From D-modules to deformation quantization modules. Pierre Schapira

Transcription

1 From -modules to deformation quantization modules Pierre Schapira raft, v5, March 2012

2 2

3 Contents 1 The ring X Construction of X Filtration on X Characteristic variety e Rham and Spencer complexes module associated with a submanifold Homological properties of X Exercises Operations on -modules Operations on O-modules Real and complex microlocal geometry Internal operations, duality and external product Transfert bimodule Non characteristic inverse images irect image Exercises Q-algebras and Q-modules Construction of ÊT X and ŴT X Star-products and Q-algebras Convolution of kernels Applications to -modules Characteristic classes for Q-modules Hochschild class for Q-modules Hochschild class for O-modules Hochschild class on symplectic manifolds inks with -modules

4 4 CONTENTS

5 Introduction The aim of these Notes is first to introduce the reader to the theory of - modules in the analytical setting and also to make a link with the theory of deformation quantization (Q for short) in the complex setting. As we shall see, the advantage of the Q-framework is that it unifies, in some sense, the theory of O-modules (when the star product is commutative) and the theory of -modules (when the star product is symplectic). This text is a short introduction, not a systematic study. In particular many proofs are skipped and the reader is encouraged to consult the literature. To our opinion, the best reference to -modules is [22], and, in fact, Chapters 1 and 2 are extracted from this book. References for -modules. Some classical titles are [20], [4] and, in the algebraic setting, [3], [5]. A nice introduction may also be found in [10], [36]. Although the ring of microdifferential operators will be introduced, its detailed study will not be treated here. References for microdifferential operators are made to [21], [37] and [31]. References for Q-algebroids and Q-modules. We refer to [25] and the references therein. References for categories, homological algebra and sheaves. The reader is assumed to be familiar with sheaf theory as well as homological algebra, including derived categories. An exhaustive treatment may be found in [24] and a pedagogical treatment is provided in [32], [33]. Among numerous other references, see [15], [23, Ch. 1, 2], [38]. History. An outline of -module theory, including holonomic systems, was sketched by Mikio Sato in the early 60 s in a series of lectures at Tokyo University (see [34]). However, it seems that Sato s vision has not been understood until his student, Masaki Kashiwara, wrote his thesis in 1970 (see [20]). Independently and at the same time, J. Bernstein, a student of I. Gelfand at Moscow s University, developed a very similar theory in the 5

6 6 CONTENTS algebraic setting (see [2]). We shall not give any reference to the vast literature on deformation quantization, referring to [25]. Indeed, most of the authors are interested in constructing or classifying Q-algebras (or Q-algebroids), but the study of modules on such algebras seems to be essentially performed in loc. cit.

7 Chapter 1 The ring X In all these Notes, we denote by K a commutative unital ring. If R is a ring, an R-module means a left R-module and we denote by Mod(R) the abelian category of such modules. We denote by R op the opposite ring. Hence, Mod(R op ) denotes the category of right R-modules. If a, b belong to R, their bracket [a, b] is given by [a, b] = ab ba. We use similar conventions and notations for a sheaf of rings R on a topological space X. In particular, Mod(R) denotes the category of sheaves of left R-modules on X. 1.1 Construction of X Coherent and Noetherian sheaves et R be a sheaf of rings on a topological space X. An R-module M is locally finitely generated if there locally exists an exact sequence R N 0 M 0. An R-module M is locally of finite presentation if there locally exists an exact sequence R N 1 P 0 R N 0 M 0, where P 0 means that P 0 acts on the right. An R-module F is pseudo-coherent if for any open subset U of X, any R U -submodule of F U locally finitely generated is locally of finite presentation. A pseudo-coherent R-module which is locally of finite presentation is called coherent. If R is coherent, then the category Mod coh (R) of coherent R-modules is a thick abelian subcategory of the abelian category Mod(R) and a 7

8 8 CHAPTER 1. THE RING X coherent R-module M is nothing but an R-module locally of finite 1-presentation. The sheaf of rings R is Noetherian if it is coherent, if for any x X, the stalk R x is a Noetherian ring and if for any open subset U of X and any family {J i } i I of coherent ideals of R U, the ideal i J i of R U is R U -coherent. O-modules et X denote a complex manifold, O X its structural sheaf, that is, the sheaf of holomorphic functions on X. Unless otherwise specified, we denote by d X the complex dimension of X. We denote by Ω p X the sheaf of holomorphic p-forms and one sets Ω X = Ω d X X. One also sets (1.1) Ω = p Ω p X. We denote by Mod(C X ) the abelian category of sheaves of C-vector spaces on X, and we denote by Hom and the internal Hom and tensor product in this category. For F Mod(C X ), we set End(F ) = Hom (F, F ). Similarly, we denote by Mod(O X ) the abelian category of sheaves of O X - modules, and we denote by Hom O and O the internal Hom and tensor product in this category. We denote by Mod coh (O X ) the full abelian subcategory consisting of coherent sheaves. The sheaf O X is Noetherian. One denotes by Θ X the sheaf of ie algebras of holomorphic vector fields. Hence, Θ X = Hom O (Ω 1 X, O X). The sheaf Θ X has two actions on Ω, that we recall. et v Θ X. The interior derivative i v End(Ω X ) is characterized by the conditions (1.2) i v (a) = 0, a O X i v (ω) = v, ω, ω Ω 1, i v (ω 1 ω 2 ) = (i v ω 1 ) ω 2 + ( ) p ω 1 (i v ω 2 ), ω 1 Ω p X. Note that i v : Ω p X Ωp 1 X is of degree 1. On the other-hand, the ie derivative v End(Ω X ) is characterized by the conditions v (a) = v(a) = v, da, a O X, (1.3) d v = v d, v (ω 1 ω 2 ) = ( v ω 1 ) ω 2 + ω 1 ( v ω 2 ),

9 1.1. CONSTRUCTION OF X 9 The ie derivative is of degree 0 and satisfies (1.4) [ u, v ] = [u,v], u, v Θ X. One has the relations (1.5) v = d i v + i v d. Using v v, one may regard Θ X as a subsheaf of End(O X ). The ring X efinition One denotes by X the subalgebra of End(O X ) generated by O X and Θ X. If (x 1,..., x n ) is a local coordinate system on a local chart U of X, then a section P of X on U may be uniquely written as a polynomial (1.6) P = a α α and we use the classical notations for multi- where a α O X, i = xi indices: = xi α m α = (α 1,..., α n ) N n, α = α α n, if X = (X 1,..., X n ), then X α = X α 1... X αn. Proposition et R be a sheaf of C X -algebras and let ι : O X R and ϕ : Θ X R be C X -linear morphisms satisfying: (i) ι : O X R is a ring morphism, (ii) ϕ : Θ X R is left O X -linear (the O X -module structure on R is given by (i)), (iii) ϕ : Θ X R is a morphism of ie algebras (the ie bracket on R being given by the commutator) (iv) [ϕ(v), ι(a)] = ι(v(a)) for any v Θ X and a O X. Then there exists a unique morphism of C X -algebras Ψ : X R such that the composition O X X R coincides with ι and the composition Θ X X R coincides with ϕ.

10 10 CHAPTER 1. THE RING X Corollary et M be an O X -module and let µ: O X End(M) be the action of O X on M. et ψ : Θ X End(M) be a C X -linear morphism satisfying: (i) µ(a) ψ(v) = ψ(av) (resp. ψ(v) µ(a) = ψ(av)). (ii) [ψ(v), ψ(w)] = ψ([v, w]) (resp. [ψ(v), ψ(w)] = ψ([v, w])), (iii) [ψ(v), µ(a)] = µ(v(a)), (resp. [ψ(v), µ(a)] = µ(v(a))). Then there exists one and only one structure of a left (resp. right) X -module on M which extends the action of Θ X. Proof. For the structure of a left module, apply Proposition to R = End(M). The case of right modules follows since the bracket [a, b] in op X is [a, b], where now [a, b] is the bracket in X. q.e.d. Examples (i) The sheaf O X is naturally endowed with a structure of a left X -module and 1 O X is a generator. Since the anihilator of 1 is the left ideal generated by Θ X, we find an exact sequence of left X -modules X Θ X X O X 0. Note that if X is connected and f is a section of O X, f 0 (i.e., f is not identically zero), then f is also a generator of O X over X. This follows from the Weierstrass Preparation emma. Indeed, choosing a local coordinate system (x 1,..., x n ), one may write f = m j=0 a j(x )x j 1, with a m 1. Then 1 m (f) = m!. (ii) The sheaf Ω X is naturally endowed with a structure of a right X -module, by v(ω) = v (ω), v Θ X, ω Ω X. (iii) et F be an O X -module. Then X O F is a left X -module. (iv) et Z be a closed complex submanifold of X of codimension d. Then HZ d (O X) is a left X -module. (v) et X be a complex manifold and let P be a differential operator on X. The differential equation P u = v may be studied via the left X -module X / X P. (See below.) (vi) et X = C n and consider the differential operators P = n j=1 2 j, Q ij = x i j x j i. Consider the left ideal J of X generated by P and the family {Q ij } i<j. The left X -module X /J is naturally associated to the operator P.

11 1.1. CONSTRUCTION OF X 11 Internal hom and tens The sheaf X is a sheaf of non commutative rings and C X is contained (in fact, is equal, but we have not proved it here) in its center. It follows that we have functors: Hom : (Mod( X )) op Mod( X ) Mod(C X ), : Mod( op X ) Mod( X) Mod(C X ). We shall now study hom and tens over O X. et M, N and P be left X -modules and let M and N be two right X - modules. (a) One endows M O N with a structure of a left X -module by setting v(m n) = v(m) n + m v(n), m M, n N, v Θ X. (b) One endows Hom O (M, N ) with a structure of a left X -module by setting v(f)(m) = v(f(m)) f(v(m)), m M, f Hom O (M, N ), v Θ X. (c) One endows N O M with a structure of a right X -module by setting (n m)v = nv m n vm, m M, n N, v Θ X. (d) One endows and Hom O (M, N ) with a structure of a left X -module by setting v(f)(m) = f(mv) f(m)v m M, f Hom O (M, N ), v Θ X. (e) One endows and Hom O (M, N ) with a structure of a right X -module by setting (fv)(m) = f(m)v + f(vm) m M, f Hom O (M, N ), v Θ X. There are isomorphisms of C X -modules; Hom (M O N, P) Hom (M, Hom O (N, P)), Hom (M O M, N ) Hom (M, Hom O (M, N )), (M O M) N M (M O N ).

12 12 CHAPTER 1. THE RING X To summarize, we have functors O : Mod( X ) Mod( X ) Mod( X ), O : Mod( op X ) Mod( X) Mod( op X ), Hom O : Mod( X ) op Mod( X ) Mod( X ), Hom O : Mod( op X )op Mod( op X ) Mod( X), Hom O : Mod( X ) op Mod( op X ) Mod(op X ). Remark Following [17] who call it the Oda s rule, one way to memorize the left an right actions is to use the correspondence left = 0, right = 1, a b = a + b and Hom (a, b) = a + b. Applying these results, we get: Proposition The functor M Ω O M induces an equivalence of categories Mod( X ) Mod( op X ). A quasi-inverse is given by N Hom O (Ω X, N ) Ω 1 X O N. Remark Suppose to be given a volume form dv on X. Then f fdv gives an isomorphism O X Ω X and we get an isomorphism X op X. The image of a section P X by this isomorphism is called its adjoint with respect to dv and is denoted by P. Hence, for a left X -module M and a section u of M, we have P u = (u dv) P. Clearly (Q P ) = P Q. If (x 1,..., x n ) is a local coordinate system on X and dv = dx 1 dx n, one checks that x i = x i and x i = xi. 1.2 Filtration on X Filtered rings and modules et X be a topological space. A filtered sheaf Fl M of K X -modules is a sheaf M of K X -modules together with a family {Fl j M} j Z of subsheaves satisfying : Fl j M Fl j+1 M, lim j Fl j M = M. The shifted filtration Fl [p] M is given by Fl [p] j M = Fl p+jm.

13 1.2. FITRATION ON X 13 A morphism of filtered sheaves Fl f : Fl M Fl N is a morphism of sheaves f : M N such that f(fl j M) Fl j N for all j. The graded sheaf gr M associated to Fl M is the sheaf j gr jm, where gr j M = Fl j M/Fl j 1 M. If Fl f : Fl M Fl N is a filtered morphism, one denotes by gr f : gr M gr N the associated morphism of graded sheaves. One denote by σ j : Fl j M gr j M the canonical morphism and calls it the symbol of order j morphism. One denotes by σ : Fl M gr M the morphism deduced from the σ j s and calls it the principal symbol morphism. (One shall be aware that σ j is an additive morphism, contrarily to σ.) Consider an exact sequence of sheaves 0 M f g M M 0 and assume that M is endowed with a filtration Fl M. The induced filtration on M is given by Fl j M = f 1 (Fl j M). The image filtration on M is given by Fl j M = g(fl j M). Note that in this case, the associated sequence 0 gr M f g gr M gr M 0 is exact. A filtered ring Fl R on X is a filtered sheaf of rings satisfying: 1 Fl 0 R and Fl i R Fl j R Fl i+j R for all i, j. A filtered R-module Fl M, or equivalently an Fl R-module, is an R- module endowed with a filtration satisfying: Fl i R Fl j M Fl i+j M. A filtration Fl M on M is locally finite free if it is locally isomorphic to a finite direct sum of Fl [i] R. A coherent filtration (one also says a good filtration ) on M is a locally finitely generated filtration, that is, Fl M is locally the image of a finite free filtration. Note that the image of a good filtration is good and that any finitely generated R-module M may be endowed with a good filtration. Namely, if R m M is an epimorphism, one endows M with the image filtration. Remark et us denote by Mod fi (K X ) the category of filtered sheaves. Clearly, the category Mod fi (K X ) is additive and admits kernels and cokernels. One shall be aware that the category Mod fi (K X ) is not abelian, even when X = pt. Indeed, consider a filtered k-module Fl M and the identity morphism u : Fl M Fl [1] M. Its kernel and cokernel are zero, although this morphism is not an isomorphism in general.

14 14 CHAPTER 1. THE RING X Total symbol of differential operators Assume X is affine, that is, X is open in a finite dimensional complex vector space E. et P be a section of X. One defines its total symbol (1.7) σ tot (P )(x; ξ) := exp x, ξ P (exp x, ξ ) = a α (x)ξ α. α m Using (1.6), one gets that σ tot (P ) is a function on X E, polynomial with respect to ξ E. This function highly depends on the affine structure, but its order (a locally constant function on X) does not. It is called the order of P and denoted ord(p ). If Q is another differential operator with total symbol σ tot (Q), it follows easily from the eibniz formula that the total symbol σ tot (R) of R = P Q is given by: (1.8) σ tot (R) = α N n 1 α! α ξ (σ tot (P )) α x (σ tot (Q)). By this formula, one gets that ord(p Q) = ord(p ) + ord(q), ord([p, Q]) ord(p ) + ord(q) 1. The ring X is now endowed with the filtration by the order, Fl m ( X ) = {P X ; ord(p ) m}. One can give a more intrinsic definition of the filtration. Filtration on X efinition The filtration Fl X on X is given by Fl 1 X = {0}, Fl m X = {P X ; [P, O X ] Fl m 1 X }. Note that { Fl 0 X = O X, Fl 1 X = O X Θ X, (1.9) Fl m X Fl l X Fl m+l X, [Fl m X, Fl l X ] Fl m+l 1 X. One denotes by gr X the associated graded ring, by σ : Fl X gr X the principal symbol map and by σ m : Fl m X gr m X the map symbol of order m.

15 1.2. FITRATION ON X 15 One shall not confuse the total symbol, which is defined on affine charts, and the principal symbol, which is well defined on manifolds. It follows from (1.8) that σ(p )σ(q) = σ(q)σ(p ) = σ(p Q). Hence, gr ( X ) is a commutative graded ring. Moreover, gr 0 ( X ) O X and gr 1 ( X ) Θ X. enote by S O (Θ X ) the symmetric O X -algebra associated with the locally free O X -module Θ X. By the universal property of symmetric algebras, the morphism Θ X gr ( X ) may be extended to a morphism of symmetric algebra (1.10) S O (Θ X ) gr X. Proposition The morphism (1.10) is an isomorphism. Proof. Choose a local coordinate system (x 1,..., x n ) on X. Then Θ X n i=1 O X i and the correspondence i ξ i gives the isomorphism S O (Θ X ) α O X α O X [ξ 1,..., ξ n ] gr X. q.e.d. enote by π : T X X the projection. There is a natural monomorphism Θ X π O T X. Indeed, a vector field on X is a section of the tangent bundle T X, hence defines a linear function on T X. By the universal property of symmetric algebra, we get a monomorphism S O (Θ X ) π O T X. Applying Proposition 1.2.3, we get an embedding of C X -algebras: gr X π O T X. In the sequel, we shall still denote by σ : X π O T X and σ m : Fl m X π O T X, the maps obtained by applying the inverse of the isomorphism (1.10) to σ and σ m. Theorem The sheaf of rings X is right and left Noetherian. Proof. This follows from Proposition and general results on filtered ring with associated commutative graded ring. (See [22, Th. A.20].) q.e.d.

16 16 CHAPTER 1. THE RING X 1.3 Characteristic variety Gabber s theorem Recall that if B is a commutative ring and I an ideal, the radical I of I is given by x I there exists k 0 with x k I. If N is a B-module, the annihilator I N of N is the ideal given by x I N xu = 0 for all u N. If 0 M M M 0 is an exact sequence in Mod(B), then clearly (1.11) IM = I M I M = I M I M. In other words, the map M I M is additive. If gr M is a graded gr A- module, then I gr M is a graded ideal. Consider now a filtered ring Fl A with gr A commutative. The ring gr A is endowed with a Poisson bracket as follows. et a and b be two elements of gr A, homogeneous say of degree p and q, respectively. Choose a and b in A such that σ p (a) = a and σ q (b) = b. One sets (1.12) {a, b} = σ p+q 1 ([a, b]). One checks easily that this bracket satisfies the Jacobi identities (1.13) {a, b} = {b, a}, {a, bc} = c{a, b} + b{a, c}, {{a, b}, c} + {{b, c}, a} + {{c, a}, b} = 0. et M be a finitely generated A-module. Recall that a good filtration on M is the image of a finite free filtration. emma et M be a finitely generated A-module, let Fl M be a good filtration on M and let gr M be the associated graded module. Then (1.14) Icar(M) := I gr M, does not depend on the choice of the filtration.

17 1.3. CHARACTERISTIC VARIETY 17 Proof. et ā I gr M of order p. There exists q such that ā p I gr M and there exists a Fl p M such that σ(a) = ā. Then a q Fl k M Fl k+pq 1 for all k, a lq Fl k M Fl k+lpq l for all k. If Fl M is another filtration, there exists r such that Fl k r M Fl km Fl k+r M. Hence, alq Fl k Fl k+lpq 1 for l >> 0. q.e.d. Theorem (Gabber s Theorem.) Assume that gr A is a commutative Noetherian Q-algebra. et M be a finitely generated A-module. Then Icar(M) is involutive, that is, is stable by Poisson bracket. Note that if ā and b belong to I gr M, then {ā, b} obviously belongs to I gr M. The difficulty is that one assumes that ā and b belong to the radical of I gr M. Poisson s structures The graded ring gr ( X ) is endowed with a natural Poisson bracket induced by the commutator in X. On the other hand, the sheaf O T X (hence, the sheaf π O T X) is endowed with the Poisson bracket induced by the symplectic structure of T X. Recall that if (x 1,..., x n ; ξ 1,..., ξ n ) is a local symplectic coordinate system on T X, this Poisson bracket is given by {f, g} = n ξi f xi g ξi g xi f. i=1 Proposition The Poisson bracket on π O T X bracket on gr ( X ). induces the Poisson Proof. et P Fl m ( X ) and Q Fl l ( X ). Then [P, Q] Fl m+l 1 ( X ) and it follows from (1.8) that (1.15) σ m+l 1 ([P, Q]) = n ( ξi σ m (P ) xi σ l (Q) ξi σ l (Q) xi σ m (P ) ). i=1 Hence, σ m+l 1 ([P, Q]) = {σ m (P ), σ l (Q)}. q.e.d.

18 18 CHAPTER 1. THE RING X Good filtration Recall that a good filtration on a coherent X -module M is a filtration which is locally the image of a finite free filtration. Hence, a filtration Fl M on M is good if and only if, (1.16) locally on X, Fl j M = 0 for j 0, Fl j M is O X -coherent, locally on X, (Fl k X ) (Fl j M) = Fl k+j M for j 0 and all k 0. emma et M be a coherent X -module, N M a coherent submodule. Assume that M is endowed with a good filtration Fl M. Then the induced filtration on N defined by Fl j N = N Fl j M is good. Proof. We do not give the proof here. q.e.d. enote by Mod gr coh (gr X) the abelian category of coherent graded gr X - modules and consider the functor : Mod gr coh (gr X) Mod coh (π O T X), gr M π O T X gr X gr M. This functor is exact and faithful. If M is a coherent X -module endowed with a good filtration, the π O T X-module gr M = π O T X gr X gr M is thus coherent and its support satisfies: supp( gr M) = {p T X; σ(p )(p) = 0 for any P Icar(M)}. In the sequel, we shall often confuse gr M and gr M. efinition The characteristic variety of M, denoted char(m), is the closed subset of T X characterized as follows: for any open subset U of X such that M U is endowed with a good filtration, char(m) T U is the support of gr M U. Theorem (i) char(m) is a closed C -conic analytic subset of T X. (ii) char(m) is involutive for the Poisson structure of T X, and in particular, codim(char(m)) d X.

19 1.3. CHARACTERISTIC VARIETY 19 (iii) If 0 M M M 0 is an exact sequence of coherent X - modules, then char(m) = char(m ) char(m ). Proof. (i) is obvious, (ii) follows from Gabber s theorem and (iii) follows from emma q.e.d. Note that the involutivity theorem has first been proved by Sato, Kashiwara and Kawai [37] using analytical tools, before Gabber gave is purely algebraic proof. Suppose that a coherent X -module M is generated by a single section u. Then M X /I, where I is the anihilator of u. There is a natural filtration on M, the image of Fl X. Put Fl j I = I Fl j X. It follows from the fact that gr X is Noetherian that the graded ideal gr I is coherent. Moreover, since gr M = gr X /gr I, we get (1.17) char(m) = {p T X; σ j (P )(p) = 0 for all P Fl j (I)}. If {P 0,..., P N } generates I it follows that char(m) j σ(p j ) 1 (0). In general the equality does not hold, since the family of the P j s may generate I although the family of the σ mj (P j ) s does not generate gr I. Example If X = A 1 (C), the affine line, the ideal generated by and x is X, but the ideal generated by their principal symbols is not O T X. Corollary et M be a coherent X -module, let p T X and assume that p / char(m). et u M. Then there exists a section P X defined in a neighborhood of π(p) with P u = 0 and σ(p )(p) 0. Proof. Consider the sub- X -module X u generated by u. It is coherent and its characteristic variety is contained in that of M. et I denotes the anihilator ideal of u in X and let P 1,..., P N denotes sections of this ideal such that σ(p 1 ),..., σ(p N ) generate the graded ideal gr I. Such a finite family exists since gr I is coherent. Since p / char( X u), there exists j with σ(p j )(p) 0. q.e.d. Remark Using the notion of characteristic cycles for coherent O X - modules, one defines similarly the notion of characteristic cycles for coherent X -modules, see [20]. Example (i) char(o X ) = T X X, the zero-section of T X. (ii) char( X / X P ) = {p T X; σ(p )(p) = 0}.

20 20 CHAPTER 1. THE RING X 1.4 e Rham and Spencer complexes If A is a ring, M is an A-module, and ϕ := (ϕ 1,..., ϕ n ) are n-commuting endomorphisms of M, one can define the Koszul complex K (M; ϕ) and the co-koszul complex K (M; ϕ). We refer to [31] for an exposition. Also recall the e Rham complex (1.18) R X (O X ) := 0 Ω 0 X d Ω 1 d X Ω d X X 0, where d is the differential. et M be a left X -module. One defines the differential d: M Ω 1 X O M as follows. In a local coordinate system (x 1,..., x dx ) on X, the differential d is given by M Ω 1 X O M, m i dx i i m and one checks easily that this does not depend on the choice of the local coordinate system. One defines the e Rham complex of M, denoted R X (M), as the complex (1.19) R X (M) := 0 Ω 0 X O M d Ω d X X O M 0, where Ω 0 X O M is in degree 0 and the differential d is characterized by: d(ω m) = dω m + ( ) p ω dm, ω Ω p X, m M. Note that R X ( X ) C b (Mod( op X )), the category of bounded complexes of right X -modules, and (1.20) R X (M) R X ( X ) M. Recall that there is a natural right -linear morphism Ω X O X Ω X. Moreover, one checks easily that the composition Ω d X 1 X O X Ω d X X O X Ω X is zero. Hence, we get a morphism in the derived category b ( op X ) (1.21) R X ( X ) Ω X [ d X ]. Proposition The morphism (1.21) induces an isomorphism in b ( op X ).

21 1.4. E RHAM AN SPENCER COMPEXES 21 Proof. Since the morphism is well defined on X, we may argue locally and choose a local coordinate system. In this case, there is an isomorphism of complexes (1.22) R X ( X ) K ( X ; 1,..., dx ) where the right hand side is the Koszul complex of the the sequence 1,..., n acting on the left on X. Since this sequence is clearly regular, the result follows. q.e.d. Applying Proposition and isomorphism (1.20), we get: Corollary et M be a left X -module. Then R X (M) Ω X M [ d X ]. et us apply the contravariant functor Hom op(, X ) to the complex R X ( X ). One sets (1.23) SP X ( X ) := Hom (R X ( X ), X ), and calls SP X ( X ) the Spencer complex. (1.24) SP X ( X ) := d x 0 X d O ΘX X O Θ X X 0, (1.25) One deduces from (1.22) the isomorphism of complexes SP X ( X ) K ( X ; 1,..., dx ) where the right hand side is the co-koszul complex of the sequence 1,..., dx acting on the right on X. Since this sequence is clearly regular, we obtain: Proposition The left -linear morphism X O X induces an isomorphism SP X ( X ) O X in b ( X ). Corollary et M be a left X -module. There is an isomorphism in b (C X ) RHom (O X, M) R X (M). Proof. Since SP X ( X ) is a complex of locally free X -modules of finite rank, one has RHom (O X, M) Hom (SP X ( X ), M) Hom (SP X ( X ), X ) M R X ( X ) M R X (M). q.e.d.

22 22 CHAPTER 1. THE RING X Proposition One has the isomorphism RHom (O X, X )[d X ] Ω X RHom op(ω X, X )[d X ] O X RHom (O X, O X ) C X. Proof. (i) One has the chain of isomorphisms RHom (O X, X )[d X ] RHom (SP X ( X ), X )[ d X ] Hom (SP X ( X ), X )[ d X ] R( X )[ d X ] Ω X. (ii) The proof is similar. (iii) The canonical morphism C X Hom (O X, O X ) induces the morphism The isomorphism C X Ω X C X RHom (O X, O X ) Hom (SP X ( X ), O X ) Ω X. is the classical Poincaré lemma. q.e.d module associated with a submanifold et Z be a hypersurface of X. One denotes by O X ( Z) the sheaf of meromorphic functions on X with poles in Z. Hence, if {f = 0} is a local equation of Z, a section u of O X ( Z) is locally written as a quotient u = g/f m, for some m N and g a section of O X. Clearly, O X ( Z) is a left X -module. One also introduces the left X -module B Z X by the exact sequence 0 O X O X ( Z) B Z X 0. If {f = 0} is a local equation of Z, then Also note that O X ( Z) (O X [1/f])/O X. B Z X H 1 [Z](O X ), where H p [Z] (O X) denotes the p-th algebraic cohomology with support in Z of O X. et Z = {f j = 0; j = 1,..., d} be a complete intersection. One sets (1.26) B Z X O X [1/f 1... f d ]/ i O X [1/f 1... f i... f d ]. One checks that this does not depend on the choice of the f js.

23 1.5. -MOUE ASSOCIATE WITH A SUBMANIFO 23 Proposition et Z be a closed smooth submanifold of X. Then B Z X is a coherent X -module and its characteristic variety is TZ X, the conormal bundle to Z in X. This result extends to the non smooth case, but the proof is then much more difficult, using the whole theory of holonomic systems. Proof. The problem is local and we may assume to be given a local coordinate system x = (x, x ) on X, with x = (x 1,..., x d ) and Z = {x = 0}. Then B Z X X /J where J is the left ideal generated by (x, x ). It follows that char(b Z X ) TZ X. The converse inclusion follows from the fact that char(b Z X ) is non empty and involutive. q.e.d. Proposition et Z be a complete intersection of codimension d and assume I Z = O X f O X f d. Then the section δ(f 1 ) δ(f d ) df 1 df d B Z X O Ω d X does not depend on the choice of the sequence (f 1,..., f d ). Proof. et (f 1,..., f d ) be another sequence defining the ideal I Z. There exists a section A Gl(O X, d) which interchanges these two sequences. The group Gl(O X, d) is generated by the transformations (i) (f 1,..., f d ) (af 1,..., f d ), with a O X, (ii) (f 1,..., f i, f i+1,... f d ) (f 1,..., f i+1, f i,..., f d ) (iii) (f 1,..., f d ) (f 1, f 2 + bf 1,..., f d ) Then, it is enough to notice that 1/af 1 1/f 2 d(af 1 ) df 2 = 1/f 1 1/f 2 df 1 df 2, 1/f 2 1/f 1 df 2 df 1 = 1/f 1 1/f 2 df 1 df 2 1/f 1 1/(f 2 + bf 1 )df 1 d(f 2 + bf 1 ) = 1/f 1 1/f 2 df 1 df 2. q.e.d. efinition Assume that Z is smooth of codimension d. We shall denote by δ Z the canonical section of B Z X O Ω d X constructed in Proposition One calls it the fundamental class of Z in X.

24 24 CHAPTER 1. THE RING X Note that δ Z belongs to d Z where Z denotes the subsheaf of Ω 1 X consisting of sections with values in the conormal bundle TZ X. enote by the diagonal in X X and by q 1 and q 2 the first and second projections X X X. The projection q 2 allows us to identify T (X X) with T X. Proposition There is a natural X op X -linear isomorphism (1.27) X which associates δ to 1 X. B X X q 1 2 O q 1 2 Ω X, Proof. Consider the left ideal J of X op X generated by the sections P -module to 1 1 P (P X ). Then X is isomorphic as a X op X ( X op X )/J. et us show that the section δ of B X X q 1 2 O q 1 2 Ω X satifies J δ = 0, which will prove that the morphism (1.27) is well defined and let us show at the same time that this morphism is an isomorphism. Both questions are local and we may choose a local coordinate system (x) on X. enote by (y) a copy of this system and by dx the section dx 1 dx n. Identifying op X to X by this section, the ideal J is identified to the left ideal of X X generated by the sections P 1 1 P where P denotes the adjoint of P with respect to dx. On the other hand, B X X q 1 2 O q 1 2 Ω X is identified to B X X which is isomorphic to ( X X )/J where J is the left ideal generated by (x y, x + y ). It remains to notice that J = J by Remark (1.1.7). q.e.d. 1.6 Homological properties of X Vanishing theorems and dimension Recall the classical theorem for O-modules: Theorem et X be a smooth manifold and let F be a coherent O X - module. Then (i) Ext k O (F, O X) is coherent for all k and is 0 for k < codim supp(f), (ii) codim(supp(ext k O (F, O X))) k. There is a corresponding theorem for -modules. Theorem et M be a coherent X -module. Then

25 1.6. HOMOOGICA PROPERTIES OF X 25 (i) Ext k (M, X) is coherent for all k and is 0 for k < codim(char(m)), (ii) codim(char(ext k (M, X))) k, (iii) char(ext k (M, X)) char(m), (iv) Ext k (M, X) = 0 for k > d X. Corollary et M be a coherent X -module. Then the support of Ext d X (M, X) has pure dimension d X. Proof. First we construct by induction a finite free filtered resolution of Fl M, that is, a filtered exact sequence of Fl X -modules Fl 1 Fl 0 Fl M 0 where the Fl j s are filtered finite free. We denote by d j the differential. Set: Then is exact. Put Fl := Fl 1 Fl 0 0, gr := gr 1 gr 0 0. gr 1 gr 0 gr M 0 j = Hom ( j, X ), = Hom (, X ) = One defines a filtration Fl j on j by setting Fl m j = {ϕ Hom ( j, X ); ϕ(fl k j ) Fl k+m X for all k}. Clearly, this filtration on j gr j. In other words, is good and moreover Hom gr (gr j, gr ) Hom gr (gr, gr ) gr. Put Z k = Ker( k d k k+1 ), I k = Im( k 1 k ) H k ( ) = Z k /I k. We endow Z k with the induced filtration and H k ( ) with the filtration image of Fl Z k. Since Ext k (M, X) H k ( ), we get a filtration Fl Ext k (M, X) on this module. Moreover Ext k (gr M, gr gr X)) H k (gr ). In order to complete the proof, we need a lemma. q.e.d.

26 26 CHAPTER 1. THE RING X emma gr H k ( ) is a subquotient of H k (gr ). Proof of emma On the other-hand, H k (gr m ) = Fl m( k ) (dk ) 1 Fl m 1 k+1 Fl m 1 ( k ) + dk 1 Fl m k 1 Fl m (Z k ). Fl m 1 (Z k ) + d k 1 Fl m k 1 gr m H k ( ) = Fl m (Z k ) Fl m 1 (Z k ) + I k Fl m (Z k ). The result then follows from Fl m 1 (Z k ) + d k 1 Fl m k 1 Fl m 1 (Z k ) + I k Fl m (Z k ). q.e.d. End of proof of Theorem It follows that (1.28) char(ext k (M, X)) supp(ext k gr (gr M, gr X))). (i) By Theorem 1.6.1, Ext k O ( gr M, O T X)) = 0 for k < codim(char(m)). By (1.28), we get that Ext k (M, X) = 0 for k < codim(char(m)). (ii) By Theorem 1.6.1, codim(supp(ext k gr (gr M, gr X))) k. By (1.28), we get that codim(char(ext k (M, X))) k. (iii) follows from the inclusion supp(ext k gr (gr M, gr X)) supp(gr M). (iv) follows from (ii) and the involutivity of the characteristic variety of Ext k (M, X). q.e.d. Example et d X = 1. Then any coherent ideal I of X is projective since Ext j ( X/I, X ) = 0 for j > 1. et t denote a local holomorphic coordinate. The left ideal of X generated by t 2 and t t 1 is projective. By Theorem 1.6.2, its characteristic is T X. Since it is contained in X, its multiplicity on T X is 1. This module does not admits a single generator, and it follows that it is not free.

27 Exercises to Chapter 1 27 Free resolutions Theorem et M be a coherent X -module. Then, locally on X, M admits a finite free resolution of length d X. In other words, there locally exists an exact sequence 0 d X 0 M 0, where the i s are free of finite rank over X and n d X. Proof. Set n = d X. Since we argue locally, we may endow M with a good filtration Fl M. We may locally find a finite free filtered resolution Fl n Fl 0 Fl M 0. On the other-hand, we know that Ext j gr (gr M, gr X) = 0 for j > n. Set K n = Ker( n 1 n 2 ) and let us endow K n with the induced filtration. Then the sequence 0 gr K n gr n 1 gr 0 gr M 0 is exact and it follows that gr K n is projective. Since projective modules over gr X are stably free, there exists a finite free X module such that gr K n gr is free and this implies that K n is a free X -module. The sequence 0 K n n 1 0 M 0 is a finite free resolution of M. q.e.d. Exercises to Chapter 1 Exercise 1.1. et A be a filtered ring and assume that gr A is commutative and has no zero divisors. et a 0, b 0 in A. (i) Prove that the sequence below in Mod(A) of is exact. 0 A/(A a A b) A/A a A/A b A/(A a + A b) 0 (ii) educe that A a A b {0}. Exercise 1.2. et Z 1 and Z 2 be two smooth submanifolds of X and assume they are transversal. Calculate RHom (B Z1 X, B Z2 X).

28 28 CHAPTER 1. THE RING X Exercise 1.3. Calculate RHom (B Z X, X ) for a smooth submanifold Z of X. Exercise 1.4. (i) Prove that the O X -module X is flat. (ii) Prove that if a X -module I is injective in the category Mod( X ), then it is injective in the category Mod(O X ). Exercise 1.5. et M, N Mod( X ). Prove that RHom (M, N ) RHom (O X, RHom O (M, N )). Exercise 1.6. Prove that Ext d X X,x (O X,x, X,x ) 0. Exercise 1.7. Recall that the weak global dimension wgld(r) of a ring R is the smallest integer d such that Tor R j (N, M) = 0 for j > d and any left R-module M and right module N. Prove that wgld( X,x ) = d X. (Hint: use [38, Ch. 4].) Exercise 1.8. Recall that the global dimension gld(r) of a sheaf of rings R is the smallest integer d such that Ext j (M, N ) = 0 for j > d and any left R-modules M and N. Prove that gld( X ) = 2d X + 1. (Hint: see [22].)

29 Chapter 2 Operations on -modules 2.1 Operations on O-modules For a complex manifold X, one denotes by Mod coh (O X ) the thick abelian subcategory of Mod(O X ) consisting of coherent modules. One denotes by b coh (O X) the full triangulated category of the bounded derived category b (O X ) consisting of objects with coherent cohomology. We shall also encounter the duality functors for O-modules: OF := RHom O (F, O X ), O F := RHom O (F, Ω X [d X ]). Recall that d X is the complex dimension of X and Ω X = Ω d X X. et X and Y be two manifolds. For an O X -module F and an O Y -module G, we define their external product, denoted F G, by F G = O X Y OX O Y (F G). Note that the functor F F G is exact. Clearly, if F b coh (O X) and G b coh (O Y ), then F G b coh (O X Y ). et f : X Y be a morphism of complex manifolds. There is a natural morphism of rings f 1 O Y O X. Using this morphism, the direct images f F and f F of an O X -module are well defined as O Y -modules. One denotes as usual by Rf and Rf! their derived functors. The inverse image of an O Y -module G is defined by f := O X f 1 O Y f 1 G. Its right derived functor is denoted f. The following result is left as an exercise. Proposition et G b coh (O Y ). Then f G b coh (O X) and there is a natural isomorphism f OG Of G. 29

30 30 CHAPTER 2. OPERATIONS ON -MOUES There is a similar result for direct images: Theorem Grauert s theorem. et F b coh (O X) and assume that f is proper on supp(f). Then Rf! F b coh (O Y ) and there is a natural isomorphism Rf! O F O Rf! F. Note that Grauert s theorem is a relative version of the Cartan-Serre s finiteness theorem and the Serre s duality theorem. 2.2 Real and complex microlocal geometry In these Notes, unless otherwise specified, a real manifold means a real analytic manifold. For a complex manifold X we denote by X R the real underlying submanifold to X. When there is no risk of confusion, we simply write X instead of X R. We denote by X the complex conjugate manifold to X. (Recall that X = X as a topological space, but the sheaf of holomorphic functions on X is the sheaf of anti-holomorphic functions on X.) Then, identifying X with the diagonal of X X, the complex manifold X X is a complexification of X R. enote by dα X the symplectic form on T X and by dα XR the symplectic form on T X R. Then dα XR = 2R dα X. et f : X Y be a morphism of real manifolds. To f are associated the tangent morphisms (2.1) T X f X Y T Y f τ T Y X X f Y. By duality, we deduce the diagram: (2.2) T X f d X Y T Y f π T Y X π X π f Y. π

31 2.3. INTERNA OPERATIONS, UAITY AN EXTERNA PROUCT31 efinition Consider a morphism f : X Y of real manifolds. (i) The conormal bundle to X in Y is the sub-vector bundle of X Y T Y given by f 1 d (T X X). (ii) et Λ T Y be a closed R + -conic subset. One says that f is noncharacteristic for Λ (or else, Λ is non-characteristic for f, or f and Λ are transversal) if, f 1 π (Λ) T XY X Y T Y Y. of course, the same notions hold for complex manifolds. emma et Λ be a closed R + -conic subset of T Y. Then a morphism f : X Y is non characteristic for Λ if and only if f d : X Y T Y T X is proper on fπ 1 (Λ). In this case: (i) f d f 1 π (Λ) is closed and R + -conic in T X. (ii) If moreover f is a morphism of complex manifolds and Λ is a complex analytic C -conic subset, then f d is finite on fπ 1 (Λ) and f d fπ 1 (Λ) is a complex analytic C -conic subset of T X. Example et Z be a closed and smooth submanifold of Y. Then f is non-characteristic for TZ Y if and only if f is transversal to Z. The next lemma will allows us to decompse a morphism into a smooth morphism and a closed embedding when studying operations on -modules. emma Consider morphisms of real manifolds X f Y h = g f. et Λ be a closed R + -conic subset of T Z. g Z and set (i) Assume that g is non characteristic for Λ and f is non characteristic for g d g 1 π (Λ). Then h is non characteristic for Λ. (ii) Assume that h is non characteristic for Λ. Then g is non characteristic for Λ on a neighborhood of f(x) and f is non characteristic for g d g 1 π Λ. 2.3 Internal operations, duality and external product For a complex manifold X, one denotes by Mod coh ( X ) the thick abelian subcategory of Mod( X ) consisting of coherent modules. One denotes by

32 32 CHAPTER 2. OPERATIONS ON -MOUES b coh ( X) the full triangulated category of the bounded derived category b ( X ) consisting of objects with coherent cohomology. If M b coh ( X), we set (2.3) char(m) = j char(h j (M)). Internal operations We denote by RHom O the right derived functor of Hom O and by the left derived functor of O acting on -modules. Hence, we get the functors : b ( X ) b ( X ) b ( X ), : b ( op X ) b ( X ) b ( op X ), RHom O (, ) : b ( X ) op b ( X ) b ( X ), RHom O (, ) : b ( op X )op b ( op X ) b ( X ). The tensor product is commutative and associative, that is, for, M, N in b ( X ) there are natural isomorphisms M N N M and (M N ) M (N ). Moreover O X M M. There are also natural functors RHom (, ) : b ( X ) op b ( X ) b (C X ), : b ( op X ) b ( X ) b (C X ). These functors are related by the formulas (2.4) and (2.5) below. Proposition For, M, N in b ( X ) and K in b ( op X ) there are natural isomorphisms (2.4) (2.5) K (M N ) (K M) N, RHom (, RHom O (M, N )) RHom ( M, N ). uality We define the duality functors on b ( X ) or b ( op X ), all denoted by and, by setting (2.6) (M) := RHom (M, X ) (M b ( X ) or M b ( op X )), (2.7) (M) := RHom (M, X O Ω 1[d X]) (M b coh ( X)), (2.8) (M) := RHom (M, Ω X [d X ] O X ) (M b coh (op X )). X

33 2.3. INTERNA OPERATIONS, UAITY AN EXTERNA PROUCT33 Proposition For M, N in b ( X ), we have (2.9) RHom (M, N ) RHom (O X, M N ). Proof. We have the isomorphism RHom (O X, M N ) RHom (O X, X ) ( M N ) Ω X ( M N ) [ d X ] (Ω X M) N [ d X ] M N RHom (M, N ). q.e.d. Proposition (i) The functor : b coh ( X) op b coh (op X ) is welldefined and satisfies id and similarly with. (ii) If M b coh ( X), then char( (M)) = char(m). Proof. (i) There is a natural morphism id. To prove it is an isomorphism, we argue by induction on the amplitude of M and reduce to the case where M is a coherent X -module. More precisely, assume H j (M) = 0 for j / [j 0, j 1 ] and the result has been proved for modules with amplitude j 1 j 0 1. Consider the distinguished triangle (d.t. for short) (2.10) H j 0 (M)[ j 0 ] M τ >j 0 (M) +1 and apply the functor. We get a new d.t. with two objects isomorphic to two objects of the d.t. (2.10). hence the third objects of these d.t. will be isomorphic. Hence, we are reduced to treat the case of M Mod coh ( X ). We may argue locally and replace M with a bounded complex of finite free X - modules. It reduces to the case where M = X. (ii) It is enough to prove the inclusion char( (M)) char(m). We argue by induction on the amplitude of M. Assume H j (M) = 0 for j / [j 0, j 1 ]. Consider the distinguished triangle (2.10) Applying the functor we find the d.t. (τ >j 0 M) M (H j 0 (M))[j 0 ] +1 Since char(m) = char(h j 0 (M)) char(τ >j 0 (M)), the induction proceeds, and we are reduced to the case where M is a coherent X -module. Then the result follows from Theorem (iii). q.e.d.

34 34 CHAPTER 2. OPERATIONS ON -MOUES External product et X and Y be two manifolds. For a X -module M and a Y -module N, we define their external product, denoted M N, by M N := X Y X Y (M N ). Note that the functor M M N is exact. Theorem et M b coh ( X) and N b coh ( Y ). Then M N b coh ( X Y ) and char(m N ) = char(m) char(n ). Proof. (o) By dévissage, one reduces to the case where M Mod coh ( X ) and N Mod coh ( Y ). (i) et us show that M N is coherent. Consider finite free resolutions of M and N : Then ( X ) N 1 P ( X ) N 0 M 0, ( Y ) M 1 Q ( Y ) M 0 N 0. ( X Y ) N 1+M 1 ( ) P 0 0 Q ( X Y ) N 0+M 0 M N 0 is a finite free resolution of M N over X Y. To conclude, apply the exact functor X Y X Y to this sequence. (ii) et us endow M and N with good filtrations Fl M and Fl N. Set Fl k (M N ) = i+j=k Fl i (M) Fl j (N ). Then {Fl k (M N )} k is a good filtration on M N and the result follows from gr (M N ) gr (M) gr gr (N ). Example et Z i be a smooth submanifold of X i (i = 1, 2). Then B Z1 Z 2 X 1 X 2 B Z1 X 1 B Z2 X 2. q.e.d.

35 2.4. TRANSFERT BIMOUE Transfert bimodule et f : X Y be a morphism of complex manifolds. We shall construct a ( X, f 1 Y )-bimodule denoted X Y which shall allow one to pass from left Y -modules to left X -modules and from right X -modules to right Y -modules. Set X Y = O X f 1 O Y f 1 Y. This sheaf on X is naturally endowed with a structure of an (O X, f 1 Y )- bimodule. We shall endow it of a structure of a left X -module by defining the action Θ X and verifying that this action satisfies the hypothesis of emma??. et v Θ X. Then f v O X f 1 O Y f 1 Θ Y. Hence f v = j a j w j, with a j O X and w j f 1 Θ Y. efine the action of v on a P O X f 1 O Y f 1 Y by setting (2.11) v(a P ) = v(a) P + j aa j w j P. If one chooses a local coordinate system (y 1,..., y m ) on Y and writes f = (f 1,..., f m ), then v(f ϕ) = m j=1 v(f j ) ϕ y j, which implies f v = m v(f j ) yj. j=1 A section P of X Y may formally be written as P = α a α(x) α y. By composing the monomorphism Y Hom CY (O Y, O Y ) with X Y = O X f 1 O Y f 1 Y we get the monomorphisms X Y O X f 1 O Y f 1 Hom CY (O Y, O Y ) Hom CX (f 1 O Y, O X )

36 36 CHAPTER 2. OPERATIONS ON -MOUES and the section 1 X Y := 1 1 X Y corresponds to the canonical morphism Note that Y being flat over O Y, f 1 O Y O X ϕ ϕ f. X Y O X f 1 O Y f 1 Y. One also introduces the (f 1 Y, X )-bimodule Y X by setting Y X = Ω X OX X Y f 1 O Y f 1 Ω 1 Y. Proposition et f : X Y, g : Y Z be morphisms of manifolds and set h = g f : X Z. Then there is an isomorphism of ( X, h 1 Z )- bimodules (2.12) X Y f 1 Y f 1 Y Z X Z. In particular, the left hand side is concentrated in degree zero. Proof. One has the isomorphisms of (O X, h 1 Z )-bimodules: X Y f 1 Y f 1 Y Z = (O X f 1 O Y f 1 Y ) f 1 Y f 1 (O Y g 1 O Z g 1 Z ) O X f 1 O Y (f 1 Y f 1 Y f 1 O Y h 1 O Z h 1 Z ) O X h 1 O Z h 1 Z O X h 1 O Z h 1 Z. (Recall that Z is flat over O Z.) Then, one checks that these isomorphisms extend as isomorphisms of ( X, h 1 Z )-bimodules. q.e.d. Proposition and f 1 Y -flat. (i) Assume f is smooth. Then X Y is X -coherent (ii) Assume f is a closed embedding. Then X Y is Y -coherent and X -flat. Proof. (i) Since the problem is local on X, we may assume that X = Z Y and f is the second projection. In this case, X Y O Z Y. Note that if x = (t, y) is a local coordinate system on Z Y with t = (t 1,..., t m ), then X Y X / X t

37 2.4. TRANSFERT BIMOUE 37 where X t denotes the left ideal generated by ( t1,..., tm ). (ii) For a local coordinate system y = (t, x) on Y such that X = {t = 0}, we have X Y Y /t Y where t Y denotes the right ideal generated by (t 1,..., t m ). q.e.d. If f is smooth, one has X Y X / X Θ f where X Θ f denotes the left ideal generated by the vector fields tangent to the leaves of f. If f is a closed embedding, one has X Y Y /I X Y where I X Y denotes the right ideal generated sections of O Y vanishing on X. Notice that any morphism f : X Y may be decomposed as f : X X Y Y where the first map is the graph (closed) embedding and the second map is the (smooth) projection. Example One has X pt O X and pt X Ω X. Proposition There is a natural isomorphism of ( X, f 1 Y )-modules X Y B Γf X Y f 1 O Y which associates δ Γf to 1 X Y X Y. Proof. Applying Proposition 1.5.4, we get X Y X Y Hence, it is enough to prove the isomorphism f 1 Ω Y Y B Y Y q 1 2 O q 1 2 Ω Y. X Y f 1 Y f 1 B Y Y B Γf X Y. This result follows from Exercise 2.1. q.e.d.

38 38 CHAPTER 2. OPERATIONS ON -MOUES Inverse and direct images of -modules efinition et f : X Y be a morphism of complex manifolds. (i) One defines the inverse image functor f 1 : b ( Y ) b ( X ) by setting for N b ( Y ): f 1 N := X Y f 1 Y f 1 N. (ii) One defines the direct image functors f, f! : b ( X ) b ( Y ) by setting for M b ( op X ): f M := Rf (M X Y ), f! M := Rf! (M X Y ). Using the bimodule Y X, one defines similarly the inverse image of a right Y -module or the direct images of a left X -module. Note that, if g : Y Z is another morphism of complex manifolds, we have (2.13) (g f) 1 f 1 g 1, (2.14) (g f) g f, (2.15) (g f)! g! f!. 2.5 Non characteristic inverse images efinition et N be a coherent Y -module. One says that f is non characteristic for N (or N is non characteristic for f) if f is non characteristic for char(n ). (See efinition ) Example (i) Since char(o Y ) = TY Y, the Y -module O Y is non characteristic for any morphism f : X Y. Note that f 1 O Y O X. (ii) See Exercise 2.1. Example Assume to be given a coordinate system (y) = (x 1,..., x n, t) = (x, t) on Y such that X = {t = 0}. et P be a differential operator of order m. Then X is non-characteristic with respect to P (i.e., for the Y -module Y / Y P ) in a neighborhood of (x 0, 0) X if and only if P is written as (2.16) P (x, t; x, t ) = a j (x, t, x ) j t 0 j m where a j (x, t, x ) is a differential operator not depending on t of order m j and a m (x, t) (which is a holomorphic function on Y ) satisfies: a m (x 0, 0) 0.

39 2.5. NON CHARACTERISTIC INVERSE IMAGES 39 emma et X, Y and P be as in Example et N = Y / Y P. Then X Y Y N m X. Proof. Notice that X Y N Y /(t Y + Y P ). By the Weierstrass preparation theorem, any Q(x, t, x, t ) Y written uniquely as may be Q(x, t, x, t ) = S(x, t, x, t ) P (x, t, x, t ) + m 1 j=0 R j (x, t, x ) j t. Hence, Q(x, t, x, t ) Y may be written uniquely as Q(x, t, x, t ) = S(x, t, x, t ) P (x, t, x, t ) + t T (x, t, x ) + m 1 j=0 P j (x, x ) j t. q.e.d. Theorem et N Mod coh ( Y ) and assume that f is non characteristic for N. Then (a) f 1 N is concentrated in degree 0, (b) f 1 N is X-coherent, (c) char(f 1 N ) f dfπ 1 char(n ). Remark In fact, there is a better result, namely char(f 1 N ) = f d fπ 1 char(n ) and the characteristic cycle of f 1 N is the image by f dfπ 1 of the characteristic cycle of N (see [20]). Proof. The map f : X Y decomposes as X h X Y p Y where h is the graph embedding and p is the projection. Using (2.13) and emma 2.2.4, it is enough to prove the result for p and for h. Hence, we shall treat separately the case where f is smooth and the case where f is a closed embedding.