3 Holonomic D-Modules

Transcription

1 3 Holonomc D-Modules In ths chapter we study unctoral behavors o holonomc systems and show that any smple obect n the abelan category o holonomc D X -modules s a mnmal extenson o an ntegrable connecton on a locally closed smooth subvarety Y o X. 3.1 Basc results Recall that the dmenson o the characterstc varety Ch(M) o a coherent D X - module M( = 0) satses the nequalty dm Ch(M) dm X and that a coherent D X -module M s called holonomc dm Ch(M) = dm X or M = 0. Notaton We denote by Mod h (D X ) the ull subcategory o Mod c (D X ) consstng o holonomc D X -modules. The next proposton mples that Mod h (D X ) s a thck abelan subcategory o Mod c (D X ). Proposton () For an exact sequence n Mod c (D X ) we have 0 M N L 0 N Mod h (D X ) M, L Mod h (D X ). () Any holonomc D X -module has nte length. In other words, the category Mod h (D X ) s artnan. Proo. The statement () s a consequence o Ch(N) = Ch(M) Ch(L). The statement () s proved usng the characterstc cycle as ollows. holonomc D X -module M consder ts characterstc cycle CC(M) = m C (M) C. C I(Ch(M)) For a

2 82 3 Holonomc D-Modules Note that dm C = d X or any C I(Ch(M)). Dene the total multplcty o M by m(m) := m C (M). C I(Ch(M)) By Theorem the total multplcty s addtve n the sense that we have m(m) = m(l) + m(n) or any short exact sequence 0 L M N 0 n Mod h (D X ). Moreover, we have m(m) = 0 Ch(M) = M = 0 by the denton o characterstc varetes. Hence the asserton ollows by nducton on m(m). Notaton We denote by D b h (D X) the ull subcategory o D b c (D X) consstng o obects M D b c (D X) whose cohomology groups are holonomc, that s, H (M ) Mod h (D X ) or Z. We easly see the ollowng rom Propostons and B.4.7. Corollary D b h (D X) s a ull trangulated subcategory o D b c (D X). Remark It s known that (see Belnson [Be]). D b (Mod h (D X )) D b h (D X) The ollowng result s the rst mportant step n the study o holonomc D- modules. Namely, we can say A holonomc D-module s genercally an ntegrable connecton. Proposton Let M be a holonomc D X -module. Then there exsts an open dense subset U X such that M U s coherent over O U. In other words, M U s an ntegrable connecton on U. Proo. Let TX X T X be the zero secton o T X and set S := Ch(M) \ TX X. I S =, then M tsel s coherent over O X by Proposton Assume that S =. Snce S s conc, the dmenson o each ber o π S : S π(s) (π : T X X) s 1 and hence dm π(s) < dm S dm X. Thereore, there exsts an open subset U X such that X \ π(s) U =. In ths case we have Ch(M U ) \ TU U = and hence M U s coherent over O U by Proposton The ollowng result, whch can be proved by dualty, s also mportant. Proposton Let M Mod qc (D X ). For an open subset U X suppose that we are gven a holonomc submodule N o M U. Then there exsts a holonomc submodule Ñ o M such that Ñ U = N.

3 3.2 Functors or holonomc D-modules 83 Proo. By Corollary we may assume that M s coherent and M U = N. Set L = H 0 (D X M). By Corollary () we have codm Ch(L) d X and hence L s a holonomc D X -module. Moreover, ts dual Ñ = D X L s also holonomc by Corollary (v). By L = H 0 (D X M) τ 0 (D X M) we have a dstngushed trangle K D X M L +1, where K = τ 1 (D X M). By applyng D X we obtan Ñ M D X K +1. Snce the dualty unctors commute wth restrctons to open subsets, we have Ñ U = D U (L U ) = D 2 U (M U ) = M U = N. It remans to show that the canoncal morphsm Ñ M s nectve. For ths we have only to show H 1 (D X K ) = 0. In act, we wll show that H (D X (τ k K )) = 0 ( < 0, k > 0) (3.1.1) (note that τ k K K or k 0). Let us rst show H (D X (H k (K )[k])) = 0 ( < 0, k > 0). (3.1.2) For k>0 we have H k (K ) H k (D X M)and hence codm Ch(H k (K )) d X k by Corollary (). Hence the asserton s a consequence o Corollary (). Now we prove (3.1.1) by nducton on k. I k = 1, then we have τ k K = H k (K )[k], and hence the asserton ollows rom (3.1.2). Assume k 2. By applyng D X to the dstngushed trangle we obtan a dstngushed trangle H k (K )[k] τ k K τ (k 1) K +1 D X (τ (k 1) K ) D X (τ k K ) D X (H k (K )[k]) +1. Hence the asserton ollows rom (3.1.2) and the hypothess o nducton. 3.2 Functors or holonomc D-modules Stablty o holonomcty We rst note the ollowng, whch s an obvous consequence o Corollary Proposton The dualty unctor D X nduces somorphsms D X : Mod h (D X ) Mod h (D X ) op, D X : Dh b (D X) Dh b (D X) op.

4 84 3 Holonomc D-Modules The ollowng s also obvous by Ch(M N) = Ch(M) Ch(N). Proposton The external tensor product nduces the unctors ( ) ( ) : Mod h (D X ) Mod h (D Y ) Mod h (D X Y ), ( ) ( ) : D b h (D X) D b h (D Y ) D b h (D X Y ). Recall that or a morphsm : X Y o smooth algebrac varetes we have unctors : Dqc b (D X) Dqc b (D Y ), : D b qc (D Y ) D b qc (D X). Moreover, s proper (resp. smooth), (resp. ) preserves the coherency and we have the unctors : Dc b (D X) Dc b (D Y ) (resp. : Dc b (D Y ) Dc b (D X)). However, nether nor preserves the coherency or general morphsms. A surprsng act, whch we wll show n ths secton, s that the holonomcty s nevertheless preserved by these unctors or any morphsm : X Y. Namely, we have the ollowng. Theorem Let : X Y be a morphsm o smooth algebrac varetes. () sends Db h (D X) to D b h (D Y ). () sends D b h (D Y ) to D b h (D X). Corollary The nternal tensor product L O X nduces the unctor ( ) L O X ( ) : D b h (D X) D b h (D X) D b h (D X). Proo. Ths ollows rom Proposton and Theorem () notng that ( ) L O X ( ) = L X (( ) ( )), where X : X X X s the dagonal embeddng. The proo o Theorem wll be completed n the next subsecton. In the rest o ths subsecton we reduce t to that o Theorem () n the case when s the proecton C n C n 1. Lemma Let : X Y be a closed embeddng. Then or M Dc b(d X) we have M Dh b (D X) M Dh b (D Y ).

5 3.2 Functors or holonomc D-modules 85 Proo. Snce s exact, we may assume that M = M Mod c (D X ). Let T Y ϖ X Y T Y ρ T X be the canoncal morphsms. Then we have ( ) Ch M = ϖρ 1 (Ch(M)), by Lemma Snce ϖ s a closed embeddng and ρ s a smooth surectve morphsm wth one-dmensonal bers, we have ( ) dm Ch M = dm Ch(M) + 1, orm whch we obtan the desred result. Next we reduce the proo o Theorem () to the case when s the proecton C n C n 1. In order to prove Theorem () t s sucent to show M Db h (D Y ) or M Mod h (D X ). By consderng the decomposton o nto a composte o a closed embeddng and a proecton we may assume that s ether a closed embeddng or a proecton. The case o a closed embeddng has already been dealt wth n Lemma 3.2.5, and hence we can only consder the case when s the proecton X = Z Y Y. Snce the problem s local on Y, we may assume that Y s ane. Take a nte ane open coverng Z = r =0 Z o Z such that Z \ Z s a dvsor on Z or each, and set X = Z Y. Then X = r =0 X s an ane open coverng o X. For 0 0 < < k r let 0,..., k : X 0,..., k = k p=0 X p X be the embeddng (note that X 0,..., k s ane by the choce o Z s). Then M s quas-somorphc to the Čech complex wth 0 C 0 (M) C 1 (M) C r (M) 0 C k (M) = 0 < < k 0,..., k (M X0,..., k ) (note 0,..., k (M X0,..., k ) 0,..., k 0,..., k M). 0,..., k 0,..., k M(= Hence t s sucent to show 0,..., k 0,..., k M) Dh b(d Y ) or any ( 0,..., k ). Thereore, we may assume rom the begnnng that X and Y are ane. Fx closed embeddngs α : X C n,β : Y C m, and consder the commutatve dagram X Y g X Y β α β C n+m C m, p

6 86 3 Holonomc D-Modules where g s the graph embeddng assocated to and p s the proecton. Lemma M Db h (D Y ) and only β M Db h (D Cm). Note that M = M = M. β β p (α β) g Snce (α β) g s a closed embeddng, we have M Mod h (D C n+m) (α β) g by Lemma 3.2.5, and hence the problem s reduced to the case when s the proecton C n+m C m. Snce C n+m C m s a composte o morphsms C k C k 1, the problem s nally reduced to the case when s the proecton C n C n 1. Let us show that Theorem () mples Theorem (). So we assume that Theorem () holds and show M D b h (D X) or any M Mod h (D Y ). By decomposng nto a composte o a closed embeddng and a proecton we may urther assume that s ether a closed embeddng or a proecton. Consder rst the case where s the proecton X = Z Y Y. Then s an exact unctor and the complex M = M[dm Z] s concentrated n the degree dm Z. Moreover, we have M O Z M and t s holonomc by Ch(O Z M) = Ch(O Z ) Ch(M) = T Z Z Ch(M), and hence M Dh b(d X). Let us consder the case o a closed embeddng : X Y. Let : U := Y \ X Y be the correspondng open embeddng. Then by the results n Secton 1.7 there exsts a dstngushed trangle M M M +1. We have M = M U Mod h (D U ), and hence () mples M D b h (D Y ). Thereore, we see by the above dstngushed trangle that M D b h (D Y ). Ths mples M D b h (D X) by Lemma Theorem () s vered assumng Theorem () Holonomcty o modules over Weyl algebras In the last subsecton the proo Theorem (), () was reduced to that o () n the case when s the proecton p : C n = C C n 1 C n 1. The am o ths subsecton s to prove t usng the theory o D-modules on C n. Set D n := Ɣ(C n,d C n) = Cx α β, α,β By where x α = x α 1 1 xα 2 2 xα n n or α = (α 1,...,α n ) and β = β 1 1 β 2 2 β n n or β = (β 1,...,β n ). The algebra D n s called the Weyl algebra. Snce C n s ane, we have equvalences o categores

7 3.2 Functors or holonomc D-modules 87 Mod qc (D C n) Mod(D n ), Mod c (D C n) Mod (D n ) gven by M Ɣ(C n, M). For N Mod(D n ) we denote the correspondng D C n- module by Ñ. AD n -module N s called holonomc Ñ s a holonomc D C n-module. Let N be a D n -module. We dene ts Fourer transorm N as ollows. As an addtve group N s the same as N, and the acton o the generators x, o D n on N s gven by x s := s, s := x s. It s easly checked that N s a let D n -module wth respect to ths acton. Ths denton o the Fourer transorm N s motvated by the classcal Fourer transorm. The Fourer transorm nduces equvalences o categores ( ) : Mod(D n ) Mod(D n ), ( ) : Mod (D n ) Mod (D n ). The correspondng equvalences or the categores o D C n-modules are also denoted by ( ) : Mod qc (D C n) ( ) : Mod c (D C n) Mod qc (D C n), Mod c (D C n). Proposton Let p : C n (= C C n 1 ) C n 1 be the proecton and let : C n 1 (={0} C n 1 ) C n (= C C n 1 ) be the embeddng. For M Mod qc (D C n) we have ( ) H k M H k (L M) p or any k. Proo. Set N = Ɣ(C n, M). Snce p s an ane morphsm, we have M Rp (DR C n /C n 1(M)) [ p M 1 p M ], and hence p ( Ɣ C n 1,H k( M) ) p Thereore, we have Ker [ N 1 N ] (k = 1), Coker [ N 1 N ] (k = 0), 0 (k = 0, 1).

8 88 3 Holonomc D-Modules ( Ɣ C n 1 (, H M) ) Ker [ N x 1 N ] (k = 1), k Coker [ N x 1 N ] (k = 0), p 0 (k = 0, 1), rom whch we obtan the desred result. Ɣ(C n 1,H k (L M)) In provng Theorem we also need the ollowng results. Proposton A coherent D C n-module M s holonomc and only M s as well. Proposton Let : (C \{0}) C n 1 C n be the embeddng. I M s a holonomc D C n-module, then so s H 0 ( M) (snce (C \{0}) C n 1 s an ane open subset o C n we have H k ( M) = 0 or k = 0). Let us complete the proo o Theorem assumng Propostons and By Propostons and and the arguments n the last subsecton t s sucent to show M Dh b(d C n 1) or M Mod h(d C n), where : C n 1 C n s as n Proposton Let : (C \{0}) C n 1 C n be as n Proposton By the dstngushed trangle M M we obtan an exact sequence 0 H 0( ) M M H 0( M +1 ) M H 1( ) M 0. Snce H 0 ( M) s holonomc by Proposton 3.2.8, we obtan M D b h (D C n) (note H k ( M) = 0 or k = 0, 1). Hence we have M Dh b(d Cn 1) by Lemma The rest o ths subsecton s devoted to provng Proposton and Proposton In addton to the usual order ltraton F, the Weyl algebra D n has another ltraton B dened by B D n := Cx α β D n. α + β We call t the Bernsten ltraton o the Weyl algebra D n. The graded algebra gr B D n assocated to the Bernsten ltraton B s commutatve and somorphc to the polynomal rng C[x, ξ] (x = (x 1,x 2,...,x n ), ξ = (ξ 1,ξ 2,...,ξ n )), as n the case o the usual order ltraton. For a D n -module M we can also dene good ltratons F on t wth respect to the Bernsten ltraton B. Any ntely generated D n -module has a good ltraton. The Bernsten ltraton has the advantage that or any good ltraton F o a ntely generated D n -module M each F M s nte dmensonal over C. Thereore, we can apply results on Hlbert polynomals to the assocated graded gr B D n -module.

9 3.2 Functors or holonomc D-modules 89 Proposton () Let F be a good ltraton on a non-zero module M Mod (D n ) wth respect to the Bernsten ltraton. Then there exsts a unque polynomal χ(m, F ; T) Q[T ] such that χ(m, F ; ) = dm C F M ( 0). () I the degree o χ(m, F ; T)s d, then the coecent o the degree d(the hghest degree) part o χ(m, F ; T)s m/d! or some nteger m>0. These two ntegers d and m do not depend on the choce o the good ltraton F. They depend only on M tsel. Proo. By dm C F M = k dm C gr F k M most o the statements are well known n algebrac geometry [Ha2, Chapter 1]. Let us show that d and m are ndependent o the choce o a good ltraton. Let F and F be good ltratons o a ntely generated D n -module M. By Proposton D.1.3 there exsts 0 > 0 satsyng F 0 M F M F + 0 M, and hence χ(m, F ; 0 ) χ(m, F ; ) χ(m, F ; + 0 ) or 0. The desred result easly ollows rom ths. We call d = d B (M) the dmenson o M, and m = m B (M) the multplcty o M. Proposton Let 0 L M N 0 be an exact sequence o ntely generated D n -modules. () We have d B (M) = Max{d B (L), d B (N)}. () We have m B (L) + m B (N) m B (M) = m B (L) m B (N) (d B (L) = d B (N)), (d B (L) > d B (N)), (d B (L) < d B (N)). Proo. Take a good ltraton F on M. Wth respect to the nduced ltratons on L and N we have an exact sequence 0 gr F L gr F M gr F N 0 o graded gr B D n -modules. The desred result ollows rom ths. Proposton For a non-zero ntely generated D n -module M we have dm Ch( M) = d B (M).

10 90 3 Holonomc D-Modules Proo. Set (M) := Mn{ Ext D n (M, D n ) = 0 }. By applyng Theorem D.4.3 to the two ltratons F and B o D n we have dm Ch( M) = 2n (M) = dm supp( gr F M), where F s a good ltraton on M wth respect to the Bernsten ltraton and gr F M denotes the correspondng coherent O C 2n-module. It s well known n algebrac geometry that we have dm supp( gr F M) = d B (M) [Ha2, Chapter 1]. By Proposton a coherent D C n-module M assocated to M s holonomc and only d B (M) = n. We can use the ollowng estmate as a useul crteron or the holonomcty o M. Proposton Let M be a (not necessarly ntely generated) non-zero D n - module. We assume that M has a ltraton F bounded rom below (wth respect to the Bernsten ltraton B o D n ) such that there exst constants c, c satsyng the condton dm C F M c n! n + c n 1 or any. Then M s holonomc and m B (M) c. Proo. We rst show that any ntely generated non-zero D n -submodule N o M s holonomc and satses m B (N) c. Take a good ltraton G on N. By Proposton D.1.3 we have G N N F +0 M F +0 M ( ) or some 0, and hence χ(n, G; ) c n! ( + 0) n + c ( + 0 ) n 1. It ollows that d B (N) n. By N = 0 and d B (N) = dm Ch(N) we obtan d B (N) = n and m B (N) c. It remans to show that M s ntely generated. It s sucent to show that any ncreasng sequence 0 = N 1 N 2 M o ntely generated submodules o M s statonary. We have shown that N s holonomc and satses m B (N ) c. Moreover, we have m B (N 1 ) m B (N 2 ) m B (N 3 ) c by Proposton , and hence the sequence {m B (N )} s statonary. Ths mples the desred result by Proposton Now we are ready to gve proos o Proposton and Proposton

11 3.2 Functors or holonomc D-modules 91 Proo o Proposton Set N = Ɣ(C n, M). By the denton o B and the Fourer transorm we have d B (N) = d B ( N). Hence by Proposton we have dm Ch(M) = dm Ch( M). Ths mples the desred result. Proo o Proposton Set N = Ɣ(C n, M). Note that Ɣ(C n,h 0 ( M)) s somorphc to the localzaton N x1 = C[x, x 1 1 ] C[x] N. Hence t s sucent to show that N x1 s holonomc. Take a good ltraton F o N and dene F N x1 to be the mage o F 2 N s x 1 s N x 1. It s easly checked that ths denes a ltraton o N x1 wth respect to the Bernsten ltraton. Moreover, we have dm C F N x1 dm C F 2 N = m B(M) (2) n + O( n 1 ) n! = m B(M)2 n n + O( n 1 ), n! and hence N x1 s holonomc by Proposton Aduncton ormulas Let : X Y be a morphsm o smooth algebrac varetes. Denton We dene new unctors by := D Y D X : Dh b (D X) Dh b (D Y ),! := D X D Y : D b h (D Y ) D b h (D X). Theorem For M Dh b(d X) and N Dh b(d Y ) we have natural somorphsms ( ) RHom DY M, N R RHom DX (M, N ), Proo. We have! R RHom DX ( N, M ) RHom DY (N, ) M. R RHom DX (M, N ) (( R X L O X D X M ) ) L D X N [ d X ] (( R X L O X D X M ) ) L D X D X Y L 1 1 N [ d D Y Y ] (( ) R X L O X D X M ) L D X D X Y L D Y N [ d Y ]

12 92 3 Holonomc D-Modules ) ( Y LOY D X M L D Y N [ d Y ] ) ( Y LOY D Y M L D Y N [ d Y ]! ( ) RHom DY M, N.! The rst somorphsm s establshed. The second somorphsm ollows rom the rst by dualty. By applyng H 0 (RƔ(Y, )) to the somorphsms n Theorem , we obtan the ollowng. Corollary For M Dh b(d X) and N Dh b(d Y ) we have natural somorphsms ( ) Hom D b h (D Y ) M, N! Hom D b h (D X ) (M, N ), Hom D b h (D X ) ( N, M ) Hom D b h (D Y ) Namely,! (resp. ) s the let adont o (resp. ). Theorem There exsts a morphsm o unctors : Dh b (D X) Dh b (D Y ).! Moreover, s proper, then ths morphsm s an somorphsm. ( N, ) M. Proo. By Hronaka s desngularzaton theorem [H], there exsts a smooth completon X o X. Snce X s quas-proectve, a desngularzaton X o the Zarsk closure X o X n the proectve space s such a completon (even X s not quas-proectve, there exsts a smooth completon by a theorem due to Nagata). Thereore, the map : X Y actorzes as X g X Y X Y p Y, where g s the graph embeddng assocated to and p = pr Y s a proecton. In ths stuaton, g and p are proper and s an open embeddng. Ths mples that we can reduce our problem to the cases o proper morphsms and open embeddngs. I s proper, we have an somorphsm = D Y D X!

13 3.3 Fnteness property 93 by Theorem So let us consder the case when = : X Y s an open embeddng. Let M D b h (D X). By Corollary we have ( Hom D b h (D Y ) M,! and hence we obtan the desred morphsm ) M Hom D b h (D X ) (M, M ) Hom D b h (D X ) (M, M ), M! M as the mage o d Hom D b h (D X ) (M, M ). 3.3 Fnteness property The am o ths secton s to show the ollowng. Theorem The ollowng condtons on M D b c (D X) are equvalent: () M D b h (D X). () There exsts a decreasng sequence X = X 0 X 1 X m X m+1 = o closed subsets o X such that X r \ X r+1 s smooth and all o the cohomology sheaves H k ( r M ) are ntegrable connectons, where r : X r \ X r+1 X denotes the embeddng. () For any x X all o the cohomology groups H k ( xm ) are nte dmensonal over C, where x :{x} X denotes the ncluson. For the proo we need the ollowng. Lemma Let M be a coherent (but not necessarly holonomc) D X -module. Then there exsts an open dense subset U X such that M U s proectve over O U. Proo. Take a good ltraton F o M. Then gr F M s coherent over π O T X. It ollows rom a well-known act on coherent sheaves that there exsts an open dense subset U X such that (gr F M) U s ree over π O T U. By shrnkng U necessary we may assume that (gr F M) U s ree over O U. Ths mples that each (F M/F 1 M) U (and hence each F M U ) s proectve over O U. Consequently M U s proectve over O U. Proo o Theorem () (). Set U r = X\X r. We wll show M Ur D b h (D U r ) by nducton on r. Assume M Ur D b h (D U r ). Let : U r U r+1, : X r \ X r+1 (= U r+1 \ U r ) U r+1 be embeddngs. Then we have a dstngushed trangle

14 94 3 Holonomc D-Modules (M Ur+1 ) M Ur+1 (M Ur+1 ) +1. By (M Ur+1 ) = r M D b h (D X r \X r+1 ) we have (M Ur+1 ) D b h (D U r+1 ). On the other hand by (M Ur+1 ) = M U r D b h (D U r ) we have (M Ur+1 ) D b h (D U r+1 ). Hence the above dstngushed trangle mples M Ur+1 D b h (D U r+1 ). () (). By Theorem we have xm Dh b(d {x}). Note that D {x} C. Hence the desred result ollows rom the act that obects o Dh b(d {x}) = Dc b(d {x}) = Dc b (Mod(C)) are ust complexes o vector spaces whose cohomology groups are nte dmensonal. () (). It s sucent to show that or any closed subset Y o X satsyng Y supp(m ) := k supp(h k (M )) there exsts a decreasng sequence Y = Y 0 Y 1 Y m Y m+1 = o closed subsets o Y such that Y r \ Y r+1 s smooth and all o the cohomology sheaves H k ( r M ) are ntegrable connectons, where r : Y r \Y r+1 X denotes the embeddng. We wll prove ths statement by nducton on dm Y. Take an open dense smooth subset V o Y, and let : V X denote the embeddng. By Kashwara s equvalence we have M D b c (D V ). Hence by Lemma there exsts an open dense subset V o V such that each cohomology shea H k ( M ) V s proectve over O V. Thereore, by shrnkng V necessary we may assume rom the begnnng that each cohomology shea H k ( M ) s coherent over D V and proectve over O V. We rst show that H k ( M ) s an ntegrable connecton. Take x V and denote by x :{x} V the embeddng. Then we have C OV,x H k ( M ) x H k+d V ( x M ) H k+d V ( xm ), where the rst somorphsm ollows rom the act that H k ( M ) x s proectve over O V,x. Hence the nte-dmensonalty o H k+d V ( xm ) mples that the rank o the proectve O V,x -module H k ( M ) x s nte. It ollows that H k ( M ) s coherent over O V, hence an ntegrable connecton. Now take an open subset U o X such that V = Y U, and let : U X be the embeddng. Dene N by the dstngushed trangle N M +1 M. We easly see that M M D b h (D X), and hence the above dstngushed trangle mples N Dc b(d X). We also easly see that supp(n ) Y \V. Moreover, or any locally closed smooth subset Z o Y \ V we have Z M Z N, where Z : Z X denotes the embeddng. Indeed, we have Z = 0 by Proposton (). In partcular, or any x Y \ V we have H ( xm ) H ( xn ). Hence by applyng the hypothess o nducton to N there exsts a decreasng sequence Y \ V = Y 1 Y m Y m+1 =

15 3.4 Mnmal extensons 95 o closed subsets o Y \ V such that Y r \ Y r+1 s smooth and all o the cohomology sheaves H k ( r M ) are ntegrable connectons, where r : Y r \ Y r+1 X denotes the embeddng. Then the decreasng sequence Y = Y 0 Y 1 Y m Y m+1 = satses the desred property. 3.4 Mnmal extensons A non-zero coherent D-module M s called smple t contans no coherent D- submodules other than M or 0. Proposton mples that or any holonomc D-module M there exsts a nte sequence M = M 0 M 1 M r M r+1 = 0 o holonomc D-submodules such that M /M +1 s smple or each (Jordan Hölder seres o M). In ths secton we wll gve a classcaton o smple holonomc D- modules. More precsely, we wll construct smple holonomc D-modules rom ntegrable connectons on locally closed smooth subvaretes usng unctors ntroduced n earler sectons, and show that any smple holonomc D-module s o ths type. Ths constructon corresponds va the Remann Hlbert correspondence to the mnmal extenson (Delgne Goresky MacPherson extenson) n the category o perverse sheaves. Let Y be a (locally closed) smooth subvarety o a smooth algebrac varety X. Assume that the ncluson map : Y X s ane. Then D X Y s locally ree over D Y and R = (hgher cohomology groups vansh). Thereore, or a holonomc D Y -module M we have H M = H! M = 0 or = 0. Namely, we may regard M and! M as D X-modules. These D X -modules are holonomc by Theorem By Theorem we have a morphsm M M n Mod h (D X ).! Denton We call the mage L(Y, M) o the canoncal morphsm! M M the mnmal extenson o M. By Proposton the mnmal extenson L(Y, M) s a holonomc D X -module. Theorem () Let Y be a locally closed smooth connected subvarety o X such that : Y X s ane, and let M be a smple holonomc D Y -module. Then the mnmal extenson L(Y, M) s also smple, and t s characterzed as the unque smple submodule (resp. unque smple quotent module) o M(resp. o! M).

16 96 3 Holonomc D-Modules () Any smple holonomc D X -module s somorphc to the mnmal extenson L(Y, M) or some par (Y, M), where Y s as n () and M s a smple ntegrable connecton on Y. () Let (Y, M) be as n (), and let (Y,M ) be another such par. Then we have L(Y, M) L(Y,M ) and only Y = Y and M U M U or an open dense subset U o Y Y. Proo. () We choose an open subset U X contanng Y such that k : Y U s a closed embeddng. Let : U X be the embeddng, and let Mod Y qc D X denote the category o O X -quas-coherent D X -modules whose support s contaned n Y. We rst show the ollowng our results: (a) For any E Mod Y qc (D X) we have H l E = 0 (l = 0). Hence H 0 = : Mod Y qc (D X) Mod qc (D Y ) s an exact unctor. (b) For any non-zero holonomc submodule N o M, we have N M. (c) M (resp.! M) has a unque smple holonomc submodule (resp. smple holonomc quotent module). (d) For a sequence 0 = N 1 N 2 M o holonomc submodules o M, we have (N 2 /N 1 ) = 0. For E Mod Y qc (D X) we have E = k E = k 1 E and supp 1 E Y. Hence (a) s a consequence o Kashwara s equvalence. Let N be as n (b). By Corollary we have Hom DX (N, ) ( M = Hom DX N, Hom DU ( N, k k ) M ) M. Snce s an open embeddng, we have = = 1. Thereore, the ncluson N M nduces a non-zero morphsm ϕ : N k M. Snce k M s a smple holonomc D U -module by Kashwara s equvalence, ϕ s surectve. Applyng k to t, we obtan a surectve morphsm N k k M M. On the other hand, we have an nectve morphsm N M = M because s exact by (a). Hence we must have N M, and (b) s proved. Suppose there exst two smple holonomc submodules L = L o M. Set N = L + L = L L. Then by (b) we have M N = L L = M M, whch s a contradcton. The asserton (c) or M s proved. Another asserton or! M s easly proved usng the dualty unctor. By (a) we have N 1 N 2 M = M, N 2 / N 1 (N 2 /N 1 ).

17 3.4 Mnmal extensons 97 Hence (b) mples (N 2 /N 1 ) = 0, and (d) s proved. Now let us nsh the proo o (). By (c) there exsts a unque smple holonomc submodule L o M. By Corollary there exst two somorphsms ( ) Hom DX M, L Hom DY (M, L) (b) Hom DY (M, M),! ( ) ) Hom DX M, M Hom DY (M, M Hom DY (M, M),! rom whch we see that the canoncal morphsm! M M s non-zero and actorzes as! M L M. Snce L s a smple module, the mage o ths morphsm should be L. Ths completes the proo o (). () Assume that L s a smple holonomc D X -module. We take an ane open dense subset Y ( : Y X) o an rreducble component o supp L so that L s an ntegrable connecton on Y (ths s possble by Proposton 3.1.6). Set M = L. We easly see by Proposton that M s smple. Moreover, by Corollary we get an somorphsm ( ) Hom DX M, L Hom DY (M, L) Hom DY (M, M) = 0,! rom whch we see that there exsts a non-zero surectve morphsm! M L. Namely, L s a smple holonomc quotent module o! M. Hence we obtan L = L(Y, M) by (). The asserton () s proved. The proo or the last part () s easy and let to the readers. Proposton Let Y be a locally closed smooth subvarety o X such that : Y X s ane, and let M be an ntegrable connecton on Y. Then we have D X L(Y, M) L(Y, D Y M). Proo. By the exactness o the dualty unctor we obtan D X L(Y, M) Im(D X M D X M) Im( = L(Y, D Y M).!! D Y M D Y M) The proo s complete.