. Here da is a bi-invariant measure on C normalized so that Since fa > 0 for all a C, the integral (f0 fa(b) da) is positive and a
da = is
1.
indeed
0
nowhere zero.
From now on we will always assume that whenever a group actions preserves a contact structure it also preserves a contact form defining this structure. This concludes the proof of Theorem 1.31. It follows from the Theorem that if an n-torus C acts effectively on the punctured cotangent bundle M = T*Tfl 0 preserving the symplectic form and commuting with dilations then it acts on the quotient B = M/IR S*Tfl preserving the corresponding contact structure. Note that 2 dim G = dim B + 1.
Definition 1.37. An effective action of a torus C on a manifold B preserving a contact structure is completely integrable if 2 dim G = dim B + 1. A contact toric C-manifold is a co-oriented contact manifold with a completely integrable action of a torus C. We are now in position to reduce Theorem 1.4 to a statement about contact toric manifolds. Consider again the action of C = on M = T*Tfl '.. 0 preserving the symplectic form and commuting with dilations. As was remarked previously M is a symplectic cone over B = S*Tfl = x By Theorem 1.31 the action of C on M induces an effective action on B. Moreover B has a C-invariant contact structure making (B, into a compact connected contact toric C-manifold. Clearly if the action of C on B is free then the original action of C on M was free as well. Therefore a proof of Theorem 1.4 reduces to
Theorem 1.38. Let (B, = ker ci) be a compact connected contact toric C-manifold and suppose the action of C is not free. Then B is not homotopy equivalent to T" x n = dim C. The rest of the notes will be occupied with a proof of Theorem 1.38. The proof uses heavily contact moment maps which are discussed in the next chapter. We end this section with a partial converse to Theorem 1.31. We have seen that given a contact form ci on a manifold B the form d(etct) on B x R is symplectic. The pair (B x R, d(eta)) is called the symplectization5 of (B, ci). It is clearly a symplectic cone. Different contact forms on B give rise to different symplectic forms on B x lit However there is a symplectic cone that depends only on the contact structure 5Sometimes (B x
is
called the symplectificatiou of (B,cx).
1.2.
Symplectic cones and contact manifolds
191
= ker o (and its co-orientation) and not on a particular choice of a contact form: Let denote the component of NO giving its co-orientation (cf. Exercise 1.23). It is not hard to check that is a symplectic submanifold of the cotangent bundle
TB with its standard symplectic structure. The action of R on T*B N 0 by dilations preserves
is a contact form BxR
and makes it into a symplectic cone. A section fi of
B
on B with = ker/3. The form /3 defines a trivialization (b,t) = et/3b. The map pulls back the symplectic form on
to d(et/3).
Finally given a symplectic cone (M, w, X) with the base B, the orbit map B and the induced contact structure on B, there is an R equivariant symplectomorphism w: M defined as follows: for a point m M let be the covector in T,,(m)B such that
w: M
= (t(X)W)m(dSb(V)) for 811 V E Tv,(m)B and a section s of w: M —+ B.
The discussion above can be summarized as: Lemma 1.39. Let (B, = ker o) be a contact manifold, let be a component of N 0, the annihilator of in B minus the zero section. The principal R bundle B is a symplectic cone. If (M, w, X) is a symplectic cone with the base B and is the induced contact structure on B, then is isomorphic to (M, w, X) as a symplectic cone.
Chapter II
Contact group actions and contact moment maps Moment maps exist in the category of contact group actions. In fact moment maps
exist for all contact actions. This is because a contact form defines a bijection between contact vector fields and smooth functions.
Definition 11.1. A vector field X on a contact manifold (B, = ker is contact if Hence for any consists of contactomorphisms. In particular C its flow B. B, the Lie bracket [X, vi is again a section of section v of the bundle Thus for a contact action of a Lie group C on (B, the vector fields induced by elements of the Lie algebra g of C are contact. Exercise 11.2. Prove that a Reeb vector field is contact. More generally prove that
a vector field X on a contact manifold (B, = ker is contact if and only if = hcx for some function h (h can have zeros). A choice of a contact form on a contact manifold (B, identifies contact vector fields with smooth functions.
Proposition 113. Let (B, =
ker be a contact manifold. The linear map from fX a(X) is one-to-one contact vector fields to smooth functions given by X and onto.
Proof. Note that the Reeb vector field corresponds to the function 1. For any vector field X on B the vector field X — Cz(X)Ya is in the kernel of which iS the contact distribution Since dale is non-degenerate, X — Q(X)Ya is uniquely determined by t(X — For any section v of B and any vector field X we have
0=
+
II. Contact group actions and contact moment maps
194
Assume now that X is contact. Then a([X, vi) = 0. Therefore, by Cartan's formula
0 = (dt(X)a + t(X)da)(v). Hence for any section v off, dcx(X,v) = —d(a(X))(v) = _dfx(v). And, of course, da(X,v) = da(X — fXYa,V) for all v. We conclude that
=
—
(111)
for any contact vector field X. Thus if X is contact, its component in the direction of the Reeb vector field is fXYC, and its component in the direction of the contact distribution is uniquely determined by (11.1). Conversely, given a function I on B there is a unique section of such
that
= The vector field X1 :=
(11.2)
is a contact vector field with a(Xf) = f.
+
0
Exercise 11.4. Given a function / on a manifold B with contact form a check that the vector field X1 := X + where is defined by (11.2), is contact. That is, show that a = ha for some function h (cf. Exercise 11.2).
Suppose now that a Lie group C acts on a manifold B preserving a contact 1-form a. Then for any vector A in the Lie algebra g of C, the induced vector field AB satisfies1 LABa = 0 and, in particular, is contact. Since the contact form a defines a 1-1 correspondence between contact vector fields and functions, it makes sense to define the a-moment map :B by
=ab(AB(b))
(11.3)
for all b E B and all A E g. Note that by Exercise 1.14 the a-moment map 'IIQ is C-equivariant. The a-moment map, as the name suggests, depends rather strongly on a: if / is any positive C-invariant function on B, then ía is another C-invariant contact form and clearly
= In particular, unlike in the symplectic case, the image of a moment map is not an invariant of the action and of the contact structure. However the moment cone C('i'a) defined by
=
E
I
e W,,(B), t E [0, oo)}
is an invariant of the action of Con 'Recall that the vector field AB is defined by
= kera). =
exp(t.4)
b.
195
Remark 11.5. Suppose a Lie group G acts on a manifold B preserving a contact form a. The action of C lifts to an action on the cotangent bundle T*B which preserves the annihilator of the contact structure = ker a. Moreover, the action T*B. of 0 which contains the image of a B preserves the component The action of C on T* B is Hamiltonian with a natural moment map T*B g5 given by
= (p,AB(b)) for all b C B, p C
is a C-invariant symplectic is a moment map for the action of
A E g. Since the submanifold
submanifold of T* B the restriction 'F: It is not hard to check that C on
=
'F
as the moment For this and other reasons it makes sense to think of 'F = U {0}. We will Note also that map for the action of G on (B, thus denote the moment cone C('F,,) by C('F). Consequently and by analogy with symplectic toric manifolds we will think = kera,W ga). of contact toric C-manifolds as triples :
:
2Note that since 4' is G-equivariant and a B TB is G-equivariant, is O-equivariant as well. This gives us an alternative proof that the a-moment map is equivariant.
Chapter III
Proof of Theorem 1.38 The rest of the lecture notes will be devoted to a proof of Theorem 1.38. Right from the beginning the proof will bifurcate into two cases: the contact manifold B is 3-dimensional and dim B > 3. If dim .8 = 3 we will argue directly using slices that the orbit space B/C is homeomorphic to a closed interval [0, 1] and then use this to compute the integral cohomology of B. This will show that B cannot be homeomorphic to S*T2 = T3. We will then consider the case where dim B > 3. In this case we have a connectedness and convexity theorem of Banyaga and Molino (see [BM1, 13M2]; for a different proof see [L2J):
be a compact connected contact Theorem 111.1. Let = '4' : toric C-manifold. Suppose dim B > 3. Then the fibers of the moment map W are C-orbits (and in particular are connected) and the moment cone C(W) is a convex Moreover if the action of C is not free. polyhedral cone in
Our proof will then bifurcate again. We will consider separately the case where the moment cone contains a linear subspace of dimension k, 0 < k 0. It is easy to see that N cannot be homotopy equivalent In the latter case we will argue following Boyer and Galicki [BC] that there
is a locally free2 S' action on B such that the quotient B/S1 is a (compact connected) syxnplectic toric orbifold. Since the action of S' is locally free there is 1The result was rediscovered a few years later by Lerman, Tolman and Woodward (Lemma 7.2 and Proposition 7.3 in ILTI). 2An action of a Lie group C on a manifold Z is locally free if all the isotropy groups are zero dimensional.
III. Proof of Theorem 1.38
198
a long exact sequence of rational cohomology groups (the Gysin sequence) tying together the cohomology of B and of B/S' .On the other hand Morse theory on the orbifold B/S' shows that all odd-dimensional rational cohomology of B/S' vanishes. Together these two facts will imply that dimq H' (B, Q) 1 . Thus B cannot be S*Tfl = x n= + 1) > 2. This completes the preview of our proof of Theorem 1.38.
111.1
Homogeneous vector bundles and slices
Suppose that C is a Lie group, H C C a closed subgroup and suppose we have a representation of H on a vector space W. Then H acts on the product G x W by h• (g,w) = (gh—',g . w) for h E H, (g,w) e C x W. The quotient C XH W := (C x W)/H is a vector bundle over C/H with typical fiber W. We denote the image of(g,w) E Cx Win GXH W by [g,w]. Note that the actionofGon Gx W given by a• (g, w) = (ag, w) commutes with the action of H and hence descends to an action C on C XH W: a• [g,w] = [ag,wj. The projection G XH W C/H is G-equivariant and the action of G on the base C/H is transitive. This makes C X H W —, C/H into a homogeneous vector bundle. Conversely if ir : E C/H is a vector bundle with an action of C by vector bundle maps making ir equivariant, then E is isomorphic to C x H W where W is the fiber of E above the identity coset 1H.3 Indeed the map
(g,w)i-+g.w is Onto and is constant along the orbits of H. It descends to a vector bundle isomorphism
Next suppose a compact Lie group C acts on a manifold M. Consider an orbit C . x c M. The group C acts on the normal bundle of the orbit z'(C . x) = x) making the projection it v(C. x) —p (C . x) equivariant. Thus :
If we choose a 0-invariant Riemannian metric on M we can identify v(G. x) with the perpendicular of T(G . x) in Furthermore, the Riemannian exponential map exp : zi(C. x) —+ M is G-equivariant. Hence, by the Tubular Neighborhood theorem we get:
Lemma 111.2. Let C be a compact connected Lie group acting on a manifold M and
let x E M be a point. A neighborhood of C•x in M is G-equivariantly diffeomorphic to a neighborhood of the zero section of the homogeneous vector bundle C x W where denotes the isotropy group of x and W = x). 3Why is there a representation of H on W? 4Choose any metric on M and then average it over the group C (cf. proof of Lemma 1.36).
111.1. Homogeneous vector bundles and slices
199
Definition 111.3. Let C be a Lie group acting on a manifold M; let x E M be a point. Denote the isotropy group of x by C1. An embedded submanifold S C M is a slice through x for the action of C on M if x S, S is C1-invariant and M given by (g,s) if the map C x S g• a descends to an open embedding Thus by Lemma 111.2 slices exist for actions of compact Lie groups: we may
choose as a slice at x the image of a small C1-invariant neighborhood of 0 in W = T1M/T1(G x) under the exponential map. Here is a typical application of the existence of slices. Locally near x the quotient M/G is homeomorphic to the quotient (C x W)/G = WIGX. Hence .
quotients of manifolds by actions of compact Lie groups are modeled on quotients of vector spaces by linear actions of compact Lie groups. The linear action of C1 on W = T1M/T1(C x) is called the slice representation at x. Here is another application of the above construction: .
Lemma 111.4. Suppose C is a compact abelian group acting effectively on a con-
nected manifold M. Then every slice representation is faithful, i.e., no slice representation has a kernel.
Proof. Suppose the slice representation of H = C1 on W = TXM/TX(G x) is not faithful at a point x M. By Lemma 111.2 it is no loss of generality to assume that a neighborhood of C x in M is the homogeneous vector bundle C/H = C XH W x and that the point x is [1,0] C xH W. If there is an element a H such that a 1 and yet a w = w for all w E W, then [g, w] = [ag, wJ = [ga, w] = [g, w] = [g, w] for all [g, w] C x H W (ag = ga since C is abelian). Thus a H fixes an open neighborhood of z. Since the set fixed by a is closed and since M is connected it follows that a fixes all of M. This contradicts the assumption that the action of C on M is effective. U
Remark 111.5. The compactness of the Lie group C is not necessary for the existence of slices. According to Palais [P] it is only necessary that its action on a manifold M be proper (see Definition 1.9 above). Thus if an action of a Lie group C on a manifold M is free and proper, then the existence of slices tell us that a neighborhood of every orbit is equivariantly diffeomorphic to a product of C with some manifold S. It is not hard to deduce from this that the orbit space M/C is a manifold and that the orbit map M M/C makes M into a principal C-bundle. We now recall a few properties of tori, which for us are compact abelian Lie groups. If C is a torus, then the (Lie group) exponential map exp : g C is a covering map. The kernel Z0 of exp is called the Integral lattice. Clearly C = g/Zc. The group ZG is isomorphic to the fundamental group of C. Also it has the property that for any X ZG the corresponding 1-parameter subgroup {exp tX I t is a circle. The dual lattice = Homz(ZG,Z) E £(Za) C Z} is the weight lattice. It parameterizes 1-dimension complex representations of C, or,
III. Proof of Theorem 1.38
200
S': Given ii E the equivalently, group homomorphisms (characters) x G = corresponding character —p S1 is defined by G= for all X E g. Given a character x: C —÷ S', its differential dXi at 1 is a weight. Recall that a complex representation of a compact abelia.n group is a direct sum of one-dimensional complex representations. Thus a complex representation
of a torus is completely characterized by a finite set of weights. The same is true for symplectic representations of tori — one defines weights with respect to some complex structure compatible with the symplectic fonn. The weights do not depend on the choice of the complex structure; they only depend on the symplectic form.
The Lemma below is a key representation-theoretic fact in the classification of symplectic and contact toric manifolds. Lemma 111.6. Suppose p: H —+ Sp(V, w) is a symplectic representation of a compact abelian Lie group H on a symplectic vector space (V, ce). Suppose that p is faithful, i.e., kerp = {1}.
Then dim set
H
dim V. If H =
dim V,
then H is connected. Moreover, the
of weights for the representation of H on (V, w) is a basis of the weight lattice of H.
Proof. Since H is compact there exists an H-invariant complex structure J on V compatible with w (i.e., w(J., J.) = w(.,•) and for any v Owe have w(Jv, v) > 0). The choice of J identifies (V, with with the standard symplectic structure dz3 A n = dimR V. This, in turn, gives us a representation of H on by unitary matrices. Thus we may assume that p is an injective group homomorphism p: H U(n). The connected component H° of H is a torus. Therefore p(H°) is contained in a maximal torus of U(n) which is the n-torus. Since p has no kernel we have dim H = dim H° = dim p(H°) n = dim V. Now suppose dim H = n. Since all maximal tori in U(n) are conjugate, we
may assume that p(H°) is the standard maximal torus, that is p(H°) is the set of all diagonal unitary matrices. Since the only unitary matrices which commute with all the diagonal matrices are the diagonal matrices, we see that we must have p(H) = p(H°). Consequently since p is faithful, = H H°, i.e., H is a torus. Finally the set of weights of the maximal torus in U(n) for its representation on is a lattice basis of weight lattice Hence the set of weights for the representation of H on (V, w) is a basis of the weight lattice of H. 0 Lemma LII.?.
Let
(B, =
be a contact toric G-inunifold. Then
1. No C-orbit is tangent to the contact structure
In particular there are no fixed points. Hence the cs-moment map W0 does not vanish at any point for any C-invariant contact form Equivalently does not contain the origin in
2. All isotropy groups are connected.
111.1.
201
Homogeneous vector bundles and slices
Proof. Let b be a point in B and let H denote its isotropy group. The group H acts on the tangent space TbB. Since the contact form a is G-invariant, its kernel at b, is an H-invariant subspace of TbB. Since the Reeb vector field the hyperplane Ya of a is unique, the vector (b) is fixed by H. Thus we have an H-equivariant splitting TbB = Note that since It is an H-invariant subspace of Tb(G . b) fl = dim B — 1 we either have V = T6(G. b) or dim V = dim(G. b) — 1. We will argue that the former case cannot occur. But first we argue that V is an isotropic subspace of the symplectic vector space Let x, z E V be two vectors. There exist vectors X, Z E 9 such that XB(b) = x and = z. Then w(x,z) = dab(XB(b),ZB(b)). Since G is abelian [Zfi,XB] = 0. Therefore 0. The function a(XB) is G-invariant, hence ZB(a(XB))
Let V =
dim
w(x, y) = da(XB, ZB) =
—
ZB(a(XB))
—
da([XB, ZBJ)
0— 0
+ 0.
This proves that V is isotropic. Since H is compact, there is an H-invariant complex structure J on compatible with w. Since V is isotropic, V fl JV = 0 and V + JV = V JV is a symplectic subepace of It is H-invariant. In fact, since C is abelian, the action of H on Tb(G . b) is trivial. Hence the action of H on V JV is trivial as well. Let W denote the symplectic perpendicular to V JV. We get a symplectic representation of H on W. We now argue that if T5(C. 6) C then dim W is less than 2 dim H and that the representation of H on W must be faithful. This by Lemma 111.6 would give us a contradiction. We would then conclude that dim W = 2 dim H, which by Lemma 111.6 implies that H is connected. Suppose now that b) C eb. Then dim V = dim(G. b). Since B is toric, dimB = 2dimG —1. Since TbB
(111.1)
dimW = (2dimG—1)—1—2dim(G.b) = 2dimG—2(dimG—dimff) = 2dimH—2. Since (111.1) is a splitting as H-representations, the slice representation of H at b is JV W. As we observed earlier the action of H on JV is trivial. Since the action of C on B is effective, the representation of H on W must be faithful. Contradiction. Therefore Tb(G. b) eb and so dimV = dim(G. b) — 1. In this case the dimension count gives us exactly that dim W = 2 dim H. 0
Lemma 111.8. Let H C T2 be a closed subgroup isomorphic to S'. Then there is another closed subgroup K C T2 isomorphic to such that T2 = K x H.
Proof. Since H is isomorphic to S' it is of the form {exptzi t IR} for some vector u = (n1, mi) E Z2 = ker{exp: R2 T2}. We may assume that n1 and I
III. Proof of Theorem 1.38
202
m2 such that n1m2 — m1 are relatively prime. Therefore there exist integers Hence the vectors (fli, m1) and (n2, m2) form a basis of Z2. Take m1 n2 = 1. 0 K = {expt(n2,m2) t E
Remark 111.9. More generally if C is a torus and H C C is a closed connected subgroup there is another closed connected subgroup K C C so that C = K x H. This is a bit harder to prove than the Lemma above.
111.2 In
The 3-dimensional case
this section
we
prove Theorem 1.38 in the case that dim B
=
3:
Lemma 111.10. Let B be a compact connected contact tone C = T2 manifold (in
particular dim B = 3). Suppose the action of C is not free. Then there exist two closed subgroups Ki, K2 C C isomorphic to S' so that B is homeomorphie to (0,ag) for all g E C, a K1 and (l,g) "-i (1,ag) where (0,g) ([0,11 x C)/ K2. In other words B is obtained from the manifold with for all p G and a boundary [0, 1] x C by collapsing circles in the two components of the boundary by the respective actions of two circle subgroups. (It may happen that K1 = K2).
Exercise 111.11. Consider the standard sphere S3 = {(zi, z2) E C2 I 1z112 + 1z2I2 = 1}. The torus C = {(A1,A2) E C2 I IAi 12 = 1, 1A212 = 1} acts on S3 by (A1,A2). Show that S3 is of the form ([0,1] x G)/ for two circle (z1,z2) = is the equivalence relation in Lemma 111.10. What subgroups K1, K2 of C where are the subgroups K1, K2?
Proof of Lemma 111.10. By Lemma 111.7 all isotropy groups for the action of C on B are connected and no isotropy group is all of C. Therefore, since dim G = 2 the possible isotropy groups are trivial or circles. And points with circle isotropy groups must exist since the action is not free. If the isotropy group of a point b E B is trivial then (by Lemma 111.2 and the dimension count) a C- invariant neighborhood U of C is equivariantly diffeomorphic to Cx I where I is an open interval and C acts on Cx I by g.(a, t) = (pa, t). Hence B so that ir o s = id where ir: B —+ B/C U/C = I and there is a map s: U/C is the orbit map, i.e., ir has a local section. Now consider a point b E B with the isotropy group Gb isomorphic to S' By Lemma 111.2 a neighborhood of b in B is C-equivariantly diffeomorphic to a neighborhood of the zero section in C x Gb W. By Lemma 111.8 there is a circle x L. Note that L C C such that G = L and that dim W = 2 and b that by Lemma 111.4 the representation of Gb on W is faithful. There is only one faithful real 2-dimensional representation of S' — it is the representation of S1 on R2 as SO(2) or, equivalently, the representation of S' on C C = Lx Gb, C XGb W = Lx W, where Lx C6 acts on Lx W by 5By Lemma 111.7 there are no more possibilities for Gb.
as U(1).
Since
=
111.2.
203
The 3-dimensional case
(Ag, p . w). We conclude that there is a G-invariant neighborhood U of
G
that
z) =
a diffeomorphism : U —p S' x D2 such for all g E C, x E U. Here (A,p) E S1 x S' C Cx 'C
acts on S' xD2cCxCby (A,p) (zj,z2) = (.Xzi,pz2).
Note that 5' x D2 = (T2 x [0, 1))! (A1,A2) E T2 and p
ES'.
We
where (A1, A2, 0)
(A,, pA2, 0) for all
conclude that
1.
2.
3. there is a local section s : U/G —' B of the orbit map ir : B
BIG;
4. the set of points in U with non-trivial isotropy groups is where again ir : U —+ [0,1) denotes the orbit map.
b=
(0),
It follows that the orbit space BIG is locally homeomorphic to either an open interval or to a half-open interval [0, 1). Hence B/C is a topological 1dimensional manifold with boundary Since B/C is compact and connected we may identify it with [0, 1]. Note that the set of points with non-trivial isotropy groups is ir'({O, 1}), where by abuse of notation ir : B [0,11 denotes the orbit map. More specifically ir'(O) = C/K1, ir'(l) = C/K2 for some circle subgroups K1,K2 C G. B, so that We now argue that ir: B [0,1] has a global section a: [0, 1] iros(t) t for all t [0,1]. We have seen that sections of ir exist locally: for every t E [0, 1] there is an interval I C [0, 1] open in [0, 1] and containing t and a map B so that it o s = idj. We want to patch these local sections into a global a:1 section. Since [0, 1] is compact, we can cover it by finitely many intervals so that on each I, there is a section —p B. Let us now assume for simplicity that there are only two intervals: jo = [0,2/3) and (1/3, 1]. The case of more than :
two intervals will be left as an exercise to the reader. Thus we have two sections [0,2/3) B and : (1/3,1] B. Since U acts freely on ir'((O, 1)) the map (1/3,2/3) x C given by = g.so(t) is a G-equivariant diffeomorphism (where G acts on the product (1/3,2/3) x C by multiplication on os0(t) os1(t) = (t,g(t)) for some the second factor). We have (t, 1) and curve g : (1/3,2/3) —+ C. Since (1/3, 2/3) is simply connected we may lift g to a curve 'y(t) in the universal cover exp: g C of C. Choose a smooth function p : (1/3,2/3) [0, 1] with p(t) 0 for t near 1/3 and p(t) 1 for t near 2/3. Now consider the curve a: (1/3,2/3) C given by a(t) = exp(p(t)-y(t)). The map s2(t) = a(t)) is a local section of it B —* [0, 1] which agrees with near 1/3 :
III. Proof of Theorem 1.38
204
and with Si near 2/3. Thus we can define a global section s: [0,1] —' B by
Iso(t)
s(t) =
t E [0,1/3], t
t 6 [1/3,0]. [0, 1] we can define a continuous map Now that we have a global section of ir : B J: [0,11 x G B, f(t,g) = g s(t). The map is onto; it descends to a bijective B where is the equivalence relation in continuous map f : ([0, 1] x G)/
the statement of the Lemma. Since ([0, 1] x G)/ homeomorphism.
is compact, the map f is a
0
Exercise 111.12. Show that if the groups K, and K2 in the statement of the Lemma
are the same, then B is S' x S2. Exercise 111.13. (This exercise is considerably harder than the one above.) Show that if the groups K, and K2 are different, then B is the quotient of S3 by a finite cyclic group. Lemma 111.14. Let G
T2 and let K1, K2 C C be two closed subgroups isomorphic
to 5'. Let B be the topological space ([0,1] x C)/ where (0,g) (0,ag) for all g E C, a E K1 and (1,g) (1,ag) for all g c C and a K2. In other words B is obtained from the manifold with boundary [0,11 x C by collapsing circles in the two components of the boundary by the respective actions of two circle subgroups. Then either H'(B,Z) = Z = H2(B,Z) or H'(B,Z) = 0 and H2(B,Z) is a finite group. In particular, B cannot be homeornorphic to the 3-torus T3.
Proof. Recall that W (G, Z) is isomorphic to the weight lattice and that the isomorphism is given as follows: A weight v defines a character xv : C S' = by the class is the element in H'(G, Z) corresponding to z.'. Here dO is the obvious 1-form on S'. Consequently if C = T2 and K3 C C is a circle subgroup, then :C C/KJ S' is a character and hence the weight v3 = (dir3)i defines an element of H'(G,Z). Thus if we identify H'(C/K,,Z) with Z and H'(G,Z) with then the map H'(C/K3, Z) H'(C, Z) becomes the map Z n '—f nil3 6 The sets U = ([0,2/3) x G)/ i-..' and V = ((1/3,1] x G)/ are two open subsets of B. We have B = U U V, Un V = (1/3,2/3) x C is homotopy equivalent
to C, U is homotopy equivalent to C/K,, V is homotopy equivalent to C/K2 and the inclusion maps U fl V '—i U, U fl V V are homotopy equivalent to projections 7r1 : C C/K,, it2 C C/K2 respectively. Hence under the above identifications of H' (U) and H'(V) with Z, the inclusions UflV U, UflV —* V induce the maps Z 3 fl ni'3 j = 1,2, respectively. We now apply the Mayer-Vietoris sequence to compute the integral cohomology of B. We have: 0 -* H°(B) H°(U) H°(V) —p H°(G) H'(B) H'(C) H2(B) H2(U)eH2(V) H2(C) H3(B) 0. :
205
Uniqueness of symplectic toric manifolds
11L3.
H°(G) is onto. Given the identifications above
Clearly the map H°(U)
(n,m) e—+ nvl+m112 E
H'(B) We therefore have 0 H3(B) 0. We conclude that • H2(B) =
H2(B)
+ Zv2)
• H1(B) = {(n,m) E Z2 I
ni.'1
+mv2 = 0}.
are differentials of projections onto quotients by circle subgroups, are independent over Z or ill = ±112. In the first case H' (B) = 0 either and and H2(B) is a finite abelian group. In the second case H'(B) = Z and H2(B) =
Since 111,
0
This finishes the proof of Theorem 1.38 in the case that dim B = 3.
111.3
Uniqueness of symplectic toric manifolds
In this section we sketch a proof of g*) be a compact connected contact Theorem 111.15. Let = toric G-manifold. Suppose the dimension k of the macimal linear subspace of the moment cone = U {0} satisfies 0 < k < dim G. Then B is homotopy equivalent to the product of a k-torus with a sphere. In particular B is not the co-sphere bundle of the n-torus, n + 1) = dim C.
The main idea of the proof is simple. We will first argue that there is an action of C on the symplectic manifold
M = T*Tk xC1 =
= {(q,p,z) E Tk x (R") xC' = T*Tk xC' 1 (p,z)
+ 1—k) >0, with moment map
M) =
M —÷
(0,0)},
such that (111.2)
We then argue that (111.2) implies that M is G-equivariantly syniplectomorphic Note that M is homotopy equivalent to Tk x to We start with the definition of a symplectic slice representation (c.f. proof of Lemma 111.7), which is essential for understanding the local structure of symplectic toric manifolds.
Definition 111.16. Let (M, w) be a symplectic manifold with a Hamiltonian action of a torus G. Then an orbit C . m is an isotropic submanifold of (M,w).6 The symplectic slice representation at m is the representation of the isotropy group 6For a proof of this easy fact see, for example, [CS].
III. Proof of Theorem 1.38
206
Tm(G m)w/Tm(G m). Here, Gm of m on the symplectic vector space V as usual, Tm(C m)" denotes the symplectic perpendicular to Tm(G m) in the symplectic vector space (TmM,
The equivariant isotropic embedding theorem (see for example [CS], Theorem 39.1) asserts that a neighborhood of an orbit G . m is determined (up to equivariant symplectomorphisms) by the symplectic slice representation at m. In fact, the topological normal bundle of the orbit G is its Lie algebra (op. cit.). is Remark 111.17. In the case of symplectic tone manifolds the dimension of the symplectic slice V at m is twice the dimension of the isotropy group Gm. Hence by Lemma 111.6 the group Cm is connected and the representation of Cm Ofl V is determined by a set of weights {v,} which forms a basis of the weight lattice of Cm. Note that the image of V under the moment map :V defined by the representation is the cone
=
0).
In particular the edges of the cone are spanned by the weights. Alternatively, the isotropy group Cm is isomorphic to Tt for some 1 and the slice representation p : Gm —i Sp(V) is isomorphic to the standard representation of Tt on With a little more work one can prove the following two propositions (their proofs can be found, for example, in [D]). PropositIon 111.18. Let (M, w, '1': M —* g') be a symplectic tome manifold, m E
M a point and U a neighborhood of C m in M. Then for a sufficiently small ball 0 about = in the set
Proposition 111.19. Let $: M —' g*) be a symplectic toric manifold, m M a point, = '1(m) and Gm the isotropy group of m. Identify Cm with the standard torus d = dim Cm, and extend it to an identification of C with T'1 x Tc, 8 c = dim C - dim Cm. A C-invariant neighborhood U of C m in M is equivariantly symplectomorphic to a neighborhood of TC x {(0,0)} in T*TC x Cd T' x (RC)* x Cd. Hence .
x
where 'i
:
T*TC x Cd
action of C = 7The
(RC)* x (R(i)*
is the moment map for the "obvious"
x Td on T*TC x representation of Tt = = (Xizi
we use Remark 111.9. 9The "obvious" action of TC on T*TC
C'
=
81-Iere
is
the lift of left multiplication.
1} on
is given by
111.3.
Uniqueness of symplectic toric manifolds
207
be a compact connected contact toric CCorollary 111.20. Let (B, = ker 'I' manifold. Suppose the dimension k of the maximal linear subspace of the moment cone = U {0} satisfies 0 < k
U
where 4': action of
{0} =
(Rk)* x xC x T' on T*Tc x C1.
x C')
is the moment map for the obmous
Proof. If C is a cone in
whose maximal linear subspace is P and if U is a of a point E P then C equals the cone on U fl C with the
neighborhood in vertex at
C={t(p—zJ)+r/lt0,pE UflC}. Since the B is contact toric does not contain the origin (cf. Lemma 111.7). —* Since B is compact and : —* is homogeneous, : {0} is proper. Hence for any c and any neighborhood U of (ii), 'IP(U) is a neighborhood of W are C-orbits (by Theorem 111.1), for any neighborhood U of an orbit C- m the set 'Ji(U) is a neighborhood of = '11(m) in In particular it contains a set of the form C) n where 0 C is a small ball about Let P C C('I') be the maximal linear subspace and let 0 E P. Then, as noted at the beginning of the proof, for any ball 0 C about ij we have
C('1') = {t(p
—
+
It
0, p E On 0, p€ W(U)}.
On the other hand, it follows from Proposition 111.19 that
{t(p —
+
t 0, p E '11(U)) =
x Cd)
RC x
for some integers c and d. It follows that c = k, d = dim C — k = I and that C('I') = 4(T*Tk x C').
0 Note that since
=
x {(0,0)}, we get
= where
M = (T*Tk x C') N
(Tic
x {(0,0)}).
Definition 111.21. A symplectic toric C-manifold the purposes of these lectures) if
4': M
g*) is good (for
III. Proof of Theorem 1.38
208
1. the fibers of 4) : M —'
2. the set 4)(M)
is
are
connected,
contractible and
3. there is an open set U C
such that 4)(M) C U and the map 4) M —p U
is proper.
The definition is designed in such a way that it includes compact symplectic toric manifolds, symplectizations of compact contact toric manifolds of dimension bigger than 3 and the manifolds of the form M = (T*Tc x C1) (Tk x {(0, O)}), 0 < k, 1. As a consequence of the definition and of Proposition 111.18 we have g*) be a good syrnplectic toric manifold. Lemma 111.22. Let (M,w, 4) : M For any point 'j E 4)(M) and any sufficiently small ball 0 centered at the set On 4)(M) determines the symplectic tome manifold (4) '(0), wI.1, -1(0), (0)) (up to a C-equivariant symplectomorphism).
Definition 111.23. Two symplectic toric C-manifolds (M, w, 4) : M —' g*) and (M',w',4)' M' are isomorphic if there is a G-equivariant diffeomorphism M M' such that u*wl = w and = 4). We denote by Iso(M, w, 4)) = Iso(M) the group of isomorphisms of a symplectic toric manifold (M, w, 4)).
Note that the last condition on c is almost redundant — if c is symplectic and C-equivariant then c c for technical convenience.
Definition 111.24. Two symplectic toric C-manifolds (M, w, 4) (M',w',4)' : M' are locally isomorphic over a set C
1. 4)(M) =
2. for any
M
g*) and
= 4)'(M') and
and any sufficiently small ball 0 C
symplectic tone manifolds (4)—' (0),
((4)')' (0),w'I(o')l(Q),
centered at
the
and are isomorphic.
Given the above definitions we see that Lemma 111.22 implies
Lemma 111.25. Suppose (M,w,4)) and (M',w',4)') are two good symplectic tone C-manifolds with 4)(M) = 4)'(M'). Then (M,w,4)) and (M',w',4)') are locally isomorphic over = 'I'(M). Therefore the proof of Theorem 111.15 reduces to
Proposition 111.26. Any good symplectic toric C-manifold locally isomorphic to a given symplectic tonic C-manifold (M, w, 4)) is actually isomorphic to (M, w, 4)).
111.3.
209
Uniqueness of symplectic toric manifolds
The proposition could be stated for a larger class of symplectic toric manifolds. We leave it to the reader to find the most general form of the statement (and prove it). The rest of the section is occupied with a proof of the proposition. is locally isomorphic to Suppose a symplectic toric C-manifold (M', w', and for any sufficiently small ball 0 C Then for any E is isomorphic to (4') '(O). Choose (0) symplectic toric manifold about the a locally finite cover {Ot}iEJ of 4>(M) by such balls. Then for each index i we have an isomorphism :
Let
=
fl 03. Define
4)1(Q\
— Jj
f
J3
etc.; we for the restriction (to keep the notation manageable we wrote will continue to omit restrictions iii the rest of the section). It is easy to see that
= id,
o
= id, and gij
o
ogki = id
(111.3)
wherever these equations make sense. and (M,w, c1) allow us to reconstruct (M',w', i"). The data
on M by ? x2 Define a relation Equations (111.3) imply that '—' is an equivaif x3 = x3 E is a symplectic toric manifold. Moreover lence relation. It follows that M/ the map F: M M' defined by = descends to a well defined map F: M/ M'. The map F is an isomorphism. is another collection of isomorphisms. Suppose of2. Clearly Let = ohT' (111.4) =
Indeed, let M =
o where E We also get a different set of data if we choose a different cover. However, all cohomology with these sets of data define one object a class in the first
coefficients in a certain sheaf. Let us now review the notions of sheaves and tech cohomology. There are many good references for this material. I will be following cwW].
Definition 111.27. A sheaf of groups S on a topological space X is an assignment
5: {opens sets in X}
groups, U
S(U)
satisfying two conditions:
1. For a pair of open sets U c W in X there is a restriction map 5(U) such that for any three sets U C W C V of X
w v_v
Pu °Pw—Pu-
:
5(W)
III. Proof of Theorem 1.38
210
Elements of 8(U) are called sections. Given a section
E 8(W) we write
calu for 2.
Given an open cover {U2} of an open set U (so that U = U collection of sections
E
and a
8(U2) such that 'PiI(JiflU)
=
E 8(U) such that
for all indices i, j there is a unique section = çoj
for all i.
The sheaf S is abelian if 8(U) is an abelian group for all open sets U.
Three examples of sheaves will be important to us. Check that they are indeed sheaves. Example 111.28. Let (M, w, 4') be a good symplectic tone C-manifold. The assignment iso: U '—i Iso(U) (U c 4'(M) open) is a sheaf on 4'(M). The group operation is composition.
Example 111.29 (Locally constant sheaf). Let H be a group and X a topological space. The assignment that associates the group H to every open connected subset of X is a sheaf, called a locally constant sheaf. It is denoted by ff. Thus ff(U) = H for every connected open set U C X. The group operation is the multiplication in H. Example 111.30. Let (M, w, 4') be a good symplectic toric C-manifold. Define a sheaf Con 4'(M) by C(U) =
G-invariant smooth functions on 4'1(U).
The group operation is addition of functions. Definition 11131. Let 81,82 be sheaves on a topological space X. A map of sheaves —* S2 is a family of group homomorphisms r:
UCXopen compatible with the restrictions: (p2)utY °
fw
fu °
for all pairs U C W of open sets in X.
Example 111.32. Consider the sheaves Iso and C defined in Examples 111.28 and 111.30 above. A section f C(U) is a C-invariant function on Its time t flow preserves the fibers of 4 and is a G-equivariant symplectoinorphism of 4'1(U). Hence is a section of Iso(U). This gives us for each open set U C 4'(M) a map ru : C(U) Iso(U), ru(f) = Moreover, any two functions 11, C(U) +12 Poisson commute [prove thisj. Hence = and therefore is a group homomorphism. Thus we get a map of sheaves i- C —* Iso. :
Uniqueness of symplectic tone manifolds
111.3.
Given a map of sheaves r
S2
211
one can define the sheaves the kernel and
image sheaves kerr and imr: for an open set U (kerT)(U) = kerru, (imr)(U) = im rU. Hence it makes sense to say that a map of sheaves is onto and more generally
talk about exact sequences of sheaves.10 Proposition 111.33. Let (M, w, be a good syrnplectic tone manifold. The map of sheaves r : C —' Iso defined above is onto. Hence Iso is an abelian sheaf.
The kernel of r is the locally constant sheaf R x Z0.
We
thus have a short
exact sequence of abelian sheaves: Iso—'O.
The proposition is due to Boucetta and Molino [BoMI. See also [LT].
111.3.1
Cech cohomology
In this subsection we "review" the notion of Cech cohomology with coefficients in an abelian sheaf. There are many good references, such as [WW], for the nontrivial facts that we list below without proofs. Let X be a topological space {U1} an open locally finite cover of X and S an abelian sheaf on X. A 0 Cech cochain is a function that assigns to each index i an element of 8(U2), i.e., the group of 0-cochains C°({U2},S) is the product flS(U1). A 1 Cech cochain assigns to an ordered pair of indices ii an element of 8(U23) where = U. n U3. Moreover we require that gj, (we now think of the groups 8(U) additively). More generally a p-cochain assigns to an ordered p + 1 tuple of indices io. . . an element where E = U20 fl. . and is skew-symmetric in the indices. The coboundary operator o: C"({U1},S) —+ is defined by
= where means that the index is omitted, and where we omitted writing the restrictions of the terms on the right hand side to One proves that
=
The cohomology of the complex denoted by !f*({U2},S) is called the Cech cohomology of the cover { U,) with coefficients in the sheaf 8. Given a refinement { } of the cover {U2 } the restrictions give rise to a chain map CP({U2}, S)— CP({1'},S), which in turn gives rise to a map in cohomology H*({U2},S) H*({V,},S). Taking the direct limit over all locally finite covers we get a well-defined cohomology group 0.
ff*(X,S) = 1LmR*({Ut},8), the
cohomology of X with coefficients in the sheaf S.
'°Warning: the map r: S2 being onto does not mean that rjj is onto for every open set U. See [WWJ or any other good book on sheaves for more details.
III. Proof of Theorem L38
212
g*) be a good symplectic toric G-manifold and Now let (M, 4': M } E a iocaliy finite cover of 4'(M) by sufficiently small balls. If = 0 means that for all triples of indices ijk we have is a 1-cochain, then
+ gi,, which is (111.3) in additive notation (where on the right hand side we omitted the E C'({01},Iso) are two 1-cochains Similarly if {gj3}, restrictions to then for some 0-cochain that differ by 0=
=
—
gjy
=
—Øjk +
+ h,
hence
=
— + which is (111.4) in additive notation. Thus the discussion above shows that to ev}, Iso) there corresponds a good symplectic toric manifold ery element of R' g*) locally isomorphic to (M,w, 4' : M —' g*). More gener(M',w', 4" : M ally one can check that there is a one-to-one correspondence between cohomology classes in H1 (4'(M), Iso) and isomorphism classes of good symplectic toric mang*). Thus to complete the proof of ifolds locally isomorphic to (M, 4' : M
Proposition 111.26 (and thereby Theorem 111.15) it remains to show that the group
H'(4'(M), Iso) is trivial for any good symplectic tone manifold M. For this we use Proposition 111.33, two properties of cohomology and a property of the sheaf C defined in Example 111.30. The first property of Cech cohomology that we need is Theorem 111.34. A short exact sequence of abelian sheaves 0 Si on a space X induces a long exact sequence in Cech cohomology
IP'(X,S3)
-+
83
0
-+
The second property that we will use is
Theorem 111.35. Let X be a simply connected topological space and H an abelian group. The Cech cohomology H* (X, H) of X with coefficients in the locally constant sheaf H is isomorphic to the singular cohomology H* (X, H) of X with coefficients in the abelian group H. We will use the following property of the sheaf C (cf. [LTJ, Proposition 7.3) Lemma 111.36. The sheaf C defined in Example 111.30 is acyctic, that is,
=
0
for all q >
0.
Now putting Theorem 111.34, Lemma 111.36 and Proposition 111.33 together we see that if (M, w, 4') is a good symplectic tone G-manifold and Iso the sheaf defined in Example 111.28 then the cohomology group (4'(M), Iso) is isomorphic
to H2(4'(M), IR x Zc). The latter group is isomorphic to the singular cohomology group x R) by Theorem 111.35. But 4'(M) is contractible, so H2(4'(M),ZG x R) = 0. Therefore H'(4'(M),Iso) = 0, which proves Proposition 111.26 and thereby Theorem 111.15.
111.4.
111.4
213
Proof of Theorem 1.38, part three
Proof of Theorem 1.38, part three
The goal of this section is to prove be a compact connected contact —b Theorem 111.37. Let (B, = ker 'I' toric G-manifold with dim B 3. Suppose there is a vector X in the Lie algebra g of C such that the function (W, X) is strictly positive on B. Then dim H1 (B, R) n = dimC = x = 1. In particular B is not the co-sphere bundle The proof of Theorem 111.37 above will complete our proof of Theorem 1.38. Since Theorem 1.38 implies the main result of the notes, Theorem 1.4, this, in turn,
will finish the proof of the main result. As was sketched out at the beginning of Chapter 4 our proof of Theorem 111.37 has several steps. The first one is a theorem implicit in a paper of Boyer and Galicki [BG]:
be a compact connected contact = kerct, 'I! toric C-manifold with dim B 3. Suppose there is a vector X in the Lie algebra g of G such that the function ('p, X) is strictly positive on B. Then there exists on B a locally free circle action so that the quotient M = B/S' is a (compact)
Theorem 111.38. Let
symplectic toric orbifold.
The second step is the argument that if M is a compact connected symplectic (M, IR) = 0 for all odd degrees q. This step uses Morse theory toric orbifold then on orbifolds. Let us now see why these two steps give us a proof of Theorem 11137. Consider the circle action produced by Theorem 111.38 and the corresponding S' orbit map B —+ M. If the circle action is actually free, then ir is a circle fibration and we have the Gysin sequence
0=
H'(M,R)
H'(M,JR)
H'(B,R)
H°(M,JR)
H2(M,R) —* (111.5)
If the action of 5' is locally free, the long exact sequence (111.5) still exists. The reason is that the Gysin sequence arises from a collapse of the Leray-Serre spectral sequence for a sphere bundle. For locally free S' actions the orbit map ir: B M is not a fibration, but the corresponding spectral sequences still collapses if we use real coefficients.11 Now, since H' (M, R) = 0 by the second step, it follows from (111.5) that dim H'(B,R)
Iii. Proof of Theorem 1.38
214
argue that a compact symplectic tone orbifold M has a Morse function with all indices even and use this to conclude that the real cohomology of M vanishes in odd degrees. We now define orbifolds and related differential geometric notions. For more
details, see Satake [Sal, Sa2] and, for a more modern point of view, Ruan [R]. The notion of orbifold was introduced by Satake in 1956 under the name of Vmanifold. Orbifolds are designed to generalize manifolds in the following sense: an n-dimensional manifold is locally modeled on an open subset of An ndimensional orbifold is locally modeled on a quotient U/F where U is an open subset of and F is a finite group acting smoothly on U. To give a precise definition we need a few preliminary notions we need to define charts, atlases and compatibility of atlases. Let U be a connected topological space, U a connected n-dimensional manifold and F a finite group acting smoothly on U. An n-dimensional uniformizing chart'2 on U is a triple (U, F, where U U is a continuous map inducing a homeomorphism between U/F and U (thus, in particular, is constant on the orbits of F). We will only consider charts where the set of points in U fixed by I' is either all of U or is of codimension 2 or greater. Note that we do not require that F acts effectively. Two uniformizing charts (Ui, F1, and (U2, F2, of U are isomorphic if there is a diffeomorphism : U1 (12 and an isomorphism A : F1 r2 such that is A-equivariant (i.e., x) = A(g) all g E Fi, x U1) and o = For example, fix a F. Define U U by = a x. Let A(g) = aga'. Then is an isomorphism. Let t: U' '—p U be a connected open subset of U. We say that a uniformizing chart (U', F', is induced from a uniformizing chart (U, F, on U if there is a monomorphism A : F' F and a A-equivariant embedding U' U such that to = o In this case A) : (U', F', (U, F, w) is called an injection. An orbifold atlas on a Hausdorif topological space M is an open cover U of M satisfying the following conditions: :
1. Each element U of U is uniformized, say by (U, F,
2. If U, U' e U and U' C U then there is an injection (U', F', 3. For any point p
with p E U3 c
(U, r,
U1 flU2, U1, U1
n
U, there is a connected open set U3 E U (and hence there are injections (U3, I'3,
and
—+
(U2, F2, cpa)).
Suppose that V and U are two orbifold atlases on a space M, that V is a refinement of U and that for every V E V and U U with V c U we have an injection
'—i
where
and
are
the respective
unifonnizing charts. We then say that V and U are directly equivalent orbifold atlases. Now take the smallest equivalence relation on the orbifold atlases on M 12Ruan calls it a uniformizing Rysiem.
111.4.
215
Proof of Theorem 1.38, part three
so that any two directly equivalent atlases on M are equivalent. We now define an orbifold to be a Hausdorif topological space together with an equivalence class of orbifold atlases. Let x be a point in an orbifold M, and let (U, F, be a uniformizing chart with x E ço(U). The (orbifold) structure group of x is the isotropy group of a point '(x) then x2 E '(x). It is well-defined as an abstract group: if in the fiber the corresponding isotropy groups are conjugate in F. Remark 111.39. It is not hard to show that if (U, F, is a uniformizing chart of U, then for any point x E U there is a neighborhood U' and a uniformizing chart is a single point i (and such that induced from as a chart centered at x. We will refer to (U', F', hence F' fixes
Let M be an orbifold with an atlas
smooth function / on M
is a collection of smooth ['1-invariant functions on U2 such that for any injection we have = f3. Naturally each :
JR. Thanks to the compatibility = defines a continuous map f, : conditions above, these maps glue together to define a continuous map from the topological space underlining M to JR. By abuse of notation we may write f M —' JR. Similarly, a differential k-form on the orbifold M is a collection of
: (U), I's, such that for any injection F2-invariant k-forms on <—p = &,. We denote the collection of all differential forms we have (Ui, ['i, on M by (M). Note that (M) has a well-defined exterior multiplication (since exterior multiplication behaves well under pull-backs) and exterior differentiation
d:
(for the same reason). In the same manner one defines
vector fields, Riemannian metrics, quadratic forms and other differential geometric objects on orbifolds. Finally we define maps of orbifolds. The reader should be aware that there are several notions of maps of orbifolds. For example Satake in his two papers gave two inequivalent definitions. The one we give below is the simplest; it is not the best. It will, however, suffice for our purposes. See Ruan's survey (RJ for a more modern point of view. Let (M, F2, }) and (N, { V,, }) be two orbifolds and let F: M N be a continuous map of underlying topological spaces. The map F is a (smooth) map of orbifolds if for every point x E M there are charts with x E and a C°° map U1 —+ C :
such
that = Foço2.
The reason for the appearance of orbifolds in these notes is the symplectic reduction theorem of Marsden, Weinstein and Meyer [MW, Me]: Theorem 111.40. Let (M, w) be a symplectic manifold with a Hamiltonian action 9* denote a corresponding moment map. Suppose of a Lie group G. Let $ : M 9* 77 E is a regular value of 4' and suppose that the action of the isotropy group Gq of on is proper. Then the action of C,7 on 4''(i) is locally free and the quotient M,, := '(77)/C,1 is naturally a symplectic orbifold.
III. Proof of Theorem 1.38
216
We will not discuss the proof of this well known theorem. Instead let me
briefly explain why quotients by locally free actions are orbifolds. The reason is the slice theorem: if the action of a Lie group G is locally free and proper on a manifold Z, then for any point z E Z there is a slice S2 for the action of G and the isotropy group G2 is finite. The quotient S2 /G2 is homeomorphic to a neighborhood of the ZIG) is Sz/Gz orbit C z in the orbit space ZIG. The triple (Se, G2, w: S2 a uniformizing chart of the orbifold ZIG. Let us now prove Theorem 111.38. Proof of Theorem 111.38. Since B is compact, the image
is compact. There-
X') is strictly posfore the set of vectors X' E g, such that the function itive on B, is open. Hence we may assume that X lies in the integral lattice ker(exp g —' C) of the torus C. Let H = {exptX t e R} be the :
I
corresponding circle subgroup of C.
Let f(x) =
X)) and let a' = Ia. The form a' is another C-
invariant contact form with ker a' = The moment map = 1 for all x B. Therefore =
defined by a' satisfies
Since the function (Wa, X) is nowhere zero, the action of H on B is lo-
cally free. Consequently the induced action of H on the symplectization (N, w) = is locally free as well. Hence any a E R is a regular value (B x IR, X) of the moment map for the action of G on the of the X-component = The manifold B x {O} Note that symplectization X). Therefore by the symplectic reduction theorem is the —1 level set of B/H is a (compact connected) symplectic orbifold. M := ((4',X))—'(—1)/H The action of C on ((f, X))' (—1) descends to an effective Hamiltonian action of C/H on M. A dimension count shows that the effective action of C/H on M is 0 completely integrable, i.e., M is a symplectic toric orbifold.
The rest of the section is a proof that real odd dimensional cohomology vanishes for compact symplectic toric orbifolds. It uses Morse theory.
111.4.1
Morse theory on orbifolds
Let us start by briefly reviewing the fundamental results of Morse theory on manifolds. Let f: M R be a smooth function. A critical point of f is a point p where
the differential df is 0. The image 1(p) of a critical point p is a critical value of is a nondegenerate f. A critical point p of f is nondegenerate if the Hessian quadratic form (in local coordinates is the matrix of second order partials (x) = xix,). The index of a nondegenerate critical (a.2°2L.); thus point p is the number of negative eigenvalues of the Hessian (counted with multiplicities.) The two and a half fundamental results of Morse theory are the two theorems and the lemma below.
111.4.
217
Proof of Theorem 1.38, part three
Theorem 111.41. Let f be a smooth function on a manifold M, and Ma the set (—oo, a]). If f1([a, b]) is compact and contains no critical points then M0
f'( is homotopy equivalent to Mb.
Sketch of proof. Fix a Riemannian metric on M. Since f has no critical points in f1([a,b]), the unit vector field X = —Vf/[jVf is well-defined. Extend X to all 0 of M. The flow of X gives a retraction of M& onto Ma. Theorem 111.42. Let I be a smooth function on a manifold M. Suppose f'([a, b]) is compact and contains exactly one nondegenerate critical point p in its interior. Then Mb has the homotopy type of Ma with a A-dimensional disk DA attached where A is the index of p. along the boundary ÔDA = We omit the proof of the theorem which is well known noting only that the key ingredient of the proof is Lemma 111.43 (Morse Lemma). Let p be a nondegenerate critical point of index A of a function f M —i IR. There is a neighborhood U of p in M and an open for all x E U. There is, embedding U —p M such that f o = f(p) + there is a change of coordinates near p so that in new coordinates is a quadratic form (up to a constant). Moreover, if a compact Lie group G acts on M fixing p and preserving f, we may arrange for U to be G-invariant and for to be G-equivarzant. Note that in this case the Hessian is a G-invariant quadratic form. Suppose now that f is a smooth function on an orbifold M. Then the 1-form df still makes sense. We define p to be a critical point of f if df9 = 0. A critical point p is nondegenerate if for any uniformizing chart (U,!', with p E U = a point E '(p) is a nondegenerate critical point of I = o (p. The index of p is the index of the F-invariant quadratic form It is not hard to believe that Theorem 111.41 holds for orbifolds with no changes and that essentially the same proof still works. Theorem 111.42 requires a small modification, see [LT].
Theorem 111.44. Let I be a smooth function on an orbifold M. Suppose f' ([a, b]) is compact and contains exactly one nondegenerate critical point p in its interior. Then Mb has the homotopy type of Ma with the quotient attached along the boundary (ôD' )/t' = /1' where A is the index of p and F is structure group of p. Here again and denote the disk of dimension A and the sphere of dimension A —
1
respectively.
Corollary 111.45. Letf: M R, p, A and F be as above. Then Hence, if 1' is orientation preserving, for q = A and 0 otherwise.
Proof. By excision H*(Mb,Ma;R) = preserving, then otherwise. Here
=
denotes the reduced cohomology.
=
R
= =R
If F is orientation for q = A and 0
0
III. Proof of Theorem 1.38
218
As a consequence of the corollary above we get
Corollary 111.46. Let f be a smooth function on a compact orbifold M. Suppose IR) = 0 for all odd indices q. all indices off are even. Then
Proof. The proof is inductive. Suppose that c is a critical value of I and suppose R) = 0 for q odd. Assume for simplicity we know that for all a < c we have (c) contains only one critical point (this keeps the notation more managethat able) and that its index is 2k for some integer k. Then using the long exact sequence (for some sufficiently small e > 0) and Corollary 111.45 for the pair = for q 2k — 1,2k, that H2k_l(Mc+E,R) we see that = The = 0 and that embeds in 0 result follows.
M —* g*) is a compact symplectic toric orbifold. Just is a simple polytope and the as for symplectic toric manifolds the image Now suppose (M, w,
in one-to-one fashion to vertices of 4'(M)
moment map sends the fixed points
X) takes (see [LT]). Therefore, for a generic vector X E g the function f = distinct values at fixed points. We will now argue that the critical points of f are exactly the fixed points. We will then argue that f is Morse and that all indices of f are even. For an action of a torus G on a compact orbifold M only finitely many isotropy groups can occur. This is a consequence of compactness and existence of slices. Therefore the set of subalgebras which are Lie algebras of isotropy groups for the action of G on M, is finite. Hence the set
U=9N xE/vt
is
open and dense. Now take X to be a vector in U. Then 0=
=
=
=0 exp tX . x = x for all t exp tX E for all t X
Since X e U we see that =0 x is fixed by C. It remains to check =g that f is Morse and that all the indices off are even. Let x E M' be a fixed point and let (U, r, be a uniformizing chart centered at x (cf. Remark 111.39). We may assume that U = cp(U) is G-invariant. Denote the symplectic form on U by Ji. The map = a U g is a moment map for an action of a torus G on U. One can show arguing as in the proof of Lemma 111.6 that r c C and that C = C/I'. In particular O has g as its Lie algebra. We now apply the equivariant Darboux theorem to U, and C. We get a C invariant neighborhood V of = (x), an open neighborhood V0 of 0 in
111.4.
219
Proof of Theorem 1.38, part three
where = T1U and a G-equivariant diffeomorphism r: V0 —* V such that U. on the vector space &o is the constant coefficient form is linear. Hence the corresponding moment map The action of C on is quadratic. In fact by choosing a C-invariant complex strucwith compatible with the symplectic form we can identify ture on A d;) so that the action of C is given by (Ca,
a• for some characters
(zj,.
..
: C —i
= (Xi(a)zl,. ..
S'. Then
= are the corresponding weights. Since r is a G-equivariant symwhere v3 = Hence plectomorphism, o r = + c for some constant c E = (4',X)or = (Fo,X)+c, i.e., =
+C.
0 for any j. Thus f o is Morse and its index at is twice the number of weights with v3 (X) <0. Therefore, by definition, f is a Morse function on M and all indices of f are even. Combining the above discussion with Corollary 111.46 we get By the choice of X, v2(X)
Proposition 111.47. Let M be a compact symplectic toric orbifold. Then for any odd index q the cohomology group HQ(M, R) is zero.
Appendix
Hypersurfaces of contact type Contact manifolds often arise as codimension 1 submanifolds of symplectic manifolds, i.e., as hypersurfaces. Definition 111.48. Let (M, w) be a symplectic manifold. A hypersurface E of M is of contact type if there is a neighborhood U of in M and a vector field X on U such that
IRX(m) for any point m E E, i.e., the vector field X is nowhere tangent to E;
1. TmM = TmE
2. the flow of X expands the symplectic form w exponentially, i.e., denotes the Lie derivative with respect to X. where as usual The vector field X with above properties is often called a Liouville vector field.
We now prove that hypersurfaces of contact type are indeed contact rnani. folds.
Proposition 111.49. Let E be a hypersurface of contact type in a symplectic manifold (M, w) and let X be a Liouville vector field defined on a neighborhood U of The 1-form = (1(X)w)JE is contact.
Note first that d(t(X)w) =
+ = is symplectic and E is of codimension I in M, the form has a 1-dimensional kernel (for any point m E E). Thus there is a vector PT,,, is symplectic and Yrn E TmE such that wm(Ym, v) = 0 for any v E TmE. Since since by assumption TmM = TmE RX(m) for all m E E we have Proof.
w
0
c4m(Ym,X(m)) =
= üm(Ym).
0 for all m E E and consequently = ker a is a codimension 1 Hence am distribution on E. It remains to show that for any m E
is non-degenerate. On the other hand dam = and = {v E TmE wm(v,X(m)) = 0}. Since Wm(V,Ym) = 0 for any v E the subspace of TmM lies in the symplectic perpendicular to the 2-plane sPanR{Ym, X(m)} in
(TmM, &m):
em C Since = dimM —2 = we have equality: Since (spana{Ym, X(m)}) is a symplectic subspace, em = (spanR{Ym, its sympleetic perpendicular is symplectic as well. C
Hypersurfaces of contact type
222
Example 111.50. Let (Q, g) be a Riemannian manifold. Let g* denote the dual metric on T*Q. The co-sphere bundle S*Q is the set of covectors in T*Q of length 1: 5*Q {(q,p) E T*Q q E Q, p E TQ, = 1}. It is a hypersurface of contact type in T*Q relative to the standard symplectic structure. The Liouville vector field is the generator of dilations. In local coordinates X (q, p) = P2
is star-shaped about the Exercise 111.51. A codimension 1 hypersurface E C the ray {tv t E (0, oo)} intersects E origin if for any nonzero vector v E transversely in exactly one point. In particular E is the image of an embedding I
t:
of the (2n — 1)-dimensional sphere.
Show that any star-shaped hypersurface is a hypersurface of contact type w= in Show that any two star-shaped hypersurfaces are dzj A isomorphic as contact manifolds. The contact structure in question is called the standard contact structure on Prove a converse: given a contact form on defining the standard —+ such that z*(t(X)w) = contact structure there is an embedding i: where X is the radial vector field on X(z) = (zj& +
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I Michêle Audin, Ana Cannas da Silva and Eugene Lerman
I
Symplectic Geometry of Integrable Systems
I
Among all the Hamiltonian systems, the integrable ones — those which have many conserved quantities — have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrabie system is a result of the fact that the system is invariant under a (semiglobal) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus act ton. This leads to symplectic toric manifolds (part B of this book), which are examples of extremely symmetric Hamiltonian systems. Physics makes a surprising come-back in part A: to describe mirror symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (part C of this book). Along the way, tools from many different areas of mathematics are brought to bear on the questions at hand, in particular, actions of Lie groups in symplectic and cohomology, contact manifolds, the Deizant theorem, Morse theory, sheaves and and aspects of Calabi-Yau manifolds. This book contains an expanded version of the lectures delivered by the authors at the CRM Barcelona in July 2001. It serves as an introduction to symplectic and contact geometry for graduate students and will be useful to research mathematicians interested in integrable systems. J
I
i
ISBN 3-7643-2167-9
I
1
1
II
Pr
