OF D I F F E R E N T I A L E Q U A T I O N S
By M. K a s h i w a r a
R e s e a r c h I n s t i t u t e for M a t h e m a t i c a l S c i e n c e s Kyoto U n i v e r s i t y
i. Let X be a c o m p l e x m a n i f o l d and f a h o l o m o r p h i c f u n c t i o n on X. Then, for a c o m p l e x of s h e a v e s F" on X, we can d e f i n e a
" v a n i s h i n g cycle sheaf" ~ F " (in D e l i g n e ' s n o t a t i o n ) on f-l(0) (See [3], [i]). The p u r p o s e of this p a p e r is to give a c o r r e s p o n d i n g h o l o n o m i c s y s t e m w h e n F" is g i v e n as a de R h a m c o m p l e x of a r e g u l a r h o l o n o m i c system.
2. Let X be a s m o o t h c o m p l e x m a n i f o l d and Y a s m o o t h s u b m a n i f o l d of X. We d e n o t e by 0 X and ~y the sheaf of h o l o m o r p h i c
f u n c t i o n s on X and the d e f i n i n g Ideal of Y. We d e n o t e by A the g r a d e d 0 x - A l g e b r a @ ~ y k t - k C O x [ t , t - l ] . Here~ ~ y k stands for 0 X
k ~ g
if k ~ 0. We d e n o t e by w:X ÷ X the space S p e c a n A over X.
T h e n X is s m o o t h and t d e f i n e s a h y p e r s u r f a c e of X i s o m o r p h i c to the n o r m a l b u n d l e T y X of Y.
Let ~ be the real m a n i f o l d ( C - ( 0 ) ) d S I w i t h the b o u n d a r y S I = c x / ~ +, w i t h the o b v i o u s p r o j e c t i o n @ + ~. For a c o m p l e x of sheaves F', we d e f i n e
(2.1) Vy(F') = i - l ~ j , p - i F ".
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Here p is the p r o j e c t i o n X - TyX = (C-{O}) x X ÷ X and j:
X-TyX ~ @ x X, w h i c h are given by t: X + ~. The map i is the i n c l u s i o n TyX ~ S I x T y X ~ @ x X given by (i mod R ~ S I.
C
By using a local c o o r d i n a t e s y s t e m ( X l , . . . , x z , . . . x n) of X such that Y is x' = (Xl,...,x E) = 0, the stalks of Vy(F') are d e s c r i b e d as follows. For (xo,v) c TyX (xo ~ cn-~,vcCZ), we have
(2.2) H J ( v Y ( [ ' ) ) ( x o , v ) = ~ HJ(u;F" ) -- •
Here, U runs over the set of open subsets of X w h i c h c o n t a i n { x : ( x ' , x " ) c C ~ × ~ n - ~ ; I x ' l < ~ , Ix"-xoI<~ , x' ~ r} for some ~ > 0 and an open cone F ~ v of ~ .
3. Let D X be the sheaf of d i f f e r e n t i a l o p e r a t o r s on X and a r e g u l a r h o l o n o m i c Dx-MOdule. We shall t h e n construct a r e g u l a r h o l o n o m i c D T y X - M O d u l e ~' such that
Vy(~ Hom DX( m, ~X )) = R Hom DTyX( I', ®TyX ).
If such an M' exists, it is unique up to an isomorphism.
denote it by V y ( ~ ) .
We shall
4. Keeping X and Y as in the p r e c e d i n g section, we shall define the f i l t r a t i o n F" = F'( D X) of D X by
(4.1) Fk( D X) = {P E DX; P ( ~ ) c I $ +k for any j}.
Then, one can show easily the f o l l o w i n g
P r o p o s i t i o n i~ (i) Fk( DX)/Fk+I( D X) is isomorphic to the sheaf of
differential operators o_nn TyX homogeneous o_~fdegree k. Hence its graduation gr F. ( ~ X ) is a subring of DTy X.
(2) There exists (locally)_ a vector field 8 tangent t__oo Y acting on Iy/I$ as the identity.
5. Now, let ~ be a coherent 9x-MOdule. A filtration F~ of m is called a good filtration of m with respect to F'( D X)
if it satisfies
(5.1) Fk( Dx)F ~ ~ _k+j ~I for any k and j
(5.2) Fk( Dx)F ~ = ~I ~k+j if j >>0 and k ~ 0
(5.3)
or if j << 0 and k < 0.
F~ is a coherent F0( Dx)-Module.
(5.4) ~ = uF~.
The following proposition is proved in [2].
Proposition 2. Let M be a regular holonomic system. Then there exist locall~ a coherent ~X sub-Module F of M and a non-zero polynomia ~ b(e ) such that
(5.5) b( 8 )F < ( D x ( d e g b ) ~ F l ( Dx))F
(5.6) m = D x F
Here Dx(m ) denotes the sheaf of differential operators of order m, and 8 is the one given in Proposition i.
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6. Let R be the abelian category of coherent Dx-MOdules satisfying the conclusion in Proposition 2. Let @ be a subset of
satisfying the following condition:
(6.1) For any a ~ @ , @ ~(a+Z) consists of a single point.
Then we have the following
Theorem I. (i) For any M ~ R, there exists a good filtration F ~ ( M ) o f M satisfying the following condition: there exists a p01ynomial b ( 0 ) such that b-l(0) C G and b( 0 - k ) F ~ ( M ) c-k+I(~G M) for any k.
Moreover such a filtration is unique.
(2) For M 6 R, g r F ~ ( M ) does not depend on the choice of G a_{s (not graded) grF.( D)-Module, W~e shall denote it b__yy gr M.
(3) M ~-~ gr ~ is an exact functor from R into the category of coherent grF. ( D )-Modules.
(4) Vy(~ Hora DX ( M, 0X)) = ~ Holm ~TyX ( DTy X grDx@ gr M, ®TyX )
Vy(~ Hom DX( @X' m)) = ~ Hom DTyX ( 0TyX' DTyX grD X @ gr M).
(5) If M i_~s regular holonomic, so is DTy X @ gr M grD We shall indicate the proof of the theorem.
Proof of (I). By using Proposition 2, there exists a good filtration F~ of M and a non-zero polynomial b such that
(6.2) b( e-k)F~ C _k+l
~I for any k.k F k ( D ) F we apply the following lemma In fact, setting F I =
Fk+I(D ).
Now, assume that b ( 8 ) in (6.2) is a product of two
k k+l -k)F~
p o l y n o m i a l s bl( 8 ) and b2( 8 ) and we set FII = F I +bl( 8
Then FII is a good f i l t r a t i o n s a t i s f y i n g b l ( 8 _ k _ l ) b 2 ( e _ k ) F ~ I ~ _ k + l
~II "
R e p e a t i n g this procedure, we can show the e x i s t e n c e of F~.
The u n i q u e n e s s of F~ is p r o v e d as follows.
L e t F I and FII be two good f i l t r a t i o n s and bl( 8 ) and b l l ( 8 ) two p o l y n o m i a l s s a t i s f y i n g b j ( e - k ) F ~ C F ~ +I and b]l(0) C G
k-N for any k.
for J = I, II. There exists N ~ i such that F C F I I
-k+lc-k-N+l 8-k+N)F Cbz(e-k+N)F NC
Then b l ( 8 - k ) F ~ < ~ I ~II and bll(
Fk-N+I
Ii . Since bl(S-k) and bll(S-k+N) have no common root, k <F~I" (3) is Flk<~ii_k-N+l. R e p e a t i n g this, we finally o b t a i n F I
p r o v e d by a similar discussion.
Proof of (2). Let G and G' be two subsets of @ s a t i s f y i n g (6.1). We shall show gr F G ~ gr FG,. We may assume G ~ I and G' = (G-{I}) U {I+i). We write gr F G for grFG ~.
Let b(8) be a p o l y n o m i a l such that b-l(0) C G and b ( 8 - k ) F ~ Fk+l Set b( 8 ) = (8-l)ma( 8 ) with a(1) # 0. Then F~, =
G "
(8-l-k m k _k+l
) FG + FG " Let us take ~,¢ 6 ~ [ 9 ] s a t i s f y i n g
(6.3)
- 0 rood ( e - l ) m ( 8 -i-i) m,~ - i mod a ( 8 ) ,
@ - 0 mod a( 8 )a( e -i), - I mod ( 8 - I ) m.
We shall define f: gr F G ÷ gr FG, and g: gr FG, ÷ gr F G as follows.
(6.4) k. k + l _ k ._k+l f: gr F G = ~ F G / F G 9 [u k ~-~ Iv k ~ gr F G, = ~ ' G ' / ~ G '
v k = 9( e - k ) u k + 4( e - k - 1 ) u k + 1
(6.5) g: gr FG, 9 Iv k ~-~ [u k ~ gr F G
by u k = v k + 4( 8 - k ) V k _ I.
T h e n one c a n e a s i l y s h o w t h a t f a n d g are i n v e r s e s to e a c h o t h e r .
(4) is s h o w n by r e d u c i n g t h e p r o b l e m to t h e f o l l o w i n g s p e c i a l c a s e , w h i c h is e a s y to p r o v e .
P r o p o s i t i o n 2. Let b( 8 ) be a n o n - z e r o p o l y n o m i a l of d e g r e e m w i t h b - l ( 0 ) C G a n d P an N × N m a t r i x of d i f f e r e n t i a l o p e r a t o r s in FI( D ) ~ D ( m ) .
Set m = D N / D N ( b ( 8 )-P). T h e n (4) in T h e o r e m i is t r u e f o r M .
(5) is p r o v e d in [2].
7. S u p p o s e t h a t Y is a s m o o t h h y p e r s u r f a c e of X g i v e n by f = 0.
Then, f o r a c o m p l e x of s h e a v e s F" w h o s e c o h o m o l o g y g r o u p s are c o n s t r u c t i b l e , o n e c a n d e f i n e ~ 4 a n d R ~ a n d can: ~ ¢ ÷ ~ Y a n d Var: RY + R @ (See [3]). If w e t a k e a v e c t o r f i e l d ~ s u c h t h a t
~f ~ i m o d Iy, t h e n 0 = f~ a n d g r F . ( D ) 0 is i s o m o r p h i c to D y.
S u p p o s e F" = R H o m D ( M, 0 X ) f o r a r e g u l a r h o l o n o m i c D x - M O d u l e X
M . T h e n we h a v e t h e f o l l o w i n g
T h e o r e m 2. A s s u m e G C @ s a t i s f i e s (6.1) a n d c o n t a i n s 0.
(1) ~ = ~ Hom D (gr~ m, @y) and Re = ~ iom D (grG 1 m, @y).
Y Y
(2) can i ss ~ i v e n b ~ f: g r ~ l ~ ~-~ g r G ~ 0 and V a r i ss g i v e n b ~
~e2Wie_ I 0 I
: gr G ~ -~ gr~ m e
2wi6_i R e m a r k I. We can r e p l a c e in (2), f and ~e
0 and ~.
2wi w i t h e
e - i f
8
R e m a r k 2. If we r e p l a c e ~ H o m D X(* , 0 X) w i t h ~ H o m D X ( @ X , * ) , t h e n (i) h o l d s by r e p l a c i n g ~ H o m D X(* , ® y ) w i t h R H o m D y ( 0 y , * ) . A c c o r d i n g l y , (2) holds by e x c h a n g i n g Var and can.
S k e t c h of proof. The t h e o r e m is e s s e n t i a l l y e q u i v a l e n t to the f o l l o w i n g o n e - d i m e n s i o n a l case. Let X = @ and Y = (0). Let V 0 and V_I be two v e c t o r spaces and let A: V 0 ÷ V_I and B: V i ÷ V 0 be two h o m o m o r p h i s m s . Let M be a D x - M O d u l e g e n e r a t e d by V 0 ~ V _ I w i t h the f u n d a m e n t a l r e l a t i o n :
xu = Bu for u ~ V _ I
~v = Av for v E V 0.
If we a s s u m e the e i g e n - v a l u e s of AB are c o n t a i n e d in G, t h e n k M = V k for k = 0, -i.
gr G
Let U be a n o n - e m p t y c o n v e x cone in @ such that U ~ 0 . T h e n we have
~ = Horn D X ( M, 0 x ( U ) ) and
R¢ = Horn DX( M, Ox(U)/ OX(C)).
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The h o m o m o r p h i s m can is g i v e n by 0 x ( U ) ÷ 0 x(U)/ ® X ( { ) . The h o m o m o r p h i s m V a r is g i v e n as follows: for ~ H O m D x ( M , ® (U)/0(X)) and s £ M , let us c h o o s e a r e p r e s e n t a t i v e u ~ ®(U) of ~(s).
Then u can be c o n t i n u e d to a m u l t i - v a l u e d h o l o m o r p h i c f u n c t i o n on
~-{0}, so that we can o b t a i n the h o l o m o r p h i c f u n c t i o n Tu d e f i n e d on U by the a n a l y t i c c o n t i n u a t i o n of u a l o n g a p a t h a r o u n d the origin. T h e n T u - u does not d e p e n d on the choice of a r e p r e s e n t a t i v e u and s ~-~Tu-u gives a h o m o m o r p h i s m f r o m M to 0 x ( U ) . This is the h o m o m o r p h i s m Vat.
Now, R ~ and ~ ¢ are i s o m o r p h i c to V~ and V*I as follows:
V$ ~ HomD (M, Ox(U)), V~ -~HOmD( M, @x(U)/OX(¢))
by V$ 9 ~ ~-~ ~ and V * 1 9 B ~-~ @ , w h e r e ~(u) = < ~ , x B A ' I B u > ,
~(v) = < ~ , x B A v > and @(u) = < B , x A B - I F ( I - A B ) u > , @(v) = - < B , A x B A F ( - B A ) v > for u 6 V _ I and v ~ V O.
R e m a r k that xlF(1) d e f i n e s well an e l e m e n t of 0 (U)/ @ (@) by the a n a l y t i c c o n t i n u a t i o n on k (e.g. xlF(k) = log x at I = 0 and xkr(k) = log x + ((log x)2/2 - y l o g x)N + ((log x)3/6 -
¥ ( l o g X)2/2 + (w2/3 + ¥ 2 / 2 ) i o g x ) N 2 at k = N w i t h N 3 = 0; y is the E u l e r c o n s t a n t ) .
Thus w i t h t h e s e i d e n t i f i c a t i o n , can is g i v e n by ~ a B ( F ( I - A B ) ) -I and V a r is g i v e n by B ~ - ~ B ( 2 w i A e W i B A / F ( I + B A ) ) . F i n a l l y it is
e n o u g h to n o t e that x, 2, 8 c o r r e s p o n d to B, A and BA (or AB-I) and (F(I-AB)) -I is i n v e r t i b l e u n d e r the c o n d i t i o n on the e i g e n v a l u e s of AB.
[ I ] [2]
[3]
References J. L. Verdier, in this volume.
M. Kashiwara, T. Kawai, Second microlocalization and asymptotic expansions, Lecture Notes in Physics, 126, pp.21-76, Berlin- Heidelberg-New York, Springer, 1980.
P. Deligne, Le formalisme des cycles evanescents, Lecture Notes in Mathematics, 340, pp.82-i15, Berlin-Heidelberg-New York, Springer, 1973.
