CHAPTER 10 D-MODULES ENRICHED WITH ASESQUILINEARPAIRING

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1 CHAPTER 10 D-MODULES ENRICHED WITH ASESQUILINEARPAIRING Summary. Our aim in this chapter is to extend the notion of (r, r)-flat sesquilinear pairing, used for the definition of a variation of C-Hodge structure (see Definition 4.1.3), to the case where the flat bundle is replaced with a D-module. The case of a holonomic D-module will of course be the most interesting for us, but it is useful to develop the notion with some generality in order to apply derived functors. In this chapter, we keep Notation However, we will only consider O X -modules and D X -modules, as coherent filtrations will not play any role here. We will use the constructions and results of Chapter 7 in this framework Introduction One of the ingredients of a variation of polarized Hodge structure is a flat Hermitian pairing (that we have denoted by Q), which is ( 1) p -definite on H p,w p. In this chapter, we introduce the notion of sesquilinear pairing between holonomic D X - modules. It takes values in the sheaf of distributions (in fact a smaller sheaf, but we are not interested in characterizing the image). This notion will not be used directly as in classical Hodge theory to furnish the notion of polarization. Instead, we will take up the definition of a C-Hodge structure as a triple (see Section 2.4.c) and mimic this definition in higher dimension. Our aim is therefore to define a category of D-triples (an object consists of a pair of D X -modules and a sesquilinear pairing between them) and to extend to this abelian category the various functors considered in Chapter Sesquilinear pairings for D X -modules and the category D-Triples 10.2.a. Distributions an currents on a complex manifold. Let X denote the complex manifold conjugate to X, i.e., with structure sheaf O X defined as the sheaf of anti-holomorphic functions O X. Correspondingly is defined the sheaf of antiholomorphic differential operators D X. The sheaf of C 1 functions on X is acted on

2 218 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING by D X and D X on the left and both actions commute, i.e., CX 1 is a left D X C D X - module. Similarly, the sheaf of distributions Db X is a left D X C D X -module: by definition, on any open set U X, Db X (U) is dual to the space Ac 2n (U) of C 1 2n-forms with compact support, equipped with a suitable topology, and the presheaf defined in this way is a sheaf. On the other hand, the space of C X (U) of currents of degree 0 on X is dual to Cc 1 (U) with suitable topology. Then C X is the right D X C D X -module obtained from Db X by the left-to-right transformation for such objects, i.e., C X =( X C X ) (OX O X ) Db X. The stupid conjugation functor M 7 M transforms O X -modules (resp. D X -modules) into O X -modules (resp. D X -modules): let us regard O X as an O X -module by setting f g := fg, andsimilarlyletusregardd X as a D X -module; for an O X -module (resp. a D X -module) M we then define M as O X OX M (resp. D X DX M). In other words, for a local section m of M, wedenotebym the same local section, that we act on by f 2 O X (resp. D X ) with the formula f m := fm. Notation From now on, the notation A X,X will mean A X C A X (A = O or D). One can easily adapt Exercise A.5.5 to prove that the C 1 -de Rham complex E 2n+ X C 1 X D X,X = E X O X,X D X,X [2n], where the differential is obtained from the standard differential on C 1 k-forms and the universal connection r X + r X on = E X 2n X,X -module. We denote by Db n p,n q X = E n p,n q X C 1 X Db X or Db X,p,q the sheaf of currents of degree (p, q) (we also say of type (n p, n q)), that is, continuous linear forms on Cc 1 differential forms of degree p, q. The distributional de Rham complex gives then a resolution of C X as a right D X,X - module: D X,X,isaresolutionofE n,n X (10.2.2) Db X[2n] OX,X D X,X CX. Let us make precise that the morphism is induced by Db n,n X D X,X = C X D X,X C X, u P 7 u P b. Sesquilinear pairings. Let us start with the case of right D X -modules. Definition (Sesquilinear pairing). (1) A sesquilinear pairing c between right D X -modules M 0, M 00 is a D X,X -linear morphism c : M 0 C M 00 C X. When M 0 = M 00 = M, wespeakofasesquilinear pairing on M. (2) The adjoint of c is c : M 00 C M 0 C X defined by c (m 00, m 0 )=c(m 0, m 00 ), with hc(m 0, m 00 ), i := hc(m 0, m 00 ), i for any test function. Weclearlyhavec = c.

3 10.2. SESQUILINEAR PAIRINGS FOR D X -MODULES AND THE CATEGORY D-Triples 219 Remark (Extension to C 1 coefficients). Let us define a right action of D X,X on M OX CX 1 by setting (m x i = m@ xi /@x i and (m xi = /@x i. Then c extends in a unique way as a CX 1 -linear morphism (M 0 OX C 1 X ) C 1 X (M 00 OX C 1 X ) C X which satisfies, for any local section of X or X, by setting c(µ 0, µ 00 ) = c(µ 0,µ 00 )+c(µ 0, µ 00 ), c(m 0 0, m ):=c(m 0, m 00 ) Conversely, given such a pairing, one recovers the original c by restricting to M 0 C M 00. Remark (The case of left D X -modules and side-changing) If M 0, M 00 are left D X -modules and c = c left : M 0 C M 00 Db X is a sesquilinear pairing, that is, a left D X,X -linear morphism, then it determines in a canonical way asesquilinearpairing ( ) c right :( X M 0 ) C ( X M 00 ) X X Db X = C X ( 0 m 0, 00 m 00 ) 7 0 ^ 00 c left (m 0, m 00 ). Conversely, from a sesquilinear pairing between right D X -modules one recovers one for left D X -modules. The compatibility with adjunction is given by the following relation: ( ) (c right ) =( 1) n (c ) right, since 0 ^ 00 =( 1) n 00 ^ 0. Let us notice the following. Lemma If M 0 and M 00 are O X -coherent (hence O X -locally free of finite rank), the pairing c takes values in C 1 forms of maximal degree (resp. functions). Proof. Weknow(seeExampleA.4.b)thatM 0, M 00 are O X -generated by their flat local sections. For such local sections m 0,m 00,thecurrent(resp.distribution)c(m 0, m 00 ) is annihilated hence is locally a constant. It follows that, for any local sections m 0,m 00, c(m 0, m 00 ) is real-analytic, so in particular C c. The category of D-triples. The category D-Triples(X) is a prototype, without any Hodge filtration, of the category of Hodge modules. It serves as a model for it, but is also a constituent of it, as we will see in Chapter 12. It is an abelian category, and possesses the basic functors we need for studying pure Hodge modules, as a consequence of the results of the previous sections. Moreover, we will define sign rules for various functors in order to eliminate various signs which have appeared in the previous formulas, and to be compatible with the notion of Lefschetz structure (see Chapter 3) when possible.

4 220 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING Definition The category D-Triples(X) has objects consisting of triples T =(M 0, M 00, c), where M 0, M 00 are D X -modules and c is a sesquilinear pairing between them (with values in Db X in the left case, and in C X in the right case), morphisms ' : T 1 T 2 consisting of pairs ' =(' 0,' 00 ), where ' 0 : M 0 1 M 0 2 and ' 00 : M 00 2 M 00 1 are D X -linear, such that for all local sections m 0 1 of M 0 1 and m 00 1 of M 00 2, ( ) c 1 (m 0 1, ' 00 (m 00 2 )) = c 2(' 0 (m 0 1), m 00 2 ). In particular, D-Triples(X) is an abelian subcategory of Mod(D X ) Mod(D X ) op. We say that an object T of D-Triples(X) is coherent, resp. R-specializable, resp. smooth, if its components M 0, M 00 are D X -coherent, resp. R-specializable, resp. O X -locally free of finite rank. Definition (Side-changing in D-Triples(X)). Let T =(M 0, M 00, c) be a left D X - triple. We set T right := (M 0right, M 00right, c right ), where c right is defined by ( ). The right-to-left side changing is defined correspondingly, so that the composition of both is the identity. Definition (Adjunction). The adjoint of an object T = (M 0, M 00, c) of D-Triples(X) is the object T := (M 00, M 0, c ). The adjoint of a morphism ' =(' 0,' 00 ) is the morphism ' := (' 00,' 0 ).WeclearlyhaveT = T and ' = '. Remark (Adjoint of a graded triple). Let T = L k T k be a graded object in D-Triples(X). We write T k as (M 0 k, M00 k, c k). The adjoint object is then T = L k Tk := L (T k ). k Remark (Side-changing and adjunction in D-Triples(X)) With the previous definitions, adjunction does not commute with side-changing when n is odd, because of the sign in ( ). Definition (Pre-polarization, see Definition ). A pre-polarization of weight w of an object T of D-Triples(X) is a morphism Q : T T which is ( 1) w -Hermitian. In the graded category, we ask Q to be graded. We say Q is non-degenerate if it is an isomorphism. Let us make this definition more explicit. We write Q=(( amorphismm 0 M 00 which satisfies c(m 0 1, Qm 0 2 )=( 1)w c (Qm 0 1, m 0 2 )) := ( 1)w c(m 0 2, Qm 0 1 ), and the sesquilinear pairing on M 0 (m 0 1,m 0 2) 7 c(m 0 1, Qm 0 2 ) 1) w Q, Q), where Q is

5 10.2. SESQUILINEAR PAIRINGS FOR D X -MODULES AND THE CATEGORY D-Triples 221 is ( 1) w -Hermitian in the usual sense. In the graded case, Q k =(Q k, ( 1) w Q k ), where each Q k is an isomorphism M 0 k M00 k which satisfies c k (m 0 k, Q k m 0 k )=( 1)w c k (m 0 k, Q km 0 k ), so the graded sesquilinear pairing M 0 k M00 k C X (or Db X ) (m 0 k,m 0 k) 7 c k (m 0 k, Q k m 0 k ) is ( 1) w -Hermitian in the graded sense. The following is shown as in Remark Lemma Any non-degenerate pre-polarized triple (T, Q) of weight w is isomorphic to a triple T 1 =(M 1, M 1, c 1 ) such that c 1 =( 1) w c 1 (i.e., with polarization (( 1) w Id, Id)). Similarly, any non-degenerate graded pre-polarized triple (T, Q ) of weight w is isomorphic to a graded triple T 1 =(M 1, M 1, c 1 ) such that c 1,k =( 1)w c 1, k : M 1, k M 1,k C X (or Db X ). In other words, working with non-degenerate pre-polarized triples of weight w amounts to working with pairs (M, c), where c is a ( 1) w -Hermitian pairing M M C X (or Db X ). Given such a pair, we associate to it the object (M, M, c) of D-Triples(X) with pre-polarization (( 1) w Id, Id). Example (Two basic examples). (1) (Left case) The triple T O X = (O X, O X, c n ) is the smooth left triple with c n (1, 1) = 1. It satisfies ( T O X ) = T O X. The identity morphism Q = (Id, Id) : TO X ( T O X ) is an Hermitian non-degenerate pre-polarization. (2) (Right case) The triple T X =( X, X, c n ) is the smooth right triple with c n ( 0, 00 )= 0 ^ 00.Wehavec n =( 1) n c n,andthemorphismq=(( 1) n Id, Id) : T X ( T X ) is a ( 1) n -Hermitian non-degenerate pre-polarization. Definition (Side-changing for a pre-polarization). Let T be a left D X -triple and let Q=(Q 0, Q 00 ) be a pre-polarization of weight w of it. We then set Q right := (( 1) n Id Q 0, Id Q 00 ), which is a pre-polarization of T right, with the natural morphisms Id Q 0, Id Q 00 : X M 0 X M 00. This defines a pre-polarization of weight w+n of T right.moreover,q is non-degenerate if and only if Q right is so. Note that, for a ( ( X M, c right ). 1) w -Hermitian left pair (M, c), the associated right pair is Remark (Lefschetz triples). The notion of (graded) Lefschetz structure (T, N) in the abelian category D-Triples(X) is obtained from Definition Using adjunction in D-Triples(X) (Definition ), we obtain as in Definition the notion of adjunction and pre-polarization of weight w of a (graded) Lefschetz D-triple (T, N).

6 222 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING Any Lefschetz D-triple (T, N) with a ( 1) w -Hermitian non-degenerate prepolarization Q is isomorphic to one of the form ((M, M, c), (N, N)), with c =( 1) w c, and Q=(( 1) w Id, Id). It is thus determined by the data (M, c, N), suchthatc is ( 1) w -Hermitian and N is skew-adjoint with respect to c Pushforward in the category D-Triples(X) 10.3.a. Pushforward of currents. Let be a C 1 form of maximal degree on X. If f : X Y is a proper holomorphic map which is smooth, thentheintegralof in the fibres of f is a C 1 form of maximal degree on Y,thatonedenotesby R f. If f is not smooth, then R is only defined as a current of degree 0 on Y, f and the definition extends to the case where is itself a current of degree 0 on X (see Appendix A.4.d for the notion of current). Exercise (Pushforward of the sheaf of currents as a right D X,X -module) Extend the notion and properties of direct image of a right (resp. left) D X,X - module, by introducing the transfer module D XY,XY = D XY C D XY. One denotes these direct images by D,D f or D,D f. In particular, D,Df C X := Rf (C X DX,X Sp XY,XY (D X,X )). Definition (Integration of currents of degree (p, q)). Let f : X Y be a proper holomorphic map and let u be a current of degree (p, q) on X. The current R f u of degree (p, q) on Y is defined by D Z f E u, = hu, fi, 8 2 E p,q c (Y ). This definition extends in a straightforward way if f is only assumed to be proper on the support of u. We continue to assume that f is proper. We will now show how the integration of currents is used to defined a natural D Y,Y morphism H 0 D,Df C X C Y.Letusfirst treat as an exercise the case of a closed embedding. Exercise Assume that X is a closed submanifold of Y and denote by : X, Y the embedding (which is a proper map). Denote by 1 the canonical section of D XY,XY.Showthatthenaturalmap Z H 0 D,D C X = C X DX,X D XY,XY C Y, u 1 7 u induces an isomorphism of the right D Y,Y -module H 0 D,D C X with the submodule of C Y consisting of currents supported on X. [Hint: usealocalcomputation.] For example, consider the case : X = X {0}, X C, with coordinate t on C and identify H 0 D,D C X with C X [@ t,@ t ].

7 10.3. PUSHFORWARD IN THE CATEGORY D-Triples(X) 223 The integration of currents is a morphism Z : f Db X,p,q Db Y,p,q, f which is compatible with the d 0 and d 00 differentials of currents on X and Y. In other words, taking the associated simple complex, it is a morphism of complexes Z : f Db X[2n] Db Y [2m]. f Let us notice that the integration of currents is compatible with conjugation. Namely, given a current u p,q 2 (X, Db n p,n q X ), its conjugate u p,q 2 (X, Db n q,n p X ) is defined by the relation hu p,q, q,p i := hu p,q, q,p i for any test form q,p. Then we clearly have Z Z (10.3.4) u p,q = f f u p,q. Since C X = (Db X ) right,right as a right D X,X -module, we can apply Exercise A.8.24(4) to get, since f is proper, (10.3.5) D,Df C X ' f (Db X[2n] f 1 O Y,Y f 1 D Y,Y )=f Db X[2n] OY,Y D Y,Y. The integration of currents R f induces then a D Y,Y -linear morphism of complexes Z (10.3.6) : D,D f C X Db Y [2m] OY,Y D Y,Y ' C Y, f where we recall that the differential on the complex Db Y [2m] OY,Y D Y,Y uses the universal connection r Y + r Y on D Y,Y, and the isomorphism with C Y is given by (10.2.2). If we star from distributions, we have a morphism Z (10.3.7) : D,D f Db X = D,D f C X [2(m n)] C Y [2(m n)]. f Exercise Extend the result of Exercise A.8.20 to the case of right D X,X -modules and show that the composed map f C X H 0 D,Df C X C Y is the integration of currents of Definition Exercise Let f : X Y be a holomorphic map and let Z X be a closed subset on which f is proper. (1) Define the sub-d X,X -module C X,Z of C X consisting of currents supported on Z. (2) Show that the integration of currents R f induces a D Y,Y -linear morphism of complexes Z : D,D f C X,Z Db Y [2m] OY,Y D Y,Y ' C Y. f

8 224 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING 10.3.b. Pushforward of a sesquilinear pairing. In order to define the pushforward of a sesquilinear pairing in the case of right D-modules, it will be convenient to start from left D X -modules as in Exercise A.8.24, and use side changing at the source to get the right definition. Let M 0left, M 00left be left D X -modules, let M 0right, M 00right be the associated D X -modules and let f : X Y be a holomorphic map which is proper when restricted to Z := Supp M 0 [ Supp M 00. Let c left : M 0left C M 00left Db X,Z be a sesquilinear pairing and let c right the corresponding right sesquilinear pairing. Our aim is to define, for every k 2 Z, asesquilinearpairing: ( ) D,Df k c right : H k ( D f M 0right ) C H k ( D f M 00right ) C Y. In order to integrate differential forms (and not poly-vector fields) we will use the formula of Exercise A.8.24(1) for computing the direct image as a complex of right D Y -modules, namely, Df M right Rf X(M left f 1 O Y f 1 D Y )[n], where the isomorphism is induced termwise by the morphism in Lemma A.5.8. Moreover, it will be convenient to compute the direct image Rf by using flabby sheaves more adapted to the computation than the Godement sheaves. According to Remark , it is enough to define the C 1 extension of D,D f k c, so we will use the formula Rf X(M left f 1 O Y f 1 D Y ) OY CY 1 f E X(M left f 1 O Y f 1 D Y ), obtained from the Dolbeault resolution k X (E (k, ), d 00 ) and by taking the associated simple complex. Lastly, we identify each term of this complex with ( ) f (E X OX M left ) OY D Y and, with this identification, the differential is given by the formula (d IdM left) Id + (Id r) Id + (Id Id) f f r Y, where r Y is the universal connection on D Y. The C 1 extension of c left is denoted by c left 1,anditinducesamorphism ( ) and by applying f, c left 1 :(E k X OX M 0left ) C 1 X (E `X O X M 00left ) Db k+` X,Z ( 0k m 0 ) 00` m k ^ 00`c left (m 0, m 00 ), f c left 1 : f (E k X OX M 0left ) f C 1 X f (E `X O X M 00left ) f Db k+` X,Z, so, by right D Y,Y -linearity, a morphism ( D,D f c left 1 ) right : f (E k X OX M 0left ) OY D Y C 1 Y f (E `X O X M 00left ) OY D Y f (Db k+` X,Z ) O Y,Y D Y,Y. The compatibility of c left with the connections on M 0left, M 00left implies that this morphism is compatible with the differentials, so that, with respect to the identifications

9 10.3. PUSHFORWARD IN THE CATEGORY D-Triples(X) 225 above and according to (10.3.5), we get a morphism of complexes of right D Y,Y - modules D,Df c right 1 :( D f M 0right OY C 1 Y ) C 1 Y ( D f M 00right OY C 1 Y ) D,Df C X,Z. Composing with the integration of currents (see Exercise ) Z : D,D f C X,Z C Y f we finally get a morphism of complexes of right D Y,Y -modules (where C Y is regarded as a complex having a single term in degree zero) that we denote by the same symbol: D,Df c right 1 :( D f M 0right OY C 1 Y ) C 1 Y ( D f M 00right OY C 1 Y ) C Y. By restricting to the holomorphic/anti-holomorphic part (and side changing from right to left in the left case), we obtain the pushforward morphism D,Df c right : D f M 0right C D f M 00right C Y. Forgetting now the right exponent on c, wedenoteby D,D f c the induced morphism ( ) D,Df k c : H k Df M 0right C H k Df M 00right C Y. Remark (Making explicit the pairing D,D f k c). We assume that f is proper on Supp M 0 and Supp M 00, so that we will consider the ordinary pushforward f.letu be an open set in Y,andlet m 0n+k 1 2 (U, f (E n+k X OX M 0left )), m 00n k 1 2 (U, f (E n k X OX M 00left )). Then R f f c left 1 (m 0n+k 1, m 00n 1 k ) belongs to (U, C Y ). If the section m 0n+k 1 1 of f (E n+k X O X M 0left ) D Y (resp. m 00n 1 k 1) is closed with respect to the differential of the complex ( ), then, denoting by [ ] the cohomology class, we get Z D,Df k c([m 0n+k 1 1], [m 00n k 1 1]) = f f c left 1 (m 0n+k 1, m 00n k 1 ) 2 (U, C Y ). Remark (Pushforward and adjunction). Let us denote by c k,` 1 the sesquilinear form ( ). Due to the relation 0k ^ 00` =( 1) k` 00` ^ 0k,wehave (c 1)`,k ( 00` m 00, 0k m 0 )= 00` ^ 0k c (m 00, m 0 ) =( k,` 1) k,`c1 ( 0k m 0, 00` m 00 ) =( k,` 1) k,`(c1 ) ( 00` m 00, 0k m 0 ). It follows that the behaviour with respect to adjunction of ( D,D f n k c left ) right (defined in a way similar to the previous formulas) is given by the relation Since we have ( D,D f n+k c left ) right =( 1) n+k ( D,D f n k c left ) right. D,Df k c right =( D,D f n+k c left ) right,

10 226 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING the behaviour with respect to adjunction is given by the formula ( ) D,Df k (c )=( 1) k ( D,D f k c) c. Pushforward of D-triples Remark (Rule of signs for the pushforward). Before defining the proper pushforward of an object of D-Triples(X) coh,wewillmodifythedefinitionofthepushforward D,Df k c of the sesquilinear pairing in order to eliminate various signs. For every k, we set (recall that "(k) =( 1) k(k 1)/2 ): ( ) Tf k c := "(n m + k) D,D f k, so that, since "(n m k) =( 1) k "(n m + k), ( ) becomes ( ) Tf k (c )=( T f k c). (The choice of "(n formula.) m + k) instead of "(k) will be justified by the side-changing Definition (Proper pushforward). Let T be an object of D-Triples(X) coh supported on Z and let f : X X 0 be a holomorphic map which is proper on Z. Then Tf k T is the object (H k Df M 0, H k Df M 00, T f k c) of D-Triples(X 0 ) coh. The total pushforward is the graded object L k T f k T, with T f k T in degree k. Example (Kashiwara s equivalence in D-Triples). Assume that : X, Y is a closed immersion and let M = M 0, M 00 be right D X -modules. We then have D M = H 0 D M = (M DX D XY ). Let 1 denote the canonical section of D XY = O X 1 O Y 1 D Y. It is a generator of D XY as a right 1 D Y -module. Any sesquilinear pairing c Y : H 0 D M 0 H 0 D M 00 C Y takes values in C Y,X and is determined by its restriction c Y X to (M 0 1) (M 00 1). Hence it takes the form D,D 0 c X.Forlocalsectionsm 0,m 00 of M 0, M 00,thecurrentofdegree0 D,D 0 c X (m 0 1, m 00 1) is the pushforward by of the current c(m 0, m 00 ). Together with the rule of signs ( ), we conclude that T 0 : D-Triples(X) coh 7 D-Triples X (Y ) coh is an equivalence of categories, compatible with adjunction. Remark (Pushforward of a ( 1) w -Hermitian pair). The behaviour of a prepolarized triple of weight w by pushforward is determined by the behaviour of the associated ( 1) w -Hermitian pair (see Lemma ). If (M, c) is a ( 1) w -Hermitian pair, then the pushforward ( D f M, T f c) graded ( 1) w -Hermitian pair.

11 10.3. PUSHFORWARD IN THE CATEGORY D-Triples(X) 227 Remark (The Lefschetz morphism). In the previous setting, let be a closed (1, 1)-form on X which is real, i.e.,suchthat =. This condition is satisfied if the cohomology class of is equal to c 1 (L ) for some line bundle L on X. The corresponding Lefschetz morphism L : D f M D f M[2] with M = M 0, M 00 (see Definition A.8.16 and Remark A.8.18) satisfies then D,Df k+2 c(l m 0, m 00 )= D,D f k c(m 0, L m 00 ), if m 0 (resp. m 00 )isalocalsectionof D f k M 0 (resp. of D f k 2 M 00 ), that is, because of "(n m + k + 2) = "(n m + k), ( ) Tf k+2 c(l m 0, m 00 )= T f k c(m 0, L m 00 ). We can thus define a Lefschetz morphism by anti-symmetrization ( ) L =(L 0, L 00 ) : T f k T T f k+2 T, ( L 0 := L on H k Df M 0, L 00 := L on H k 2 Df M 00. It is functorial with respect to T and satisfies L = L. Let (M, c) be a ( 1) w -Hermitian pair (corresponding to a pre-polarized triple of weight w). Then ( ) impliesthat,fork > 0, thesesquilinearpairing is ( 1) w+k -Hermitian. ( T f c) ( k) = T f k c(, L k ): Tf k M T f k M C Y 10.3.d. Pushforward of left D-triples. In a way analogous to ( ), we first define a sesquilinear pairing ( ) D,Df n m+k c left : H n m+k ( D f M 0left ) C H n m k ( D f M 00left ) Db Y. According to Exercise A.8.24(1), we have ( D f M left ) right Rf X(M left f 1 O Y f 1 D Y )[m] By using ( ) we obtain a morphism of complexes of right D Y,Y -modules ( D,D f c left 1 ) right :(( D f M 0left ) right OY C 1 Y ) C 1 Y (( D f M 00left ) right OY C 1 Y ) D,D f C X,Z [2(m n)] and by composing with the integration of currents, we finally get a morphism of complexes of right D Y,Y -modules ( D,D f c left 1 ) right :(( D f M 0left ) right OY C 1 Y ) C 1 Y (( D f M 00left ) right OY C 1 Y ) C Y [2(m n)], which restrict to the pushforward morphism, after going from right to left in the target Y : D,Df c left : D f M 0left C D f M 00left Db Y [2(m n)].

12 228 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING At the cohomology level, and omitting the left exponent on c, wedenoteby D,D f c the induced morphism ( ) D,Df n m+k c : H n m+k Df M 0left C H n m k Df M 00left Db Y. Remark (Pushforward and side-changing). With respect to the side-changing functor of Remark , the right pairing ( ) is obtained by side changing from ( ), if c right is obtained by side-changing from c left. This follows from the definition of both sesquilinear pairings, since both are defined from ( D,D f c left 1 ) right, namely, ( ) by applying the side-changing at the source, and ( ) at the target. In other words, for c = c left, If we define T f k c in the left case by ( D,D f n m+k c) right = D,D f k (c right ). ( ) Tf k c := "(k) D,D f k, the side-changing formula become ( ) ( T f n m+k c) right = T f k (c right ). Remark (Pushforward and adjunction). As in Remark , we use that to obtain D,Df n m+k c left = ( D,D f n+k c left ) right left, ( ) D,Df n m+k (c )=( 1) n m+k ( D,D f n m k c). Now ( ) becomes ( ) Tf n m+k (c )=( 1) n m ( T f n m k c). Then, if c is ( 1) w -Hermitian, then T f c is graded ( 1) w+n m -Hermitian. As a consequence, the pushforward of a ( 1) w -Hermitian pair (left case) is a graded ( 1) w+n m - Hermitian pair. The definition of the pushforward of a D-triple in the left case is similar to Definition The only difference is the grading, which is shifted by n m. Wethus set Tf n m+k T left =(H n m+k Df M 0, H n m k Df M 00, T f n m+k c). The total pushforward is the graded object T f n m+ T = L k f n m+k T T, with Tf n m+k T in degree k. Similarly, the pushforward of a left ( 1) w -Hermitian pair (M, c) is the graded pair ( T f n m+ M, T f n m+ c), which is a graded ( 1) w+n m -Hermitian pair. Lastly, for k > 0, thelefschetzmorphisml induces a sesquilinear pairing ( T f c) (n m k) = T f n m k c(, L k ): Tf n m k M T f n m k M Db Y which is ( 1) w+n m+k -Hermitian.

13 10.4. SPECIALIZATION IN D-Triples 229 Example (Kashiwara s equivalence in D-Triples left ) , we obtain that In the setting of Example T n m is an equivalence of categories. : D-Triples(X) left coh 7 D-Triples X (Y ) left coh Specialization in D-Triples 10.4.a. Specialization of a sesquilinear pairing. Let g : X C be a holomorphic function on X and let M 0, M 00 be D X -modules which are specializable along g =0.Assumethatcis a sesquilinear pairing between M 0 and M 00 with values in C X (right case) or Db X (left case). We wish to define sesquilinear pairings between the D X -modules g, M 0 and g, M 0 with values in C X or Db X. We start with the case where g is the projection X = X 0 C C and M 0, M 00 are specializable D X -modules along X 0 equipped with a sesquilinear pairing. We will denote by t the coordinate on C. In order to define a sesquilinear pairing on nearby cycles, we will use a Mellin transform device by considering the residue of c(m 0, m 00 ) t 2s at various values of s. It is important to notice that, while we need to restrict the category of coherent D X - modules in order to define nearby and vanishing cycles (i.e., to consider R-specializable coherent D X -modules only), the specialization of a sesquilinear pairing between them does not need any new restriction: any sesquilinear pairing between such D X -modules can be specialized. We assume that M 0, M 00 are right D X -modules which are R-specializable along X 0. Let c : M 0 C M 00 C X be a sesquilinear pairing. Fix x o 2 X 0. For local sections m 0,m 00 of M 0, M 00 defined in some neighbourhood of x o in X, thecurrentofdegree0 c(m 0, m 00 ) has some finite order p on some neighbourhood nb X (x o ). For 2Res>p, the function t 7 t 2s is C p,soforeverysuchs, c(m 0, m 00 ) t 2s is a section of C X on nb X (x o ). Moreover, for any test function with compact support in nb X (x o ), the function s 7 c(m 0, m 00 ) t 2s, := c(m 0, m 00 ), t 2s is holomorphic on the halfplane {2Res > p}. We say that c(m 0, m 00 ) t 2s depends holomorphically on s on nb X (x o ) {2Res>p}. Let (t) be a real C 1 function with compact support, which is 1 near t =0. In the following, we will consider test functions o (t), where o is a test function on a neighbourhood nb X0 (x o ) of x o in X 0. 7 Proposition Let M 0, M 00, c be as above. Then, for every x o 2 X 0, there exists an integer L > 0 and a finite set of real numbers satisfying t,exp(2 i ) M 0 x o 6=0 and t,exp(2 i ) M 00 x o 6=0, such that, for every element m 0 of M 0 x o and m 00 of M 00 x o, the correspondence ( ) o Y (s ) L c(m 0, m 00 ) t 2s, o (t)

14 230 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING defines, for every s 2 C, a section of C X0 respect to s 2 C. on nb X0 (x o ) which is holomorphic with The proposition asserts that the current of degree 0 o 7 c(m 0, m 00 ) t 2s, o extends as a current of degree 0 on nb X0 (x o ) depending meromorphically on s, with poles at s = k + (k 2 N) at most, and with a bounded order. We note that changing the function will modify the previous meromorphic current of degree 0 by aholomorphicone,as t 2s is C 1 for every s away from t =0. The proposition is a consequence of the following more precise lemma. Lemma Let x o 2 X 0 and let 0, 00 2 R. There exist L > 0 and a finite set of real numbers satisfying ( ) t,exp(2 i ) M 0 x o, t,exp(2 i ) M 00 x o 6=0, and 6 min( 0, 00 ), such that, for any sections m 0 2 V 0M 0 x o and m 00 2 V 00M 00 x o, the correspondence ( ) o 7 Y (s ) L c(m 0, m 00 ) t 2s, o (t) (t) defines, for every s 2 C, a section of C X0 respect to s 2 C. on nb X0 (x o ) which is holomorphic with Proof. Letb m 0(S) = Q 2R(m 0 ) (S ) ( ) be the Bernstein polynomial of m 0 (see Definition ), with ( ) bounded by the nilpotency index L of E z. It is enough to prove that the product Q 2R(m 0 ) (s ) ( ) of factors can be used in ( ) (recall that the function has no zeros and has simple poles at the non-positive integers, and no other poles). Indeed, arguing similarly for m 00 and using that the set of roots R(m 00 ) of b m 00(S) is real, one obtains that the product of factors indexed by R(m 0 )\R(m 00 ) can also be used in ( ). It is then easy to check that Conditions ( ) on are satisfied by any 2 R(m 0 ) \ R(m 00 ). We note first that, for every germ Q 2 V 0 D X,xo and any test function on nb X (x o ), the function Q ( t 2s ) is C p with compact support if 2Res>p. Applying this to the Bernstein operator Q = b m 0(E) P for m 0 (see Definition ), one gets 0= c(m 0, m 00 ) [b m 0(E) P ], t 2s (10.4.3) = c(m 0, m 00 ), [b m 0(E) P ] ( t 2s ) = b m 0(s) c(m 0, m 00 ), t 2s + c(m 0, m 00 ), t 2s t 1 for some 1, which is a polynomial in s with coefficients being C 1 with compact support contained in that of. As t 2s t is C p for 2Res +1>p,wecanargueby induction to show that, for every and k 2 N, (10.4.4) s 7 b m 0(s + k 1) b m 0(s) c(m 0, m 00 ) t 2s,

15 10.4. SPECIALIZATION IN D-Triples 231 extends as a holomorphic function on {s 2Res>p k}, andthus,lettingk 1, s 7 Y (s ) ( ) c(m 0, m 00 ) t 2s, extends as an entire function. We apply this result to = o (t) to get the lemma. Remark The previous proof also applies if we only assume that c is D X,X -linear away from {t =0}. Indeed, this implies that c(m 0, m 00 ) [b m 0(E) P ] is supported on {t =0}, and(10.4.3) onlyholdsforre s big enough, maybe p. Then, (10.4.4) coincides with a holomorphic current of degree 0 defined on {s 2Res>p k} only for Re s 0. But, by uniqueness of analytic extension, it coincides with it on Re s>p. Acurrentofdegree0on X 0 which is holomorphic with respect to s can be restricted as a current of degree 0 by setting s =. Byasimilarargument,thepolarcoefficients at s = of the meromorphic current of degree 0 c(m 0, m 00 ) t 2s, (t) exist as currents of degree 0 on nb X0 (x o ). Lemma Let [m 0 ] be a germ of section of t,exp(2 i ) M 0 near x o and [m 00 ] a germ of section of t,exp(2 i ) M 00 near x o. Then the polar coefficients of the current of degree 0 c(m 0, m 00 ) t 2s, (t) at s = do neither depend on the choice of the local liftings m 0,m 00 of [m 0 ], [m 00 ] nor on the choice of, and take value in C X0. Proof. Indeed, any other local lifting of m 0 can be written as m 0 +µ 0, where µ 0 is a germ of section of V < M 0.Bythepreviouslemma, c(µ 0, m 00 ) t 2s, (t) is holomorphic at s =. Wenotealsothatadifferentchoiceofthefunction does not modify the polar coefficients. According to the lemma, for 2 [ 1, 0), wegetawell-definedsesquilinearpairing (10.4.7) gr V M 0 C gr V M 00 ([m 0 ], [m 00 ]) 7 gr V (c) CX0 i 2 Res s= c(m 0, m 00 ), t 2s (t), where m 0,m 00 are local liftings of [m 0 ], [m 00 ]. The nilpotent operator N:=2 i(e ) is skew-adjoint with respect to this pairing, in the sense that (10.4.8) gr V (c)(n[m 0 ], [m 00 ]) = gr V (c)([m 0 ], N[m 00 ]). This is a consequence of the following properties: is real, t@ t t 2s = t@ t t 2s, t@ t (t) and t@ t (t) are zero in a neighbourhood of t =0.

16 232 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING Exercise (see Remark ). Show that gr V (c) induces pairings (` 2 Z): gr M` gr V (c) := gr M` gr V M 0 C gr M`grV M 00 C X0 and, for ` > 0, P`gr V (c) :=P`gr V M 0 C P`gr V M 00 C X0 by composing with N` on the M 00 side. Exercise (Adjunction and nearby cycles). Show that gr V (c )= (gr V c). [Hint: usethat and are real.] Exercise Show that, if M 0 or M 00 is supported on X 0,theright-handsideof ( ) is always zero,and the residue formula(10.4.7)returns the value zero for every 2 R. Example (The smooth case). We set M = M 0 or M 00.AssumethatM 0, M 00 are O X -locally free of finite rank, and let c : M 0 C M 00 C X be a sequilinear pairing. Let m 0,m 00 be horizontal local sections of M 0, M 00 (i.e., local sections annihilated xi ). Then the current c(m 0, m 00 ) satisfies c(m 0, m 00 )@ xi = c(m 0, m 00 )@ xi =0for every i, hence is a locally constant function by the Poincaré lemma for distributions. Since M = X C M r,weconcludethatc takes values in the sheaf of C 1 forms of maximal degree. For local sections 0 µ 0 and 00 µ 00,wethushavec( 0 µ 0, 00 µ 00 )= c r (µ 0, µ 00 ) 0 ^ 00, where c r : M 0r M 00r C is the induced sesquilinear pairing on the underlying local systems. Assume that X = H t. Then M = V 1 M, gr V M =0for 62 N,and gr V 1M = M/tM. For local sections as above, set 0 = o 0 ^ dt and 00 = o 00 ^ dt. Then from Exercise we obtain, by choosing (t) =µ( t 2 ), gr V 1c( 0 o µ 0, 00 o µ 00 )=( 1) n 1 c r (µ 0, µ 00 ) H 0 o ^ 00 o, since i 2 Res s= 1 c( 0 o ^ dt µ 0, o 00 ^ dt µ 00 ) t 2s, o = i 2 Res s= 1 c r (µ 0, µ 00 ) H Z =( 1) n 1 c r (µ 0, µ 00 ) H (t) Z t 2s o (t) 0 o ^ dt ^ 00 o ^ dt o 0 o ^ 00 o. We now take up the setting of Exercise (4), namely we consider the local embedding : X = H t {0}, X 1 = H t x. Let M 0, M 00 be coherent D X -modules which are R-specializable along H and let c : M 0 M 00 C X be a sesquilinear pairing between them. We deduce a sesquilinear pairing D,D 0 c between D 0 M 0 and D 0 M 00. Let us denote by o : H {0}, X the inclusion. We consider the V -filtrations along (t) in X and X 1.

17 10.4. SPECIALIZATION IN D-Triples 233 Lemma (Independence of the embedding). With these assumptions, we have for all 2 [ 1, 0), gr V ( D,D 0 c) = D,D 0 o gr V (c) and gr V ( T 0 c) = T 0 o gr V (c). Proof. The second equality follows from the first one, since the codimension of H in X and H x in X 1 is the same. Recall that if is any test function on X 1 and m 0,m 00 are local sections of M 0, M 00,sothatm 0 1,m 00 1 are local sections of M 0 [@ x ], M 00 [@ x ],then h D,D 0 c(m 0 1, m 00 1), i := hc(m 0, m 00 ), X i. For m 0,m 00 in V M 0,V M 00, atestfunctiononh x and (t) as above, we thus have (arguing for Re s 0 first and then using analytic continuation) hgr V ( D,D 0 c)([m 0 1], [m 00 1]), i = i 2 Res s= h D,D 0 c(m 0 1, m 00 1) t 2s, (t)i = i 2 Res s= hc(m 0, m 00 ) t 2s, H (t)i = hgr V c(m 0, m 00 ), X i. Exercise (Non-characteristic restriction of a sesquilinear pairing) Assume that X = H t and let M 0, M 00 be coherent D X -modules such that the hypersurface H = {t = 0} is non-characteristic for them (see Section 7.5). Let c : M 0 C M 00 C X be a sesquilinear pairing. Then gr V 1c, asdefinedby(10.4.7),is asesquilinearpairingbetween t,1 M 0 = M 0 /tm 0 and t,1 M 00 = M 0 /tm 00.Wedenote it by D H c. Check that D Hc only depends on the embedding H, X and not on the product decomposition X ' H t. Conclude that it is a well-defined sesquilinear pairing D H c : D H M0 C D H M00 C H. Proposition (Uniqueness along a non-characteristic divisor) Let M 0, M 00 be coherent D X -modules and let H be an hypersurface which is noncharacteristic for them. If two sesquilinear pairings c 1, c 2 : M 0 C M 00 C X coincide when restricted to the open set X r H, then they coincide. Proof. The question is local, so we can assume that X = H t and we can shrink t if needed. Set c = c 1 c 2 and let m 0,m 00 be local sections of M 0, M 00 defined on some neighbourhood nb(x o )=nb H t of x o 2 H {0}. Let 2 Cc 1 (nb(x o )), and let p be the order of c(m 0, m 00 ) on the compact set Supp. We aim at proving that hc(m 0, m 00 ), i =0. We consider the current on t defined by 7 c(m 0, m 00 ) ( ):=hc(m 0, m 00 ), i for 2 C 1 c ( t). It is enough to prove that c(m 0, m 00 ) =0(by choosing 1 on the projection to t of Supp ). This current has order 6 p and is supported at the origin, hence can be

18 234 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING written in a unique way, by using the Dirac current 0 at the origin, as c(m 0, m 00 ) = X c a,b ( ) t b t, c a,b( ) 2 C. 06a+b6p We will prove that all the coefficients c a,b ( ) vanish. This is obvious if = t q t r q,r with q + r>pand if q,r is C 1, so that we can reduce to the case where does not depend on t, t. We claim that there exists N large enough such that m 0 satisfies an equation of the form NY m 0 b(t@ t ):=m 0 t + k) =m k=1(t@ 0 t X p+1 P j (t, x )(t@ t ) j, j where x are local coordinates on H. Indeed, H is also non-characteristic for the coherent sub-module m 0 D X,andthefiltrationm 0 V k D X is comparable with the V -filtration V (m 0 D X ),sothereexistsn such that V N 1 (m 0 D X ) m 0 V (p+1) D X. Since m t N 2 (m 0 D X )=V 1 (m 0 D X ),wehavem t N t N 2 V N 1 (m 0 D X ),hence the assertion. We thus have c(m 0, m 00 ) b(t@ t )=0.Since t b t (t@ t + k) =(a + k) t b t,we conclude that for every a, b, c a,b ( ) QN k=1 (a + k) =0,soc a,b( ) =0. We also have an analogue of Corollary for sesquilinear pairings. Proposition Let M 0, M 00 be two holonomic D X -modules which are S-decomposable and let (Z i ) i2i be the family of their strict components. Then any sesquilinear pairing c : M 0 Z i C M 00 Z j C X vanishes identically if Z i 6= Z j. We will first prove a similar result related to the S-decomposition along a function. Lemma Let g : X C be a holomorphic function and let M 0, M 00 be two coherent D X -modules which are R-specializable along (g). Assume that one of them, say M 0, is a minimal extension along (g), and the other one, say M 00, is supported on g 1 (0). Then any sesquilinear pairing c : M 0 M 00 C X vanishes indentically. Proof. ByKashiwara sequivalence ,wecanassumethatg is the projection X 0 C C, andwechooseacoordinatet on C. Weworklocallynearx o 2 X 0. Consider c as a morphism M 0 Hom DX (M 00, C X ).FixlocalD X -generators m 00 1,...,m 00 ` of M 00 x o.bykashiwara sequivalence7.6.1,thereexistsq > 0 such that m 00 k tq =0for all k =1,...,`.Letm 0 2 M 0 x o and let p be the maximum of the orders of c(m 0 )(m 00 k ) on some neighbourhood of x o.ast p+1+q /t q is C p,wehave,foreveryk =1,...,`, c(m 0 )(m 00 k )tp+1+q = c(m 0 )(m 00 k )tq tp+1+q t q = c(m 0 )(m 00 k tq ) tp+1+q t q =0, hence c(m 0 )t p+1+q 0. Applying this to generators of M 0 x o shows that all local sections of c(m 0 x o ) are killed by some power of t.

19 10.4. SPECIALIZATION IN D-Triples 235 As M 0 is a minimal extension along (t), weknowfromproposition7.7.2(2)that V <0 M 0 x o generates M 0 x o over D X. It is therefore enough to show that c(v <0 M 0 x o )=0. On the one hand, we have c(v M 0 x o )=0for 0: indeed, t : c(v M 0 x o ) c(v 1 M 0 x o ) is an isomorphism for <0,henceactsinjectivelyonc(V M 0 x o ),therefore the conclusion follows, as t is also nilpotent by the argument above. Let now <0 be such that c(v < M 0 x o )=0,andletm 0 be a section of V M 0 x o ; there exists > 0 such that, setting b(s) =(t@ t ),wehavem 0 b(s) 2 V < M 0 x o, hence c(m 0 )b(t@ t )=0;ontheotherhand,wehaveseenthatthereexistsN such that c(m 0 )t N =0,hence,puttingB(s) = Q N 1 `=0 (s `), italsosatisfiesc(m0 )B(t@ t )=0; notice now that b(s) and B(s) have no common root, so c(m 0 )=0. Proof of Proposition The assertion is local on X, sowefixx o 2 X and we work with germs at x o. Assume for example that Z i is not contained in Z j and consider a germ g of analytic function, such that g 0 on Z j and g 6 0 on Z i.then we can apply Lemma to M 0 Z i and M 00 Z j. Definition (Sesquilinear pairing on nearby cycles). Let g : X C be a holomorphic function. Assume that M 0, M 00 are R-specializable along (g). Forasesquilinear pairing c : M 0 M 00 C X and for every 2 S 1 and 2 [ 1, 0) such that =exp(2 i ), wedefine(notethat T 0 g c = "( 1) D,D 0 g c = D,D 0 g c) ( ) g, c := ( 1) n 1 gr V ( T 0 g c) : g, M 0 g, M 00 C X. Remark (The basic example). The sign ( 1) n 1 is justified by the calculation in Example Namely, if X = X 0 C, and if M 0 = M 00 = X, with c n being the natural pairing (2) in dimension n, thengr V 1 X ' X0 and ( 1) n 1 gr V 1c n = c n 1. Setting now g = t and denoting by : X 0, X the inclusion, we have gr V 1( T 0 g c n )= T 0 gr V 1c n, according to Lemma , so with our definition, g,1 c n = T 0 c n 1. Remark (Properties of g, c). The following properties are obviously obtained from similar properties for gr V T 0 c. (1) g, c(nm 0, m 00 )= g, c(m 0, Nm 00 ) (m 0 2 g, M 0 x o, m 00 2 g, M 00 x o ). (2) We have induced pairings gr M` g, c : gr M` g, M 0 gr M` g, M 00 C X and, for every ` > 0, P` g, c :P` g, M 0 P` g, M 00 C X is induced by gr M` g, c(, N` ). (3) g, (c )= ( g, c). Remark (Sesquilinear pairing on vanishing cycles). It is possible to give the definition of a sesquilinear pairing t,1c : t,1m 0 t,1 M 00 C X for every M 0, M 00 which are coherent and R-specializable along t =0.However,suchageneraldefinition is not needed for our further purpose, since we will mainly have to work with S-decomposable D X -modules. We now focus on this case. Assume first that M = M 0, M 00 is a minimal extension along X 0 (see 7.7.3). Then t,1m =Im[N: t,1 M t,1 M]. Then, for [µ 0 ] 2 t,1 M 0 and [µ 00 ] 2 t,1 M 00,and

20 236 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING [µ 0 ] = N[m 0 ], [µ 00 ] = N[m 00 ],weset t,1c([µ 0 ], [µ 00 ]) := t,1 c([m 0 ], N[m 00 ]) = t,1 c(n[m 0 ], [m 00 ]), where the latter equality is due to (1), as well as the well-definedness, since t,1c([m 0 ], N[m 00 ]) = 0 for [m 0 ] 2 Ker N. Then all properties of Remark also holds for t,1c instead of t, c. Assume now that M 0, M 00 are S-decomposable. Then c only pairs components with the same pure support (see Proposition ). If the pure support is not contained in X 0, we apply the previous construction. If the pure support is contained in X 0, then t,1 M = M for M = M 0, M 00. Then we obviously set t,1c := c b. Specialization of objects of D-Triples(X). We say that an object T = (M 0, M 00, c) of D-Triples(X) is R-specializable along (g) if M 0, M 00 are so. We then define, for 2 S 1, ( ) g, T := ( g, M 0, g, M 00, g, c). Then g, is a functor from the full subcategory of R-specializable objects of D-Triples(X) to the category of objects supported on g 1 (0). Example In the setting of Remark , and using Definition (2) for T X,wethushave t, ( T X )= T 0 ( T X0 ),if : X 0 {0}, X = X 0 C denotes the inclusion. Remark (Rule of signs for the nilpotent endomorphism) We did not define the nilpotent operator on g, T since we wish to emphasize the rule of signs in its definition. This rule is motivated by two properties we want to realize. As for the Lefschetz operator L L (2.3.10), we wish that the nilpotent comes from geometry, i.e., is defined from the monodromy. Definition is the raison d être of the definition of N as 2 i(e ). Let N=(N 0, N 00 ) be the nilpotent operator to be defined on g, T. Due to the skew-adjointness of N with respect to g, c,weneedtoanti-symmetrizetheaction of N on g, M 0 and g, M 00. We are thus led to define the nilpotent operator N on g, T as ( ) N:=(N 0, N 00 ), N 0 = N on g, M 0, N 00 =Non g, M 00. With this definition, N is indeed a morphism g, T g, T. Remark (The graded Lefschetz object attached to ( g, T, N)) The monodromy filtration of N exists in the abelian category D-Triples(X), and we have, according to Remark (2), gr M` g, T = (gr M` g, M 0, gr M` g, M 00, gr M` g, c), P` g, T =(P` g, M 0, P` g, M 00, P` g, c) (` > 0).

21 j * SPECIALIZATION IN D-Triples 237 Remark (Specialization of a pre-polarization of weight w) Let Q=(( 1) w Q, Q) :T T be a pre-polarization of weight w of T. Then (( 1) w 1 g, Q, g, Q) is a pre-polarization of weight w 1 of g, T. Remark (Specialization of a ( 1) w -Hermitian pair). Let (M, c) be a ( 1) w - Hermitian pair. Assume that M is R-specializable along (g). Then ( g, M, g, c) is a ( 1) w 1 -Hermitian pair and N is skew-adjoint with respect to g, c and, for ` > 0, (P` g, M, P` g, c) is a ( 1) w 1+`-Hermitian pair. Example Let us take up Example (2) with X = t,the( 1)-Hermitian pair ( X, c). With respect to the coordinate t, wehavegr V 1 X = C and, for m 0 = a 0 dt, m 00 = a 00 dt with a 0,a 00 2 C (so that [m 0 ]=a 0, [m 00 ]=a 00 in gr V 1 X ), gr V 1c([m 0 ], [m 00 ]) := a 0 a 00 i 2 Res s= 1 Z t 2s (t)dt ^ dt = a 0 a 00. Definition (Middle extension quiver of a R-specializable D-triple) Assume that T is R-specializable along (g). with (1) We say that it is a minimal extension along (g) if M 0, M 00 are so (see 7.7.3). (2) If T is a minimal extension along (g), wedefine In other words, g,1t := ( g,1 M 0, g,1m 00, g,1c) g,1m 0 := Im N 0, g,1m 00 := Coim N 00 := M 00 / Ker N 00, g,1c := g,1 c Im N 0 Coim N 00. g,1t =ImNin the category D-Triples(X). Let us make this definition more explicit. Since g,1 c(n 0 m 0, m 00 )= g,1 c(m 0, N 00 m 00 ), we have g,1 c(n 0 m 0, m 00 )=0for m 00 2 Ker N 00,and g,1 c is well-defined. We define the middle extension diagram when T is a minimal extension along (g): ( ) g,1 T where the epimorphism can : g,1 T can var g,1t, g,1 T is given by can 0 =N 0 : M 0 Im N 0, can 00 =N 00 : M 00 / Ker N 00, M 00, and the monomorphism var : g,1t g,1 T is given by var 0 =incl. :ImN 0, M 0, var 00 =proj. : M 00 M 00 / Ker N 00. Remark (Middle extension quiver of a R-specializable ( 1) w -Hermitian pair) Let (M, c) be a ( 1) w -Hermitian pair which is R-specializable along (g) and such that M is a minimal extension along (g). Wethushave g,1m =Im[N: g,1 M g,1 M].

22 238 CHAPTER 10. D-MODULES ENRICHED WITH A SESQUILINEAR PAIRING For local sections m 0,m 00 of M, weset Then g,1 c is ( 1) w 1 -Hermitian. g,1c(nm 0, Nm 00 ):= g,1 c(m 0, Nm 00 ). Definition (S-decomposable D-triples). We say that a coherent D-triple T is S-decomposable if its components M 0, M 00 are so. It then has a decomposition T = L i T Z i with T Zi having pure support the irreducible closed analytic subset Z i X (see Proposition ) c. Specialization of left D-triples. While the question of signs is better behaved for the pushforward functor in the case of right D X -modules, the situation is reversed for the specialization functors. Let us start with the case of a projection X = X 0 C C and R-specializable left D X -modules M 0, M 00 equipped with a sesquilinear pairing c : M 0 M 00 Db X between them. For every 2 ( 1, 0] and each test form o of maximal degree on X 0, the formula ( ) gr V c([m 0 ], [m 00 ]), o := Ress= 1 t 2s c left (m 0, m 00 i ), o ^ (t) 2 dt ^ dt defines sesquilinear pairing gr V c : gr V M 0 gr V M 00 Db X0. This is proved as for (10.4.7). Since and are real, and since the form i 2 dt ^ dt is real, we have ( ) gr V (c ) = (gr V c). On the other hand, N is skew-adjoint with respect to c (same proof as for (10.4.8) in the right case). Definition (Sesquilinear pairing on nearby cycles, left case) Let g : X C be a holomorphic function. Assume that M 0, M 00 are R-specializable along (g). Forasesquilinearpairingc : M 0 M 00 Db X and for every 2 S 1 and 2 ( 1, 0] such that =exp( 2 i ), wedefine ( ) g, c := gr V ( D,D 1 g c) = gr V ( T 1 g c) : g, M 0 g, M 00 Db X. Given a left D-triple T =(M 0, M 00, c) which is R-specializable along (g), weset ( ) g, T := ( g, M 0, g, M 00, g, c). Remark (Side-changing for g, c). If X = H t and T =(M 0, M 00, c) is a left D-triple which is R-specializable along H, then,forholomorphicforms o, 0 o 00 of maximal degree on X 0,foraC 1 function o on X 0 and for Re s 0, wehave hc right (( 0 o ^ dt) m 0, ( 00 o ^ dt) m 00 ), o We deduce that, for = 1, (t) t 2s i =( 1) n 1 hc(m 0, m 00 ), o 0 o ^ 00 o (gr V c) right =( 1) n 1 gr V (c right ). (t) t 2s dt ^ dti.

23 10.5. LOCALIZATION AND DUAL LOCALIZATION OF A SESQUILINEAR PAIRING 239 As a consequence, for every 2 S 1,wehave ( ) g, (c right )=( g, c) right. Indeed, we have g, (c right )=( 1) n 1 gr V ( T 0 c right ) (Definition ) =( 1) n 1 gr V (( T 1 c) right ) (see ( )) = gr V ( T 1 c) right (see above for X, X C) =( g, c) right. Example Let : X 0 {0}, X 0 C denote the inclusion. Using Definition (1) for T O X C,wethushave t, ( T O X C )= T 1 ( T O X ). Indeed, this follows from Example , and from the side changing formulas ( ) and ( ), since T X =( T O X ) right. Example (The smooth case). Let us take up the notation of Example in the left case. We have M = O X C M r and c takes values in C 1 functions. For local horizontal sections µ 0,µ 00,wehavec(1 µ 0, 1 µ 00 )=c r (µ 0, µ 00 ). If X = H t,then gr V M =0for /2 N and gr 0 V M = M/tM. Forlocalholomorphicfunctionsf 0,f 00 and local horizontal sections µ 0,µ 00,indicatingbyanindexo the restriction to t =0,and for a C 1 test form o on H of maximal degree, we obtain according to Exercise hgr 0 V c(f 0 o µ 0 o, f 00 o µ 00 o), o i = c r (µ 0, µ 00 ) o Z f 0 of 00 o o, Localization and dual localization of a sesquilinear pairing 10.5.a. Moderate distributions. We refer to [Mal66, Chap. VII] for the results in this subsection. Let D be a reduced divisor in X and let O X ( D) be the sheaf of meromorphic functions on X with poles along D. The subsheaf Db X,D of Db X consists of distributions supported on D (i.e., vanishing when applied to any test form with compact support in X r D). On the other hand, let j : X r D, X denote the open inclusion. By definition, there is an exact sequence of left D X,X -modules 0 Db X,D Db X j Db XrD. The image of the latter morphism is the sheaf on X of distributions on X r D which are extendable as distributions on X. It can be characterized as the subsheaf of j Db XrD consisting of distributions which can be tested along C 1 forms of maximal degree on X r D having rapid decay along D. It is denoted by Db mod X D (sheaf on X