Monographie N° 32 de L'Enseignement Mathematique Serie des Conferences de !'Union Mathematique Internationale, N° 7
INTRODUCTION TO MICROLOCAL ANALYSIS M. KASHIWARA
Imprimerie SRO - KUNDIG Geneve, 1986
INTRODUCTION TO MICROLOCAL ANALYSIS
L'ENSEIGNEMENT MATHEMATIQUE
I-\IVFRSITF DF C',F\FV'F
© 1986 by L'Enseignement Mathematique Tous droits de traduction et reproduction reserves pour tous pays
INTRODUCTION TO MICROLOCAL ANALYSIS by Masaki KASHIWARA
CONTENTS
0.
Introduction
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Systems of differential equations
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Micro-differential operators
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3.
The algebraic properties of 6
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4.
Variants of
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The vanishing cycle sheaf
6.
Micro-differential operators and the symplectic structure of the cotangent bundle . . . .
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Quantized contact transformations
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Functorial properties of micro-differential modules
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Regularity conditions
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Structure of regular holonomic-modules
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Application to the b-function
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References
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INTRODUCTION TO MICROLOCAL ANALYSIS') by Masaki KASHIWARA
§ 0.
INTRODUCTION
0.1. In this lecture, we explain the micro-local point of view (i.e. the consideration on the cotangent bundle) for the study of systems of linear
differential equations. 0.2.
The importance of the cotangent bundle in analysis has been recognized
for a long time, although implicitly, for example by the following consideration. We consider a linear differential operator
P(x, 0) = Y aa(x)r7a
with
as = (r7/8xi)a' ...
a solution to P(x, 8)u(x) = 0. If we suppose that u(x) has a singularity along a hypersurface f (x) = 0, then the
for a = (ai , ...,
simplest possible form of u(x) is ci(x)f(x)s+' +
u(x) = co(x)J(x)s +
Then setting P,(x, (0.1.1)
...
we have
P(x, r7)u(x) = s(s -1) ... (s - m + 1)co(x)Pm(x, d f) f (x)s m + ... + (s +J) ... (s +j -- m + 1)Cj(x)Pm(x, d f)
+ (terms in co, ..., c;_i)f(x)s-' + ...
Therefore Pm(x, d f) must be a multiple of j (x)
(i.e.
Pm(x, d f) = 0 on
f - '(0)). In this case, f - '(0) is called characteristic.
Thus the hypersurface f -'(0) is not arbitrary and the singularity of the solution to Pu(x) = 0 has a very special form. ') Survey lectures given at the University of Bern in June 1984 under the sponsorship of the International Mathematical Union.
6
0.3.
If Pm(x, E) 0 0 for any non-zero real vector E, then P is called an
elliptic operator. In this case, one can easily solve P(x, 8)u(x) = f (x) for any f (x), at least locally. We start from the plane wave decomposition of the 8-function. S(x) = const.
(0.3.1)
f
w(E)
<x, i;>"
where is the invariant volume element of the sphere Sn-1. By formula (0.1.1), we can solve
P(x r7)K(x v) =
by setting K(x, y) _
1
cj (< x, 4> -
)"and determining cj recur-
sively. Then K(x, y) = const
f K(x, y,
satisfies
P(x, r7)K(x, y) = S(x - y) by (0.3.1).
If we set u(x)
K(x, y) f (y)dy then u(x) satisfies P(x, r7)u(x) = f (x).
In fact
P(x, 8)u(x) =
f P(x, r7)K(x, y) f (y)dy = f S(x - y) f (y)dy = f (x)
.
By these considerations, M. Sato recognized explicitly the importance of the cotangent bundle by introducing the singular spectrum of functions and microfunctions [Sato]. For a real analytic manifold M, let 4, be the sheaf of real analytic functions and 9M the sheaf of hyperfunctions. Let n: T*M -> M be the cotangent bundle of M. Then he constructed the sheaf 0.4.
W,, of microfunctions and an exact sequence
0"4_" , MM The action of a differential operator P(x, a) on
0. -4M
on (em. Moreover P: `P'M -> `'M is an isomorphism outside
{(x, ) e T*M; Pm(x, E) = 0} .
extends to the action
7
In the situation of § 0.2, u(x) = co(x) f (x)s + ... satisfies supp sp(u(.x)) ± d f (x)}. Therefore Pm(x, d f) must be zero if P(x, o)u(x) = 0. In fact otherwise the bijectivity of P: 'M - lir,,M implies sp(u) = 0. 0.5.
Such a method of studying functions or differential equations locally on the cotangent bundle is called microlocal analysis. After Sato's discovery of microfunctions, microlocal analysis was studied intensively in Sato-KawaiKashiwara [SKK]. 0.6.
Also L. Hormander [H] worked in the C'-case. Since then, microlocal
analysis has been one of the most fundamental tools in the theory of linear partial differential equations.
§ 1.
SYSTEMS OF DIFFERENTIAL EQUATIONS (See [0], [Bj])
1.1. Let X be a complex manifold. A system of linear differential equations can be written in the form No ( 1 . 1. 1 )
Z P,; (x, a)uj = 0
,
i = 1. 2, ..., Ni
J=1
Here u,, ..., uNo denote unknown functions and the P,, (x, a) are differential operators on X. The holomorphic function solutions of (1.1.1) are simply the kernel of the homomorphism (1.1.2)
P: C,Xo
X'
_
which assigns (v,, ..., vN1) to (U,, ..., UNo ), where v, _
Pij(x, a)u1.
Let us denote by 9x the ring of differential operators with holomorphic coefficients. Then (1.1.3)
P:1/X
gNo
..., Y-Q;P;NO) is a left 9x-linear homomorphism. If we denote by .i# the cokernel of (1.1.3), then . becomes a left 9, module and 3fom,X (./Z, (Ox) is the kernel of (1.1.2). This means
given by (Q L , ..., QN,) to (EQ,P; i ,
that the set of holomorphic solutions to Pu = 0 depends only on
. /7.
For this reason we mean by a system of linear differential equations a left -/X-module.
Let us take a local coordinate system (x1 , ..., x") of X. Then any
1.2.
differential operator P can be written in the form P(x, a) _ Y aa(x)8a
(1.2.1)
aeN^
where as = r7la1/axi ... 3x'^, I a I = al + ... + a" and the aa(x) are holomorphic functions. For j e N, we set
Y
Pi(x,
kal -l
and we call {Pj(x, E)} the total symbol of P. The
where
largest m such that P.: 0 is called the order of P and P. is called the principal symbol of P and denoted by 6(P). Let us denote by T*X the cotangent bundle of X, and let (x1....' xn; S1, e Sn)
be the associated coordinates of T*X. It is a classical result that if we consider 6(P) as a function on T*X, then this does not depend on our choice of the local coordinate system (x1 , ..., x").
Let M be a real analytic manifold, and X its complexification, e.g.,
1.3.
M = R" c X = C". Let P be
a differential
operator on X. When
0 for (x, E) E R" x (R"\{0}), P is called an elliptic differential operator. In this case, we have the following result. 6(P) (x, 4)
PROPOSITION 1.3.1.
Pu
and
If
u
is real analytic, then
is a hyperfunction (or distribution) on M u is real analytic. More precisely if we
denote by d the sheaf of real analytic functions on M and by 92b) the sheaf of hyperfunctions (resp. distributions) on M, .9b/sad) is a sheaf isomorphism. P: ../sad -.M/sad (resp. P : -9b/d (resp.
.4 then
This suggests that if 6(P) (x, i;) 0, we can consider the inverse P-1 in a certain sense. Since (x, i;) is a point of the cotangent bundle, P-1 is attached to the cotangent bundle.
In fact, as we shall see in the sequel, we can construct a sheaf of rings
.X on T*X such that IX c n*g,, where it is the canonical projection T*X -> X. Moreover if P E -9X satisfies 6(P) (x, ) 54 0 at a point (x, i;) e T*X, then P-1 exists as a section of &x on a neighborhood of (x, E,).
This can be compared to the analogous phenomena for polynomial rings, as shown in the following table.
9
C[x1, ..., x.] C"
the sheaf Uc, of holomorphic functions
§ 2.
2.1.
MICRO-DIFFERENTIAL OPERATORS (See [SKK], [Bj], [S], [K2])
Let X be an n-dimensional complex manifold and let itx: T*X - X
be the cotangent bundle of X. Let us take a local coordinate system of X and the associated coordinates (x, .., 1, ..., .) of T X For a differential operator P, let {Pj(x, E)} be the total symbol of P as in § 1.2. We sometimes write P = EP;(x, a). Let Q = EQ;(x, a) be another differential operator. Set S = P + Q and (x 1 , ...,
.
R = PQ. Then the total symbols {S; } and {Rj } of R and S are given explicitly by
Si = P; + Q;
(2.1.1)
1
(2.1.2)
R R,
-(07j) (aXQk) i
l=j+k-Ian a aEN^
and ax = (a/axi)a' ... (a/ax.)a^ The total symbol {Pj(x, E)} of a differential operator behaves as follows under coordinate transformations. Let (x1 , ..., and (z 1 , ..., be two local and (fir, ..., .) be related by coordinate systems. Let where a
.
KK
Sk =
ax's
KK
i
Sj
axk
and (x"1, ..., x".; Z 1 , ..., are the associated local 1, coordinate systems of the cotangent bundle T*X. Let P be a differential operator on X and let {Pj(x, E)} and {F;(z, i)} be the total symbols of P with respect to the local coordinate systems (x1, ..., and (zl, ..., i.e. (x1 , ..., x.;
respectively. Then one has (2.1.3)
Pl (x,
v!ai!...a,! <, ax'x>
)
j(x,
)
- 10 Here the indices run over j c Z, v c N, al , ..., a, e N" such that
l+v-jaoc,1. For
lcc
denotes Y 700:z. 2.2.
in
The total symbol {P;(x, t;)} of a differential operator is a polynomial i,. We shall define microdifferential operators by admitting P; to be
holomorphic in t;. For ? E C, let L,*X(A.) be the sheaf of homogeneous holomorphic functions of degree k on T*X, i.e., holomorphic functions f(x, t,) satisfying
(Y ;alai; - X)f(x, ,) = 0 . Definition 2.2.1.
For X E C we define the sheaf 1' (k) on T*X by
P_ ; E F(Q; (9r*X(?-1)) ));EN ; and satisfies the following conditions (2.2.1)}
0 H {(PA_;(x,
(2.2.1)
for any compact subset K of 0, there exists a CK > 0 such that sup I P,,_; I < CK'(j!)
for all
j>0
K
Remark. The growth condition (2.2.1) can be explained as follows. For a differential operator P = EP,(x, 0), we have
P(x, a) (< x, r; > + p)µ = EP; (x, )
r(µ(µ)
(<X, > + p)µ
For P = (P,,_;(x, E)) E '(?,) we set, by analogy
P(<x, > +p)N =
P,,-;(x, )
F(µ)
r(µ-k+j+ 1)
(<x, >
+p)µ-"+;
.
Then the growth condition (2.2.1) is simply the condition that the right hand side converges when 0 < I < x, 4 > + p I
<< 1.
Now, we have the following PROPOSITION 2.2.2 ([SKK], Chap. II, § 1, [Bj] Chap. IV, § 1).
contains 'X(k-m) as a subsheaf for m c N.
(0)
olX(A.)
(1)
Patching by rule (2.1.3) under coordinate transformations, 1X(X) becomes
a sheaf defined globally on T*X.
T*X.
(2)
By rule (2.1.1), S,(X) is a sheaf of 'C-vector space on
(3)
By rule (2.1.2), we can define the "product" homomorphism.
.ff A) 0 c
XQ,+ µ),
O.i)
which satisfies the associative law. (4)
'x(0)
In particular,
60x = v f x(m)
and
become sheaves of (non
mEz
commutative) rings on T*X, with a unit. The unit is given by (P3(x, E)) with Pj = 1 for
for j
j=0
and
Pi = 0
0.
We define the homomorphism o
: -ffx(X)
()T*x(A.)
(PI_J)PA.
by
Then, o is a well-defined homomorphism on T* X (i.e. compatible with coordinate transformation) and we have an exact sequence 0
A-1) ' 'ffx(X)
6T*x(A.)
0.
Now we have the following proposition, which says that the ring
ffx
is a kind of localization of -9x . PROPOSITION 2.2.3. (1) (2)
For P E 1(X) and Q E'(µ), we have 6a+µ(PQ) = ([SKK] Chap. II, Thin. 2.1.1) If P e S(X) satisfies ox(P) (q) : 0 at q e T*X, then there exists Q E ff(-X) such that PQ = QP = I.
The relations between 61x and _9x are summarized in the following theorem. THEOREM 2.2.4 ([SKK], Chap. II, § 3). (i)
(ii)
'-9x as a subring and is.fiat over rt fix. 'x I T*X "' 9x, where T *X is the zero section of T*X.
- 61x
contains
I f
Tc
(iii) For a coherent
fix-module
,
the characteristic variety of .K
coincides with the support of 61x ® 71X , X la x
12
§ 3.
THE ALGEBRAIC PROPERTIES OF & (See [SKK], [B1])
In the preceding section, we introduced the notion of micro-differential operators. The ring of of micro-differential operators has nice algebraic properties similar to those of the ring of holomorphic functions. Let us recall some definitions of finiteness properties. 3.1.
Let sad be a sheaf of rings on a topological space S. An sad-module .& is called of finite type (resp. of finite presentation) if for any point x c X there exists a neighborhood U and an exact sequence 0 - . t I u - sa p I u (resp. 0-. u '- sop I u '- sd 9 I U).
Definition 3.1.1. (1)
I
(2)
(3)
is called pseudo-coherent, if any submodule of finite type defined on an open subset is of finite presentation. If .t is pseudo-coherent and of finite type, then .t is called coherent. .t is called Noetherian if . # satisfies the following properties : .
(a)
. h' is coherent.
(b)
For any x e X, M. is a Noetherian .W module (i.e. any increasing sequence of sax submodules is stationary).
(c)
For any open subset U, any increasing sequence of coherent (sad 1 u)-submodules of .
I u is locally stationary.
As for the sheaf of holomorphic functions, we have
THEOREM 3.1.1 ([SKK] Chap. II, Thm. 3.4.1, Prop. 3.2.7). Let denote the complement of the zero section in
T*X
T*X.
(1)
9x and ofx(0) are Noetherian rings on T*X.
(2)
ffx is flat over n-'-9x.
(3)
61x(X) I i *x
(4)
For p c T*X, ofx(0)p is a local ring with the residual field C.
(5)
A coherent fix-module is pseudo-coherent over ffx(0).
is a Noetherian fx(0) I T*x-module.
§ 4.
VARIANTS OF f (See [SKK], [Bj], [S])
We have defined the sheaf of rings S. However we can introduce other sheaves of rings, similar to ff, which makes the theory transparent. 4.1.
13
4.2.
The sheaf I = lim /&(-m) is called the sheaf of formal micromeN
differential operators. This is nothing but the sheaf similar to &, obtained by dropping the growth condition (2.2.1).
We can define the sheaf d" of micro-differential operators of infinite order ([SKK]). For an open S2 c C", we set
4.3.
F(2; 'M = {(Pj)jez ; Pi C_ '-((2; OTIM) satisfying the following conditions (4.3.1) and (4.3.2)}. (4.3.1)
For any compact set K c 0, there sup Ip,I < CK'(-j)! forj < 0.
is
a CK > 0 such that
K
(4.3.2)
For any compact set K c Q and any c > 0, there
exists a
CK, > 0 such that
sKPP;I 4.4.
CK.Eji
for
j > 1.
We can also define the sheaf SR on T*X
z
(See
[KS] Chap. II, [SKK]). Here n = dim X, Cz x",. is the sheaf of holomorphic
forms on X x X which are n-forms with respect to the second variable, and µe is the micro-localization with respect to the diagonal set of X x X (See [SKK] Chap. II for the details).
We have ffx c 61X' c &X, 'x c (?X- Moreover, 61X , 6X and 1x are faithfully flat over &x. The sheaf ix is Noetherian. The sheaf olX contains 4.5.
&x(X)'s compatible with the multiplication.
If we denote by y the projection map T*X -> T*X/C*, then for j 0 and 1" = y-ly*&R 4.6.
4.7.
In [SKK], 1', 1, and 9°° are denoted by .if,
0
and Y.
To explain the differences between tR and e, we shall take the following example. Let X be a complex manifold and Y a hypersurface of X. 4.8.
We shall take local coordinates (x 1, ..., x") of X such that Y is given by x1 = 0. The 92x-module -9x/-9xx1 + 12xa; is denoted by Z4YIx. Set j> 1
14
WYIx = e® ® ° YIx , `eYIx = ?x ®`6'Ylx ex
a-lgx
`By'lx = offX ®`eYlx
and
ceRI
ex
Oez
x=
0 WYIx. ex
Then we have, setting p = (0, dx1), xo = 0 ,
YlX,P = {a + b log x1; a e Cx, xa[l/x1], b e Lx,x0}/Ox,
X0
((1,X,XO[1/x11/OX,XO) ®Lx,XO
'Ylx, p = {a + b log x1; a e 6rx,x0[1/x1], b c CX1Y,xO1/vx,xO ((9x, 0[1/xl]/(OX,xo) ®Ox1Y,xO
Here Gxll. = lim (P/x''& x is the sheaf of formal power series in the x 1-direction. WYJX.
P = {a + b log x1; a e (j*1
b c- (9X, XO}/(9x,
1Gx)x0,
xO
where j is the open embedding X\Y c, X. `IX,P = lim O(U)/&X xo
.
U
Here U ranges over the set of open subsets of the form
{xeX;xI
THEOREM 4.8.1 ([KK] Thm. 5.2.1). Let be a holonomic ax-module. Then there exists a (unique) regular holonomic ax-module Mreg such that .
& - ®./ PL = 0- - (3 J!'C feg . ex 9X
THEOREM 4.8.2 ([SKK] Chap. II, Thm. 5.3.1). Let X and Y be complex manifolds and let T Y Y be the zero section of T*Y. If . & is an ffx Y-module whose support is contained in T *X x T *Y, Y, then there exists a (locally) coherent fix-module P such that
9f.Y ® Ji = 9f. Y ® (2®(9Y) eX%Y
g+X.Y
Here ® denotes the exterior tensor product. (See § 8).
15 ---
§ 5.
THE VANISHING CYCLE SHEAF
Let M be a real manifold and f : M -+ R a continuous map. For a sheaf F on M, f_,(0) is called the (j-th) vanishing cycle sheaf of F. Here R+ = It E R; t 0}. This measures how the cohomology groups 5.1.
of F change across the fibers of f. Its algebro-geometric version is studied by Grothendieck-Deligne ([D]). 5.2.
Let (X, LX) be a complex manifold. Let f : X --* R be a C°`-map and Let s be the section
consider the vanishing cycle sheaf .''f-1(R+)(&X) I of f -1(0) -> T*X given by d f Then we have .
PROPOSITION 5.2.1 ([KS1] § 3, [K2] § 4.2). f-1(R+)(C(X) f- ,(C has a structure of an sLet P be a differential operator. If 6(P) does not vanish on s(f -1(0)), then P has an inverse in s- 1,X by Proposition 2.2.3. Therefore we obtain 19X-module.
COROLLARY 5.2.2.
P:
If
6(P) sf-,(o)
6,
then
f-1R+((9X) f-l(0)
f-1(R+)(C(:X)
I f-1(0i
is bijective. 5.3.
More generally, let .# be a coherent -9X-module, and set
F' = RComl,X(
, 61x) .
Then the preceding corollary shows that RF f- IR+(,5 ) I f-,(o) = 0
if
s(f -1(0)) n Ch(Jf) = G
.
Here Ch. # denotes the characteristic variety of .#. 5.4. To consider vanishing cycle sheaves is very near to the "microlocal" consideration. In this direction, see [K-S2].
§ 6. MICRO-DIFFERENTIAL OPERATORS AND THE SYMPLECTIC STRUCTURE ON THE COTANGENT BUNDLE
The ring ffX is a non-commutative ring. This fact gives rise to new phenomena which are not shared by commutative rings such as the ring of 6.1.
- 16
holomorphic functions. They are also closely related to the symplectic structure of the cotangent bundle.
Let us recall the symplectic structure on the cotangent bundle. Let Ox denote the canonical 1-form on the cotangent bundle T*X of a complex manifold. Then dOx gives the symplectic structure on T*X. The Hamiltonian map H: T*(T*X) - T(T*X) is given by 6.2.
(6.1.1)
= and
it e T*(T*X)
for
v c- T(T*X).
For a function f on T *X, H(df) is denoted by H f and the Poisson bracket If, g} is defined as H f(g). If we denote by the Euler vector field (i.e. the infinitesimal action of C* on T*X), then we have .
.
'
' = H(-OX) .
With a local coordinate system (xl,..., coordinate system (x1, ..., x,,; E1....,
of X and the associated local
of T*X, we have
ox = 1i;,dx; , dOX =
H : dE, i--. a/ax;, dxi i-4 -
_ {f, g} _
a
f
ag
ax;
-
ag
afJ ax;
This structure is deeply related to the ring of micro-differential operators. The first relation between them appears in the following 6.3.
PROPOSITION 6.3.1.
For P E 1(?) and Q E &(µ), set
[P, Q] = PQ - QP E 01(k + µ -1) . Then
6a+p-1([P, Q]) = {6a,(P), 6,,(Q)}
.
17 -
An analytic subset V of T*X is called involutive if f I v = g I v= 0 implies If, g} I v = 0. The following theorem exhibits a phenomenon which has no analogue in the commutative case. THEOREM 6.3.2 ([G]). Let .# be a coherent 61X-module defined on an open subset 0 of T*X and let . be a 6'X(0) I n-module which is a union of coherent 60X(0)-modules. Then V = {p E S2; 2 is not coherent over .ffX(0) on any neighborhood of p} is an involutive analytic subset of Q.
COROLLARY 6.3.3 ([SKK] Chap. II, Theorem 5.3.2, [M]). coherent SX-module .iH, Supp . is involutive.
For any
Since any involutive subset has codimension less than or equal to dim X, we have COROLLARY 6.3.4.
The support of a coherent ff X-module has codimension
dim X.
After some algebraic calculation, this implies THEOREM 6.3.5 ([SKK] Chap. II, Theorem 5.3.5). For any point p E T*X, gx, v has a global cohomological dimension dim X.
An analytic subset A of T*X is called Lagrangean if A is involutive and dim A = dim X. A coherent gx-module is called holonomic if its support is Lagrangean. 6.4.
§ 7.
7.1.
QUANTIZED CONTACT TRANSFORMATIONS
In the previous section, we saw that the symplectic structure of
T*X is closely related to micro-differential operators via the relation of commutator and Poisson bracket. In this section, we shall explain another relation. Definition 7.2.1. Let X and Y be complex manifolds of the same dimension. A morphism cp from an open subset U of T*X to T*Y is
called a homogeneous symplectic transformation if cp*0y. = 0X.
- 18
We can easily see the following
If cp is a homogeneous symplectic transformation, then cp is a local isomorphism and is compatible with the action of C*. (7.2.1)
(7.2.2)
Assume Y = C" and let (yl , ..., yn; m ...., rl.) be the coordinates of
T* Y, so that OY = YTljdyj.
Set pj = rlj o cp and qj = yj o cp. Then we have (7.2.3.1)
{pj, pk} = {qj, qk} = 0, {pj, qk} = Sj.k for j, k = 1, ..., n.
pj is homogeneous of degree 1 and qj is homogeneous of degree 0 with respect to the fiber coordinates. (7.2.3.2)
Conversely assume that functions {q1, ..., qn, pl, ..., pn } on U c T*X satisfy (7.2.3.1) and (7.2.3.2). Then the map cp: U - T*Y, given by
(7.2.4)
Uaxr-+(q1(x),...,gn(x);p1(x),...,p.(x))eT*Y, is a homogeneous symplectic transformation. We call (q1, ..., qn; P1, ..., Pn) a homogeneous symplectic coordinate system.
Let let (a)
THEOREM 7.2.2 ([SKK] Chap. II § 3.2, [K2] § 2.4, [Bj] Chap. 4 § 6). cp: T*X D U -. T* Y be a homogeneous symplectic transformation,
px be a point of U and set p, = cp(px). Then we have There exists an open neighborhood
isomorphism D : cp transformation). (b)
1
U'
'Y I u, -' 'x I u- (we call
px
and a C-algebra
((p, 1)
a quantized contact
of
If I : cp-1.ff y - xI u is a C-algebra homomorphism then for any m, D
gives an isomorphism
cp-'5Y(m) 4 -1x(m) I u.
Moreover the
following diagram commutes:
(c)
Let D and
D'
be two C-algebra homomorphisms cp-141Y -' .1x I u
- 19
Then there exist k E C, a neighborhood U' of pX and P E F'(U; 1X(X)) such that o (P) is invertible and Pcb(Q)P-1
(Q) =
Q E w-lgr I
.
is unique and P is unique up to constant multiple. Let Y = C" and let U be an open subset of T*X.
Moreover (d)
for
A.
If Pj E F'(U; ,ff,e(1)) and Q j E ['(U; .1x(0)) (1 <j 5 n) satisfy
[Pj, Pk] = [Qj, Qk] = 0
(7.2.5)
[Pd, Qk] = Sjk
then there exists a unique quantized contact transformation ((p,') such that ap(p) = (6o(Q1) (p), ..., "0(Q") (p), o 1(P1) (p), ..., 61(P") (p))
and (D(yj) = Qj, I(avj) = Pj. We call {Q1i..., Q", P1, ..., P"} 7.3.
quantized canonical coordinates.
We shall give several examples of quantized contact transformations. Example 7.3.1.
If P(a) is a constant coefficient micro-differential operator
of order 1, then (x1 +[P, x1], x2+[P, x2], ..., xn+[P, xn], ax,, ..., axn) gives quantized canonical coordinates.
More generally if P is a micro-differential operator of order 1 and exp tHQ,(P) exists, then exp tP gives a quantized contact transExample 7.3.2.
formation D, by solving the equation dt *,(Q) = [P, ,(Q)] with the initial condition 1,(Q) = Q for t = 0. Example 7.3.3. (Paraboloidal X = C1 +" = {(t, x) E 1. l X C"},
Q = {(t, x; T, E) E T*X; T
transformation
S I E G, let Y Ca
given by
p.
36).
Set
0}, G = Sp(n; C)
= {g E GL(2n; C); `gJg = J}
For g =
[K2]
with
J=
`h9 be the quantized contact transformation
-20ax Haas - Rxa,
X H yaxa ' + Sx a, H a,
tit +2{ a, 2+
ar
+ <`Y(3x, ax> Of 1 + <x, 60x> }
.
Then we have `Y9,T92 = Z9i92.
§ 8.
FUNCTORIAL PROPERTIES OF MICRO-DIFFERENTIAL MODULES
(See [SKK]) 8.1.
External Tensor Product.
Let X and Y be complex manifolds and let p1 and p2 be the projections T*(X x Y) -> T*X and T*(X x Y) -+ T* Y, respectively. Then 9x>Y contains p 1 'fix ®p z 'of y as a subring. For an .1x-module M and an of Y-module .A1, C
we define the 9x x Y-module M ® .N by (pi'm
0
.A ®A' =,1'xxY
(8.1.1)
Then one can easily see
(9 p2'.iY). C
P1 19x ®P2 1ry C
PROPOSITION 8.1.1. (i)
(ii)
8.2.
®.' is an exact functor in .# and in = Supp fl x Supp N'.
If if
is 9x-coherent and 9, x ,-coherent.
.'
is
.N'
and Supp (,#®.N)
gy-coherent, then
.4 ® .'
is
For a complex submanifold Y of a complex manifold X of codimension 1,
the sheaf lim 9xtox((x//'", fix) has a natural structure of -9x-module, which is denoted by .VYlx. Here f is the defining ideal of Y. The homomorphism (9 -> o1xtox((9Y, ft ) -' Qz 0 -4Y,x gives the canonical section ox
c(Y, X) of S2z ®MYlx. If we take local coordinates (x1, ..., ox
that Y is defined by x1 = ... = x, = 0, then we have
of X such
21
X/Y_ -9 Xxj + I _qxaj j>I j-I
YIX =
If we denote by S the canonical generator of the left hand side, then c(Y, X) corresponds to dx1 A ... A dx1 0 S. We set 14YIX
WYIX = e® ®
TC
n-19x
Therefore locally we have
exl I exxj + I
WYlx
j>d
j_< d
Then 16Ylx is a coherent 9X module whose support is T1*.X.
8.3.
For an invertible Ox-module 2', 2 ®of x ® 2'® 6x
has a natural
6x
structure of sheaves of rings, by the composition rule (S®Q®s®-1) = S ® PQ ® S®
(s®P®s®-1)
1
°
for an invertible section s of 2' and P, Q e 9x.
Then the category Mod ('x) of left fix-modules and the category Mod (2' ®9x ®£®-1) of left (. ®9x ®.®-1)-modules are equi6x
6x
6x
6x
valent by the functor
(2'0 x®2'® ')
Mod 6x
6x
6x
Let wx be the canonical sheaf on X, i.e. the sheaf of differential forms with top degree. Let a be the antipodal map of T*X, i.e. the 8.4.
multiplication by - 1. Then we have the anti-ring isomorphism. wX®46X®wX®-1 ^->a of x. (8.4.1) 1
6x
6x
This homomorphism is given by using a local coordinate system (x1....,
as follows. For P = YPj(x, a) e 9x we define P* = YP*(x, a), called the formal adjoint of P ([SKK] Chap. II, Th. 1.5.1), by (8.4.2)
P*(x, -E) = I j='-IMI
1 ICI (
a1
mEN"
This is well-defined and satisfies (8.4.3)
(P*)* = P
a1a1P;(x,
)
22 -
(PQ)* = Q*P*
(8.4.4)
Then the isomorphism (8.4.1) is given by
dx 0 P ® (dx)®' --* P*
(8.4.5)
where dx = dx1 A ... A dx e co,. This is independent of coordinate transformations.
The isomorphism (8.4.1) can be explained as follows. Let Ox be the diagonal set of X x X, and let p3 be the j-th projection from Te*x(X x X) 8.5.
to T*X for j = 1, 2. Then the p; are isomorphisms and P2 - P 1 1 = a. Let q, be the j-th projection from T*(X x X) to X(j=1, 2). Then c(Ox, X x X)
gives the canonical section of qz lwx ®
'Axlxxx'.
-1 q2 Ox
Since W,xlxxx is a
Pi '.Ox-module, this section gives a homomorphism P1 1
'x -, q2 lwx ® nxIxxx. q2
&x
It turns out that this is an isomorphism and the right multiplication of ( 9 x on Sx corresponds to the (9x-module structure of q z lwx 0 4ex(x x x -1
q2
via q2. Thus we obtain P1 '((,)x (9) Ox (9 Ox
we-1)
!9x
Ox
-' qi lwx ® (Wnxlxxx. ql lOx
This last being isomorphic to p2 19x, we obtain wx®gx®w®-1 _'p1p219x'_- a-ldox Ox
8.6.
Ox
By 8.3 and 8.4, if .# is a left 'x,,-module for an open set U of
T*X, then wx 0 a-1.# is a right (.6x,av)-module. Ox
8.7.
For a left coherent .6x-module . , 9xt gx (.I, tx) is a right coherent Therefore 61xt jex (M, 9x) ® w e -1 is a left 9x-module by § 8.6. 9x-module.
Ox
If .I is holonomic then [i xt gx (M, fix) = 0 for j j4l n = dim X (See [SKK], [Kl]). Set .I* = gxtnx (M, 'x) ®o 1 . Then .I* is also a holonomic Ox
6'x-module.
We call JI* the dual system of M. We have and . is an exact contravariant functor on the category of holonomic ofx-modules. i
- 23 --
Let X and Y be complex manifolds, and let p1: T*(X x Y) - T*X T*Y be the canonical projections. Let p2 denote and P2: T*(X x Y) 8.8.
P2 -a. Let _'t_ be a left 9x .,-module defined on an open subset 0 of T*(X x Y). Then, by § 8.6, wY ®1' has a structure of (p i I&" pa- I&Y)0Y
bi-module. For an gy-module A',
')
= P I *((wY (& &Y
0
P'2- "Al
a-1
P2
Iffy
has a structure of 9x-module. We have the following
Let
THEOREM 8.8.1.
S2,
and
U.
UY
be open subsets of T*(X x Y),
T*X and T* Y, respectively. Let -C be a coherent ,A-
a coherent (S y I uy)-module. Assume
(i)
pi : p i 1 Ux n Supp .7l' n p'- 1 Supp .N' -. Ux
is
)-module and
(9'x
a finite morphism.
Then we have (a) (b)
9 o .J 2
I-Y
0 P2
.# = PI*((wy ® .1') Supp
P2
I P1 Ivx =
0 for j:0.
1 V)
1 ux is a coherent 6x-module.
I rY
= Ux n PI (Supp .1'np i 1 Supp .N).
We denote p1*((WY ® ') (r) Y
8.9.
1-4/-)
09Y
BY
(c)
, Pi
(wi, ®
(g
pa-1.,4) by I
Pa-1
.
'
Y
Let f : X -. Y be a holomorphic map and let A f be the graph of f,
i.e. {(x, f (x)) E X x Y; x e X}, then .%' _ Wo fix x y is a coherent S X x Y-module
whose support is Tef(X x Y). Now let eo be the canonical map X x T*Y Y
T*X and p the projection X x T*Y. Then we have the following Y
diagram
T*X
:-f
X x T*Y
P:
T*Y
Y
(8.9.1)
id
21
II
11
T*X
Tof(X X Y) Pi
a P2
id.
T*Y
-24We set 'x-Y = w® ® 'e,jxXY and consider this as a sheaf on X x T*Y Oy
Y
by the above isomorphism. Then lx-Y is a (w -1 'x, p-'
)-bi-module.
For an 'Y-module .,Y,
J J£' - N = Rw*p-1(ex, ® p-1.N'). p-16°y
We shall denote this by f *-4/' and call it the pull-back of .N'. Then Theorem 8.8.1 reads as follows. Let Ux and UY be open subsets of T*X respectively. Let .N' be a coherent (1,I v)-module. Assume
THEOREM 8.9.1. T* Y, (i)
and
p f 1(Supp .N) n r of 1(Ux) --+ Ux is a finite morphism. Then we have
0 for j
(a) J orpf 'y (.'x-Y, (b)
0.
.11 = w f*(-Ox.Y ® p f 1.N')1 vx is a coherent Sx-module. _1
pf ey (c)
Supp M = w f p f 1 Supp N' n Ux.
8.10.
Similarly let g : Y --. X be a holomorphic map and let A, be the
graph of g, i.e. {(g(y), y) e X x Y; y c Y}. Then we have the isomorphisms
T*X (8.10.1)
Y x T*X
F-
T*Y
t1
II
T*X
T** g (X
<-Pt
We set gx-Y = COY ®(CAgJX x Y
11
x Y)
aP2
id.
T*Y
and regard this as a sheaf on Y x T*X. x
Oy
Then 'x-y is a (p-19x,6) -1.6Y)-bi-module. For an gy-module . 1- we have
I16o9Ixxy°A'
Rp*w-1(gx-Y 0
c7)-1.A,)
CO- 1,Y
We shall denote this by f9 .N'. Then Theorem 8.8.1 applies to this case 9
and we have
- 25 THEOREM 8.10.1. Let Ux and U, T* Y, respectively. Let .N' be a coherent (i)
be open subsets of T*X and ,)-module. Assume
pg: wg '(Supp .N1) n p9 1(Ux) - Ux is a finite morphism. Then we have
(a) (b)
9_o,"a "y(9x_y, wgi-41^) = 0 for j A 0.
= pg*(.0x-r ® wg 1.K)1 ux is a coherent 6x 1 ux-module. Cog
(c)
f"y
P'(69-, Supp .N'n Ux).
Supp 9.
REGULARITY CONDITIONS (See [KK], [K-O])
Let us recall the notion of regular singularity of ordinary differential equations. Let P(x, a) _ Y aj(x)a` be a linear differential operator in one 9.1.
j_< m
variable x. We assume that the a.{x) are holomorphic on a neighborhood of
x = 0. Then we say that the origin 0 is a regular singularity of Pu = 0 if (*)
ordx=oaj{x) > ordx=oam(x) - (m-j) .
Here ordx=o means the order of the zero. In this case, the local structure of the equation is very simple. In fact, the -9x-module -9x/-9xP is a direct sum of copies of the following modules : Ox = _9xl_9xa, -4[o)Ix = -9x/-9xx, _9xl_9x(xa-k)m+1 (meN), _9xl_9xa(xa)m+1
_9x/_9x(xa)m+1x
(?eC,meN),
(meN).
If we denote by u the canonical generator, then we have Pu - 0. By multiplying either a power of a or a power of x, we obtain N
Y b.{x) (xa)ju = 0
j=o
N-1
with bN(x) = 1. Hence . = Y (9(xa)ju = Y (9(xa)ju is a coherent (9-subj=0
j=o
which satisfies (xa). (-_ .. We shall generalize this property module of to the case of several variables. 9.2.
Let X be a complex manifold, 9) an open subset of T*X and V
a closed involutive complex submanifold of Q. Let us define
-26/V = fu E9(1)In; 61(P)IV = 0} and let .9v be the subring of 9X I n generated by f v. For a coherent 60X-module .#, a coherent sub- 'X(0)-module 2' of # is called a lattice of .' if A' = 9x.. The following proposition is easily derived from the fact that 9(0) is a Noetherian ring. PROPOSITION 9.2.1 ([K-O] Theorem 1.4.7). Let . &,I a-module. Then the following conditions are equivalent.
For any point
(1)
p e S2,
there is a lattice
be a coherent
Ao of .'
on a neigh-
borhood of p such that /v.fo = Mo For any open subset U of Q and for any coherent 9(0)-submodule 2'
(2)
of
.
I u, S'v2' is coherent over '(0) I u.
Definition 9.2.2. If the equivalent conditions of the preceding proposition are satisfied, then we say that .' has regular singularities along V.
Remark that if .# has regular singularity along V, then the support of A' is contained in V. Let us denote by IRv(. #) the set of points p such that .' has no regular singularities along V on any neighborhood of p. The following theorem is an immediate consequence of Gabber's Theorem 6.3.2. THEOREM 9.2.3.
IRv(A') is an involutive analytic subset of ..
In fact, if we take a lattice A'0 of A', then T*X\IRv(.') is the largest open subset on which t'v.'o is coherent over 9(0).
If an 9-module .# has regular singularities along an involutive sub-
9.3.
manifold V then A' is, roughly speaking, constant along the bicharacteristics
of V. More precisely, let Y and Z be complex manifolds and X = Y x Z. Let zo e Z and let j be the inclusion map Y c, X by y i--+ (y, zo). Then we have THEOREM 9.3.1.
Let
.# be a coherent 9x module.
Assume that A' A' is isomorphic to
has regular singularities along T*Y x TZZ. Then j*,# ®(9Z. Note that any involutive submanifold V of T*X with 0x I v : 0 is transformed by a homogeneous symplectic transformation to the form T*Y x TZZ.
- 27
Noting that any nowhere dense closed analytic subset of a Lagrangean variety is never involutive, Theorem 9.2.3 implies the following theorem. 9.4.
THEOREM 9.4.1.
Let .t be a holonomic &X-module. Then the following
conditions are equivalent.
There exists a Lagrangean subvariety
(i)
A
such that
.4Y
has regular
singularities along A.
For any involutive subvariety regular singularities along A.
(ii)
(iii)
A
which contains
Supp .11,
..#
has
There exists an open dense subset 0 of Supp .A' such that .A' has regular singularities along Supp .t on S2. If these equivalent conditions are satisfied, we say that 41 is a regular
holonomic 61X-module.
The following properties are almost immediate. THEOREM 9.4.2.
0 --+ M' -..t - ..#" -. 0 be an exact sequence of three coherent .x -modules. If two of them are regular holonomic then so is the third. If . is regular holonomic, its dual .%* is also regular holonomic. Let
(i)
(ii)
We just mention another analytic property of regular holonomic modules, which generalizes the fact that a formal solution of an ordinary differential equation with regular singularity converges. THEOREM 9.4.3 ([KK] Theorem 6.1.3). ax-modules, then gxtjX(.1,
If .t
holonomic
xt'.A1)
and A' are regular oxt j,(M, ofX ® .') and 9X
X ® V) are isomorphisms. 'X
§ 10.
STRUCTURE OF REGULAR HOLONOMIC & -MODULES
(See [SKK], [KK]) 10.1.
Let A be a Lagrangean submanifold of T*X. We define fA and
O'A as in § 9.2. is
Then A(-1) = gA (-1) is a two sided ideal of i A and SA/SA(-1) the sheaf of a sheaf of rings which contains (9A(0) =
homogeneous functions on A.
M0)//A(-1),
28Let us take an invertible (9A-module 2' such that 2®Z
wA ®wz Cx
Such an 2' exists at least locally. For P = P1(x, a) + Po(x, a) + ... E / we define, for cp E (9A and an invertible section s of 2', L(P)
1 yr((p)+-Lp s (W)= ' 2
LHP1(s®z®dx)
s®z ®dx
I aZPi +\ Po--> 2 j= ax;a; /
S. cp
Here dx = dxl A ... A dx E co, and s®z ® dx is regarded as a section of CO,. The Lie derivative LHP of HP1 operates on (OA as the first order 1
differential operators so that LHP (s®z®dx) is a section
of wA and
1
LHP1(s®z®dx)/s®z ® dx is a function on A.
We thus obtain L: f A -. d'ndc(2). Then this does not depend on the choice of local coordinate system and moreover it extends to the ring homomorphism L: 6A -> 'ndc(2). Since the image is contained in the differential endomorphism of 2', we obtain the ring homomorphism
L:[
'A-Q2'®-9A®2®-1
CA
CA
'A/'A(- 1) coincides with the subsheaf of consisting of differential endomorphisms of 2 homoBy
PROPOSITION 10.1.1.
2' ® 9A 0 2® 1 CA
L,
CA
geneous of degree 0.
If we take 31(x, a) + 90(x, a) +
fAE
such that d81
...
OX mod 'AS21 and 1
2
X
az `41
$o(x, E) mod .1A
then L(9) gives the Euler operator of Y. Such a 9 is unique modulo
10.2.
Let .,# be a regular holonomic 9X-module whose support is
A.
Let .moo be a coherent sub-$A-module of M which generates #. Such an
.&o is called a saturated lattice of Al. Then P = Mo/&(-1).A is an A/'A(-1)-module, which is coherent over UPA(0).
Since a coherent sheaf with integrable connection is locally free, we have LEMMA 10.2.1.
.#
is a locally free (A(0)-module of finite rank.
29 --
1), 9 can be considered as an Since 9 belongs to the center of endomorphism of --Yom, A/rA(- 1)(Ji, 2'), which is a locally constant sheaf
on A. Its eigenvalues are called the order of AZ with respect to AY 10.3.
.
Let us take a section G (-_ C of C --p C/Z. Then there exists a unique
saturated lattice .#o such that the orders of
with respect to 'Wo are
contained in G (See [K4]). Then
F=
omeA/"A(- n(
,
2')
and
M = exp 2TnS E slut(J) does not depend on the choice of G. THEOREM 10.3.1 ([KK] Chapter I, invertible
2'
(9A-module
such that
§ 3).
Assume that there exists an Then the
2°2 = (')x (o)©-1
category of regular holonomic &X-modules with support in
to the category of (., M)'s
A
.
is equivalent
where . is a locally constant CA-module
and M e slutc(.F). 10.4. If u c- ./#, then the solution to L(P)cp = 0 for P E gA with Pu = 0 is called a principal symbol of u and denoted by 6(u). The homogeneous degree of 6(u) is called the order of u. In the terminology of § 10.2, the principal symbol is a section of .*'omg^lgn(_ 1)(9'Au/&x(- 1)u1 .') and the order is the eigenvalue of 9 inomgn/gn(-1)(exu/(ffx(-1)u, 2'). 10.4. When the characteristic variety is not smooth, we don't know much about the structure of holonomic systems. In this direction, we have
THEOREM 10.4.1 ([K-K] Theorem 1.2.2).
subset of an open subset
S2
Let Z be a closed analytic
of T *X. n = dim X, and let .1l and .1'
be holonomic 6X 1 n-modules. (i)
If dim Z < n-1, then F(s2; JComgX(.#, l)) - r(Q\Z,
OmSx(.
..4'))
is injective. (ii)
If dim Z < n - 2, then F(S2 ;
.
'omgX(.&, .N')) -. I S2\Z ; *E om,x(J&, -4/))
-30is an isomorphism.
In particular if Supp c Al v A2 and if dim (A1nA2) _< n-2, then is a direct sum of two holonomic x-modules supported on Al and A2, respectively.
Here is another type of theorem. THEOREM 10.4.3 ([SKKO]).
Let
.i = 'u = 9'//
9-module defined on a neighborhood of
p e T*X.
be a holonomic Assume
Supp ./#
= Al v A2 and and Al n A2 are non-singular and dim Al = dim A2 = n, dim (A1nA2) = n-1. (ii) Tp- Al n Tp A2 = Tp.(A1nA2) for any p' in a neighborhood of p in Al n A2. (iii) The symbol ideal of f coincides with the ideal of functions vanishing on Al u A2. Setting k = ordA,u - ordA2u - 1/2, we have has a non-zero quotient supported on Al a i has a non-zero (a) (i)
A1, A2
.i
submodule supported on A2 p k e Z. (b)
. #p is a simple 6'p module a k 0 Z.
Sketch of the proof. By a quantized contact transformation, we can transform p, A1, A2 and f as follows: p = (0, dx1) Al J= {((Yx, E); x1 = 2 = ... = n Y= 01
A2 = {(x, S), X1 = X2 = S3 = ... = Sn = 0}
d=
'i(X1a1-A.) +(X282-µ) + Y_ (60, j>2
In this case, we can easily check the theorem.
§ 11.
APPLICATION TO THE b-FUNCTION (see [SKKO])
As one of the most successful application of microlocal analysis, we shall sketch here how to calculate the b-function of a function under 11.1.
certain conditions.
- 31
Let f be a holomorphic function on a complex manifold X. Then, is proved ([Bj], [Be] [K1]) that there exist (locally) a non zero polyit 11.2.
nomial b(s) and P(s) c- _9[s] = -9 0 C[s] such that P(s) f (x)s+' = b(s) f (x)s de(
C
for any s e N. Such a polynomial b(s) of smallest degree is called the b-function of f (x) and is denoted by b f(s). For the relations between the b-function and the local monodromy see [M1], [K3].
Set / = {P(s) e _i [s] ; P(s)f' = 0 for s e N} and AV' _ -q[s]//- We by f s. Then t: J^ 3 P(s) f shall denote the canonical generator of P(s + 1) f f 5 e .N gives a J-endomorphism of .A and t.A' = !2[s] f s+ 1 11.3.
Here f" 1 = f f s e .N. In this terminology b f(s) is the minimal polynomial of s e 'nd_q(.N'/t. V).
For 7,.E C, we set #,, _ ms[s]/(/ + -9 [s] (s - A)) and denote by f' the canonical generator of .Wa. Then f'`+1 -4 f f' defines a !2-linear homomorphism #,,+1 -> -Wa Let W be the closure of
11.4.
{(s, x, ) e C x T*X ; E = sd log f (x), f (x)
in C x T*X. Set Wo = W n {s=0}
0}
T*X. Then we can prove
PROPOSITION 11.4.1 ([K1]). (i)
(ii)
N is a coherent 2X module and Ch (iv) = p(W), where p is the projection from C x T*X to T*X. For any X e C, J4 is a regular holonomic gx-module and
Ch (,) = W,. (iii)
.iV/t.'
is
a
regular holonomic
-9X-module
and
Ch (.K/t-41)
= Wo n (n. f)-'(0). 11.5.
In the sequel, for the sake of simplicity, we assume that there exists
a vector field v such that v(f) = f. Therefore we have vk(f s) = skf s Hence V is a -9-module generated by f'. If we set 3 _ 9 n /, then .41 -_ -i// and / = -9 [s] (s-v) + _q [s] 1. 11.6.
The following lemma is almost obvious but affords a fundamental
tool to calculate the b-function.
- 32 LEMMA 11.6.1.
Let 2 be an tx-module and w a non-zero section
of Y. For ? E C, we assume (i)
v(w) = a,w
(ii) )w = 0 (iii) fw = 0. Then we have b f(X) = 0. Proof. There is a P E -9 such that b1(s)f5 = Pf s+ i Hence (b f(v) - Pf)f' = 0, which implies b f(v) -- Pf c- ?. Since b f(v)w = b f(X)w we have
0 = (b f(v) - Pf)w = b1(X)w. This implies b f(A.) = 0.
11.7.
Let I be the symbol ideal of ). Then the zero set of / is W,
and the zero of / + (26(v) is W,. Let A be an irreducible component of W,. If / + (9T*X6(v) is a reduced ideal at a generic point p of A then we call A a good Lagrangean.
If A is a good Lagrangean, then W is non-singular on a neighborhood of a generic point p of A and 6 = 6(s) w has non zero-differential. Let p: W --* X denote the projection. We define m(A) e N as the degree of zero of f o p along A, and set fA = (f op/6'°(A))IA. Let co be the non-vanishing
n-form on X. Then (p*w) A d6 is an (n+1)-form on W. Let µ(A) be the degree of zeros of (p*w) A d6 along A, and let rl be the n-form on A given by
p*w A d6 6R(A)
If we set 'CA = rl ® w®
=rlAd6. A
E wA ® w ®-'
,
then this is independent of the
choice of w. We have
PROPOSITION 11.7.1 ([SKKO]). ?L e C, .W, is a simple holonomic point p of A and we have (i)
a)
6(.f = f A In particular
If A is a good Lagrangean, then for any system on a neighborhood of a generic
KA
ord f' = - m(A)), - µ(A)/2 .
33 (ii)
There exists a monic polynomial invertible micro-differential operator
bA(s)f s= PA f f S
bA(s)
PA
in
of degree of order
m(A) m(A)
and an such that
S ®.w`
6(PA)IA = J n1.
and
Remark that fA and WA are homogeneous of degree -m(A) and - µ(A), respectively. Remark also that the minimal polynomial of s c &nde(off (D .N/t.,V)1 A D
is bA(s). In fact, if Pf s+ 1 = b(s) f S in 6' (D X, then (P P A 1 bA(s) - b(s)) f' = 0.
This implies that P PA-1bA(v) - b(v) E 4). Hence 6(P Pn lbA(v)b(v)) I w = 0.
If ord P PA 1bA(v) = ord P > deg b, then 6(P) K. = 0. Therefore P = P + P" with P" E Si and 6(P') < 6(P). Hence P',f'-- = b(s)f'. Thus, we may assume ord P < deg b. Then 0 = 6(b(v) - P P A 1bA(v)) I W = b(6) - (6(P)1 Wf AbA(6))
.
This shows that b(s) is a multiple of bA(s).
COROLLARY 11.7.2.
If every irreducible component of
Lagrangean, then b f(s) is the least common multiple of the
Wo
is good
bA(s).
11.8. Let Al and A2 be two good Lagrangeans. We assume the following conditions for a point p c- Al rr A2: (11.8.1)
dimPA1 n A2 = n-1 and A1, A2 and Al n A2 are non singular on a neighborhood of p.
(11.8.2)
For any point p' on a neighborhood of p in A, n A2, we have Tp Al n TP.A2 = TP.(A1nA2).
(11.8.3) / + (96(v) coincides with the defining ideal of Al u A2 with the reduced structure.
In this case we say that Al and A2 have a good intersection. We have the following theorem.
34 Let Al and A2 intersection. If m(A1) >, m(A2), then
be good Lagrangeans with a good
THEOREM 11.7.3.
m(Ai)-m(A2)-1
il k=O
I
(ordA2P_ordAlf5++k) bf(s).
In order to prove this let us take k e C such that k = ordA, f' - ordAZ f A - 1/2 e N
and
(11.8.4)
k' = ordA, f a+ 1 - ordAZ f1 + 1 - 1/2 e N. Recall that
k = (m(A2)-m(A1))X - 2 (µ(A2)-µ(A1)-1/2)
and k' = k + (m(A2)-m(A1)). Then by Theorem 10.4.3, to has a non-zero
quotient 2' whose support is A 1. Let w e 2' be the image of f' e #,. Let a : # , - 2' be the canonical homomorphism and 0: _Wx + 1 -' M1 be the homomorphism given by f'+1 F- f f'. Then, since V o N, .tea+1 has no non-zero quotient supported in A1. Hence a o (3 = 0. Therefore fw = a(3(f'`+1) = 0. Thus we can apply Lemma 11.6.1 to conclude that b f(),) = 0. If k c Z with 0 < k < m(A1) - m(A2) then
m(A1)
1
m(A2)
(k + 2
(µ(A1)-µ(A2)-1))
satisfies (11.8.4). This shows that b f(s) is a multiple of m(At)-m(A2)- 1
F1 k=0
(m(A1)-m(A2))s -
2(µ(A1)-µ(A2)-1) + k) I
m(A,)-m(A2)- 1 const.
j k=0
(ordA2f5 - ordA,fs + 2 + k
If we refine this argument, we can prove THEOREM 11.8.2 ([SKKO]).
If Al and A2 are good Lagrangeans with
a good intersection and if m(A1) >_ m(A2) then bA,(S)
b A2O S
'(Al)-m(A2)-1
= const.
H
k=0
(ordA2f' - ordA, f5 +
I 2
+k
35 Example 11.8.3. (i)
X = Mn(C) = Cn2
and
f (x) = det x.
0
s+1
-s-1/2 s+2 n
- 2s - 4/2
b f(s) = f (s +j). i=1
s+n
-ns-n2/2
Here O means a good Lagrangean which is the conormal bundle to an a-codimensional submanifold. O-O means that the two corresponding good Lagrangeans have a good intersection.
The polynomial attached to the intersection is the ratio of the corresponding bA-functions, calculated by Theorem 11.8.2. The polynomial attached
to the circle is the order of f'. (ii)
X = C", f (X) = x j + ... + x 0
OO
S+1 - s - 1/2
s + n/2 (
-2s-n/2
(iii) X = C3 f = x2y + ZZ
b(s) = (s + 1) (s + n/2)
- 36 0
- s - 1/2
(2s + 2) (2s + 3)
-3s-2
br(s) = (s + 1)2(s + 3/2)
37
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