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A RAPID INTRODUCTION TO HODGE MODULES, OR 0 TO 60 IN 3 HOURS.

DONU ARAPURA

1. Classical Our somewhat daunting task is understand what a (mixed) Hodge module is. Let’s start with the case where the base variety is a point, where it’s already com- plicated enough. Then a (mixed) Hodge module is the same thing as a (mixed) . Before getting to the definition, we will first do a simple ex- ample. Let L ⊂ Cn be a lattice, so that the quotient X = Cn/L is a complex torus. The first de Rham H1(X, C) has a basis given by the classes of dx1, dy1, dx2,.... A different basis is dz1, dz2,... together with their conjugates. If we set F 1 ⊂ H1(X, C) to be the span of the dz’s, then we see that 1 1 1 H (X, C) = F ⊕ F This is a basic example of a Hodge structure. Recall also that in order for X to be an abelian variety, i.e. a torus which also a projective variety, we need the existence of a Riemann form or polarization. For our purposes, this is a skew symmetric form 1 1 Q : LQ × LQ → Q such that F ⊥ F and Q(iu, v¯) is hermitian positive definite. A pure Hodge structure of weight n consists of a finitely generated group HZ, descending filtration p+1 p ...F ⊆ F ⊆ ...H := HZ ⊗ C such that for every p, n−p+1 H = F p ⊕ F

A rational or real Hodge structure is defined as above, but HZ is replaced by a Q or R vector space. An equivalent, and probably more familiar, formulation is that there exists a bigrading M H = Hpq p+q=n such that Hpq = Hqp. We can go back and forth via M 0 F p = Hp q p0≥p and q Hpq = F p ∩ F We can define a morphism of Hodge structures to be homomorphism of abelian groups preserving the filtrations/bigradings. The two descriptions yield isomorphic categories. Using the second, it is easy to see that

Date: August 11, 2018. 1 2 DONU ARAPURA

Proposition 1.1. The category of pure Hodge structures of fixed weight is abelian, and the functors GrpF are exact. To appreciate the last proposition, it is good to keep in mind that the category of filtered vector spaces is not abelian1. The basic examples of Hodge structures are given below.

Theorem 1.2 (Hodge). If X is a smooth complex projective variety, then Hn(X, Z) carries a canonical Hodge structure of weight n, with pq ∼ q p (A) H = H (X, ΩX ) We’ll outline the proof, and refer to [A, GH, V, W] for the details. One starts by choosing a K¨ahlermetric g, such as a Fubini-Study metric with respect to an embedding X ⊂ PN . The metric determines inner products on the spaces of differential forms. So we can form adjoints d∗, ∂¯∗ to the exterior derivative and Cauchy-Riemann operators. We form the Laplacians ∗ ∗ ¯¯∗ ¯∗ ¯ ∆ = dd + d d, ∆∂¯ = ∂∂ + ∂ ∂ Then the analytic part of the Hodge theorem can be summarized as follows: Theorem 1.3. Each de Rham cohomology class has a unique representative which is harmonic in the sense that ∆α = 0. Furthermore

(B) ∆ = 2∆∂¯ holds when the metric is K¨ahler. n Thus we have an isomorphism between Hn(X, C) and the space Harm (X, g) of harmonic n-forms. Since ∆ is real, α is harmonic if and only ifα ¯ is harmonic. The K¨ahleridentity (B) implies α is harmonic if and only if all its (p, q) parts are harmonic. Thus we can decompose M Harmn(X, g) = Harmpq(X, g) into harmonic (p, q)-forms, and this gives a Hodge structure. Furthermore (A) holds by Dolbeault’s theorem. However, this Hodge structure depends on g. To make it canonical, we observe that by the holomorphic Poincar´elemma, we have an exact sequence of sheaves d 1 d 0 → C → OX → ΩX → ... This gives an isomorphism i ∼ n ∼ n • H (X, Z) ⊗ C = H (X, C) = H (X, ΩX ) where the right side is hypercohomology. See Gelfand-Manin or Weibel [GM, W] for an explanation of this and other constructions from used here. Now define the canonical Hodge filtration by p n ∼ n ≥p F H (X, C) = im H (X, ΩX ) Obviously, this does not depend on a metric, but one does need to compare it to the previous construction to see that this really is a Hodge structure.

1In an morphisms have kernels and cokernels, and a morphism is an isomor- phism if has zero kernel and cokernel. In the category of filtered spaces, the first property is fine but the last fails. HODGE MODULES 3

Given a bounded complex C• of modules with finite filtration F , it is generally not true that p i • ∼ i p • (C) GrF H (C ) = H (Gr (C )) The correct relationship is given by a spectral sequence pq p+q p p+q (D) E1 = H (Gr C) ⇒ H (C) pq The right side of (C) should be replaced by E∞ in general. We say that the filtration is strict if any of the equivalent conditions hold: Lemma 1.4. The following are equivalent for a complex of modules. (a) (C) holds (b) The spectral sequence (D) degenerates i.e. E1 = E∞. (c) The maps Hi(F pC) → Hi(C) are injective.

Proof. Easy exercise.  We can extend the above definition and proposition to a complex of sheaves by replacing cohomology by hypercohomology. For a complex , we obtain a spectral sequence pq q p p+q E1 = H (X, ΩX ) ⇒ H (X, C) called the Hodge to de Rham or Fr¨olicher spectral sequence. Proposition 1.5. When X is a smooth projective vairiety, the last spectral se- quence degenerates, consequently the filtration is strict i.e. i ≥p i • H (X, ΩX ) → H (X, ΩX ) is injective. Proof. Theorem 1.2 implies that n X pq dim H (X) = E1 p+q=n

So we must have dim E∞ = dim E1. The second statement follows from the previous proposition.  We note that purely algebraic proofs of the proposition were found by Faltings and Deligne-Illusie [DI] in the 1980’s. Nevertheless, analytic arguments are still needed to deduce the full Hodge theorem. A polarization on a Hodge structure H of weight n is a -valued form Q, Q0 0 symmetric/skew-symmetric for n even/odd such that Hpq ⊥ Hp q and Q(Cu, v¯) is hermitian positive definite, where C acts by multiplication by ip−q on Hpq.A Hodge structure is polarizable if it possesses a polarization. The following fact is an easy generalization of the Poincar´ereduciblity theorem. Proposition 1.6. The category of pure polarizable rational Hodge structures of fixed weight is semisimple, i.e. every Hodge structure is a direct sum of simple Hodge structures. The reason for this is simple. The orthogonal complement of a sub Hodge struc- ture, with respect to a polarization, gives a complementary Hodge structure. We want to explain that the cohomology of projective manifold carries a po- larization depending on an embedding X ⊂ N . If H ⊂ N is a hyperplane, the P PC 4 DONU ARAPURA class [X ∩ H] ∈ H2(X, Q) is well defined. By Poincar´eduality, we can identify this with a cohomology class [X ∩H] ∈ H2(X, Q). As a de Rham class it is represented by the K¨ahlerform of the Fubini-Study metric. Cup product with this class, or wedge product with the K¨aherform, will be denoted by L. Let n = dim X denote the dimension as a complex manifold (so 2n is the real dimension). The following is proved with the help of further K¨ahleridentities. Theorem 1.7 (Hard Lefschetz). (1) We have isomorphisms i n−i ∼ n+i L : H (X, Q) → H (X, Q) (2) Let P n−i(X, Q) = ker Li+1 : Hn−i(X, Q) → Hn+i+2(X, Q). This is called primitive cohomology. Then i i i−1 2 i−2 H (X, Q) = P (X, Q) ⊕ LP (X, Q) ⊕ L P (X, Q) ⊕ ... This rather complicated statement has a number of important consequences. For example, it implies that the even (and odd) Betti numbers form an increasing sequence b0 ≤ b2 ≤ b4 ... up to bn. Another consequence, involving the Leray spectral sequence will be explained later on. Theorem 1.8. The pairing Z 1 k(k−1) n−k Q(α, β) = (−1) 2 L α ∧ β X defines a polarization on P k(X) Corollary 1.9. Hk(X) is polarizable. If X is an arbitrary, possibly singular or nonprojective, variety, then it is no longer true that H∗(X) carries a natural pure Hodge structure. The correct gener- alization is due to Deligne who defined the category of mixed Hodge structures. A (rational) mixed Hodge stucture H carries two filtrations, the weight filtration W• • W and and Hodge F . W• is defined over Q. The quotient Grk H = WkH/Wk−1H, with filtration induced from F is required to be a pure Hodge structure of weight W k. A mixed Hodge structure is polarizable if each Grk H is polarizable in the above sense. A pure Hodge structure can be regarded as a mixed Hodge structure W where Grk H = 0 for all but one k. A morphism is Q-linear map preserving both filtrations. Theorem 1.10 (Deligne). An arbitrary complex algebraic variety carries a canoni- cal polarizable2 mixed Hodge. When X is smooth and projective, this coincides with the pure Hodge structure given above. While we won’t say anything about the proof, it may be good to have a simple example to understand what W measures. Let E be the an . Choose two points p, q ∈ E, and pinch them together to obtain a nodal curve X. Let π : E → X be the projection. The homology H1(X, Z) = π∗H1(E) ⊕ Zγ, where γ is any loop passing through the node. This not canonical, what is canonical is the subspace π∗H1(E). Dually, we have canonical subspace 1 1 W0 = ker H (X, Z) → H (E, Z)

2Deligne doesn’t state this explicitly in [D2], but it follows from his proof. HODGE MODULES 5

Furthermore the subspace F 1H1(E, C) can be naturally lifted to H1(X, C). Taken together, these subspaces determine the mixed Hodge structure on H1(X). The weight filtration also has an arithmetic intepretation involving the Galois action on ´etalecohomology; see [D3].

2. Cohomology of families of varieties Given a family of smooth projective varieties f : Y → X over a complex mani- S i −1 fold, we can consider the family of Hodge structures H (Yx), Yx = f (x). The first thing to note Y → X is a C∞ fibre bundle by Ereshmann’s theorem, this i implies that the R f∗Q which is the sheaf associated to the presheaf U 7→ Hi(f −1U, Q), is locally constant. It need not be constant however. The nontrivial- i ity is measured by the monodromy representation π1(X, x) → Aut(H (Yx, Q)). At this point, we should recall Theorem 2.1 (Riemann-Hilbert I). Let X be a complex manifold and k a field. There is an equivalence between the categories of: (1) locally constant sheaves of k-vector spaces (also called local systems), (2) k-linear representations of the fundamental group, (3) and when k = C, holomorphic vector bundles V with an integrable connec- 1 tions ∇ : V → ΩX ⊗ V . The last item needs a bit more explanation. A connection is a C-linear map sat- isfying the Leibnitz rule ∇fv = df ⊗v +f∇v. In local coordinates, ∇ is determined P by the endomorphisms ∇∂i given by ∇v = dxi ⊗ ∇∂i v. Integrability is the con- dition that ∇∂i commute. Existence and uniqueness theorems for PDE guarantee that if ∇ is integrable, then ker ∇ is a locally constant sheaf of the same rank as V .

Conversely, given a locally constant sheaf L of C-vector spaces, OX ⊗C L is a holo- morphic vector bundle. It carries an integrable connection ∇ such that ker ∇ = L. Integrability has another interpretation which we recall. The sheaf of rings DX of holomorphic differential operators is locally generated by x1, . . . , xn, ∂1, . . . , ∂n subject to the Weyl relations [xi, xj] = [∂i, ∂j] = 0 and [xi, ∂j] = δij. Integrability is precisely the condition for the action of vector fields on V , given by ∂i ·v = ∇∂i v, to extend to a left DX -module structure. i Returning to our example, V = OX ⊗Q R f∗Q carries an integrable connection i ∇ such ker ∇ = R f∗C. This is called the Gauss-Manin connection. The Hodge filtrations form subbundles F p ⊆ V . These are not generally stable under ∇. Instead we have a weaker property

p 1 p−1 Theorem 2.2 (Griffiths transversality). ∇(F ) ⊂ ΩX ⊗ F We sketch a proof when X is a curve, since similar ideas will occur later on. We realize i • V = R f∗ΩY/X ≥p as relative de Rham cohomology. The filtration ΩY/X induces

p i ≥p F = im R f∗ΩY/X We note that, as before, this is strict. Associated to ∗ 1 •−1 • • 0 → f ΩX ⊗ ΩY/X → ΩY → ΩY/X → 0 6 DONU ARAPURA is a connecting map i • 1 • R f∗ΩY/X → ΩX ⊗ Rf∗ΩY/X Katz and Oda showed that this precisely ∇. By comparing to the connecting map for ∗ 1 ≥p−1 ≥p ≥p 0 → f ΩX ⊗ ΩY/X → ΩY → ΩY/X → 0 we see that F get shifted as above under ∇. Griffiths transversality has a natural interpretation in the context of D-modules. The ring DX has filtration by order. FkDX is the subsheaf of operators with locally −` at most k partial derivatives. If we set F`V = F V . Then Griffiths transversality is just the compatibility condition

(E) FkDX · F`V ⊂ Fk+`V

We note also that F•V satisfies the sort of finiteness conditions that make it a good filtration in the sense of D-module theory. We now give a name to this sort of structure. A rational variation of Hodge structure of weight i on a manifold Y consists of a locally constant sheaf L of Q- p vector spaces, subbundle F ⊂ V = OX ⊗ L satisfying Griffiths transversality such • ∼ that for each x,(Lx,Fx ⊂ Vx = C ⊗ Lx) is a pure Hodge structure of weight i. For all the deeper properties, it is important to require the existence of a polarization, which a quadratic form Q on the L which gives a polarization, in the previous sense, on all the fibres. Polarizability is automatic for variations arising from geometry. Theorem 2.3 (Schmid). The monodromy representation of a polarizable variation of Hodge structure is semisimple. Finally, we have the following: Theorem 2.4 (Deligne). If f : Y → X is a smooth projective map, then the Leray spectral sequence pq p q p+q E2 = H (X,R f∗Q) ⇒ H (Y, Q) degenerates. This is deduced in [D1] from the hard Lefschetz theorem and the following more abstract statement: Theorem 2.5 (Deligne). If an object A in a bounded of an abelian category admits a degree 2 endomorphism satisfying hard Lefschetz, then it splits noncanonically M A =∼ Hi(A)[−i] i

3. Zucker’s theorem We want to briefly explain the one piece of analysis that goes into Saito’s theory. Suppose that we have a smooth algebraic curve X. This can be completed to a smooth projective curve X by adding a finite number of points E. Let j : X → X denote the inclusion. Given a local system L on X, the intersection cohomology group is defined as i i+1 IH (X, L) := H (X, j∗L) HODGE MODULES 7

NB: In the literature, there are several different conventions about how to index intersection cohomology [BBD, GM]; to be consistent with later sections, we follow the rule that it should be symmetric about 0. In this case, the degrees are −1, 0, 1. Theorem 3.1 (Zucker). If (L,V,F •, ∇,Q) is a polarized variation of Hodge struc- ture on X of weight n, then IHi(X, L) carries a polarized Hodge structure of weight n + i + 1. To prove this, Zucker chooses a metric, with K¨ahlerform ω, which looks like the ∗ Poincar´emetric on ∆ √ −1dz ∧ dz¯ 2|z|2(log |z|2)2 around each point of E. The sheaf Ei(V ) of V -valued i-forms is the tensor product ∞ i i of the sheaf of C i-forms E with V . Let E (V )(2) be the sheaf of measurable V -valued i-forms α such that α and ∇α (understood as a distribution) are both locally L2 in the sense that Z Z ||α||2ω, ||∇α||2ω < ∞ K K for compact sets K ⊂ X. The pointwise norm || − ||2 is defined using the polariza- tion. The key step is showing that

• j∗L ⊗ C → E (V )(2), • i is a resolution. Since the sheaves E (V )(2) are fine, this implies that IH (X, L) ⊗ C can be identified with i • H (Γ(X, E (V )(2))) Standard Hodge theoretic arguments show that this is isomorphic to the space of harmonic L2 forms. Then an extension of the K¨ahleridentities due to Deligne finishes the job. With this notation, we can be more explicit about the Hodge filtration. We filter • p • E (V )(2) so that sections of F E (V )(2)|X are locally in dz ⊗ F p−1V + dz ∧ dz¯ ⊗ F p−1V + 1 ⊗ F pV + dz¯ ⊗ F pV Theorem 3.2 (Zucker). The Hodge filtration on IHi(X, L) is induced by the above filtration, and it is strict.

4. D-modules Before making the jump to Hodge modules, we need to quickly review a few concepts from D-module theory; referring to [HTT] for more details. First of all, we should point out we mainly work with left modules, but right modules are more natural in some contexts. On a smooth variety, the canonical module ωX is naturally a right module; tensoring with it or its dual can be used to switch between left and right modules. Let us recall the construction of the direct image. Although this operation is easier for right modules, we stick to left modules. Since any map can be factored as the inclusion of the graph followed by a projection, it is only necessary to understand these two cases. Given a closed immersion f : X → Y of smooth varieties and a 8 DONU ARAPURA

DX -module M, we can form a new DY -module f+M. If X is defined by an equation t = 0, then ∞ M n (F) f+M = M∂t n=0 n n n+1 where the symbol ∂t is formal. We have ∂t(m ⊗ ∂t ) = m ⊗ ∂t , but the other operators act less obviously. If f : X = Y × T → Y is a projection, then direct image only really makes sense in the setting of derived categories, or roughly as a complex considered up to quasi-isomorphism. The derived direct image

(G) f+M = Rf∗DRX/Y (M) where 1 DRX/Y (M) = (M → ΩX/Y ⊗ M → ...)[dim T ] is the relative de Rham complex. The shift [dim T ] means that the complex starts in degree − dim T . Rather than working with arbitrary D-modules, we restrict our attention to the class of regular holonomic modules. For the sake of expedience we give a nonstandard definition. Let X be a smooth variety. A vector bundle (V, ∇) with integrable connection on X is regular if there exists a smooth compactification X such that E = X − X has simple normal crossings, and an extension to a vector bundle V on X with an operator 1 ∇ : V → ΩX (log E) ⊗ V Let us say that a D-module M on a smooth variety is regular holonomic if it posses a finite composition series such that every simple factor restricts to a regular integrable connection on an open subset of its support. The (derived) category of these modules is stable under direct images and other standard operations. Given a regular holonomic module M, the complex obtained by applying the de Rham functor 1 K = DR(M) = (M → ΩX ⊗ M → ...)[dim X] has the following properties: (1) K is cohomologically constructible and bounded, i.e. X can be partitioned into Zariski locally constant sets on which Hi(K) restricts to a locally constant sheaf of finite rank, and all but finitely many Hi(K) are zero. (2) semiperversity: dim supp Hi(K) ≤ −i (3) The Verdier dual DK = RHom(K, C)[2 dim X] in the bounded derived category is also semiperverse. Any object in the derived category of sheaves of vector spaces satisfying these 3 conditions is called a of C vector spaces. Aside D-module theory, perverse sheaves also arise in intersection cohomology [GM]. The complex IC(L)

3The terminology is unfortunate. As the inventors themselves observe in the introduction to [BBD], “Les faisceaux pervers n’etant ni de faisceaux, ni pervers...” [Perverse sheaves are neither sheaves, nor perverse...] HODGE MODULES 9 which computes intersection cohomology with coefficients in a local system is per- verse. When the base is a curve, IC(L) = j∗L[1] occurred implicitly in the previous section. The category of perverse sheaves P erv(X) is known to be abelian. Theorem 4.1 (Riemann-Hilbert II (Kashiwara)). The functor DR gives an equiv- alence between the categories of regular holonomic modules and the category of complex perverse sheaves. Many of the operations on the D-module side have analogues on the perverse side. For instance, the operation M • → Hi(M •) which takes a complex of regular holonomic modules to it cohomology corresponds to perverse cohomology pHi : b Dc(X) → P erv(X) on the constructible derived category. 5. Hodge modules We are ready to introduce Hodge modules. These should be viewed as variations of Hodge structures with (controlled) singularities. When passing from variations to Hodge modules, filtered bundles with connection are replaced by a filtered regular holonomic D-modules, and local systems by a perverse sheaves. The perverse sheaf is defined over Q, and its role is to supply the rational structure. So to give a Hodge module, we need a regular holonomic D-module M with filtration F (satisfying (E) and appropriate finiteness conditions), a rational perverse sheaf L, and a fixed isomorphism α : DR(M) =∼ L ⊗ C These are subject to some delicate inductive axioms in the original paper [S1], that we postpone discussing. For the moment, we only spell out the alternative description, deduced after the fact, in the second paper [S2]. Theorem 5.1 (Saito). Let X be a smooth variety. The category HM(X, n) of Hodge modules4 of weight n on X is a subcategory of quadruples (M,F, L, α) de- scribed above, which is abelian and semisimple. Given an irreducible closed subva- riety Z ⊂ X, a polarizable variation of Hodge structure of weight n + dim Z on a smooth open subset of Z can be extended to object of HM(X, n). All simple objects are of this form. We want to explain how a variation of Hodge structure is extended to a Hodge module. We start with a special case now, and finish the story in the next section. Let X be smooth projective, let E ⊂ X be a divisor with simple normal crossings, and let j : U → X denote the inclusion of X − E. We assume that we are given polarizable variation of Hodge structures (L, ...) on U. We recall that V = OX ⊗ L is a filtered DU -module. We want to extend this X. We do this in steps. First, we use a result of Schmid that the connection ∇ has regular singularities, so that we have an extension of (V, ∇) to a log connection (V, ∇) on X. This extension is not unique, but we can make it so by choosing the eigenvalues of the residues of ∇ S to lie in the interval (−1, 0]. Note that the action of ∇ turns V (∗E) = n V (nE) into a DX -module. Let M ⊂ V (∗E) be the sub DX -module generated by V . This is a regular holonomic module extending V in the minimal way in the sense that it has no subquotients supported on D. This is part of the datum that will define a Hodge module.

4We are assuming that Hodge modules are polarizable. Saito denotes this category by either MH(X, n) or MH(X, n)p in his papers; MH stands for “modules de Hodge” presumably. 10 DONU ARAPURA

Next we define the Hodge filtration on M by X (H) FpM = FiDX (j∗Fp−iV ∩ V ) ⊂ M i≥0 Finally 1 DR(M) = M → ΩX ⊗ M → ... is isomorphic to the intersection cohomology complex IC(L)⊗C. So the Q-structure is given by the isomorphism DR(M) =∼ IC(L) ⊗ C This constitutes the basic example of a Hodge module.

6. Direct images of Hodge modules The main goal of Saito’s first paper [S1] was to construct and establish the properties of the direct image of a Hodge module. We first explain how to take direct images of filtered D-modules. We consider the two cases dealt with previously. If f : Y → X is a closed immersion with X defined by t = 0, the direct image was given by (F). If M is filtered by F•, we define M i Fpf+M = Fp−iM ⊗ ∂t Next suppose that f : X = Y × T → Y is a projection, the derived direct image was given by (G). We filter this by 1 Fpf+M = Rf∗(FpM → ΩX/Y ⊗ Fp+1M → ...)[dim T ] At the end of the day, we want modules rather complexes, so we take cohomology to i p i get H f+M. Under Riemann-Hilbert, the corresponding perverse sheaf is H (f∗L). We filter the module by i i FpH f+M = im H Fpf+M Theorem 6.1. (Saito) Let f : Y → X be a projective map and (M,F, L) ∈ HM(Y, n). Then (1) The filtration is strict in the sense that i i FpH f+M → H f+M is injective. i i p i (2) (H f+(M),F•H f+(M), H (f∗L)) forms a Hodge module on X of weight n + i. (3) The hard Lefschetz theorem holds. More precisely, cup product with a rela- tively ample class induces an isomorphism −i ∼ i H f+(M) = H f+(M)(i) The notation (i) is a so called Tate twist; it has the effect of shifting the weight on the right by −2i, so that it equals the weight on the left. The only analytic input in the proof is Zucker’s theorem §3. The proof is too complicated to go into, except for one case. When Y is a curve and X a point, part (2) of the theorem says that IHi(Y, L) carries a Hodge structure of weight n + i, and this is exactly Zucker’s theorem 3.1. In this case, part (1) can be deduced from theorem 3.2, and part (3) is also straightforward. HODGE MODULES 11

Now we want to finish the story started earlier. Suppose that (V,...) is a polar- ized variation of Hodge structure defined on a smooth locally closed subset U ⊂ X. We want to explain how to extend this to a Hodge module. Let Z = U¯, and let f : Z˜ → X be a resolution of singularities of Z which is an isomorphism over U and such that E = Z˜ − U has simple normal crossings. Then, as explained above, 0 0 we can extend V to a Hodge module M˜ on Z˜. Then M = H f+(M˜ ) gives an extension of V to X. If M 00 ⊂ M 0 is the largest submodule, supported on Z − U, then M = M 0/M 00 gives the minimal extension. When the previous theorem was combined with Deligne’s theorem 2.5, Saito ob- tained a Hodge theoretic proof, and generalization, of the decomposition of Beilin- son, Bernstein, Deligne, and Gabber [BBD]. Their original proof used characteristic p methods. A more elementary proof was found later by de Cataldo and Miglior- ini. We refer to their survey [dCM], for more information about the and its significance. Theorem 6.2 (Saito). Let f : Y → X be a morphism of complex projective vari- eties. If L is a perverse sheaf which is part of a Hodge module, then Rf∗L decom- b poses in Dc(X) as a direct sum of translates of perverse sheaves which arise from Hodge modules. In particular, this holds when Y is smooth and L = Q. b In this statement, Rf∗L decomposes in Dc(X). This can be refined to a Hodge theoretic statement using Saito’s category of mixed Hodge modules introduced in his second paper [S2]. Mixed Hodge modules are related to Hodge modules in the same way that mixed Hodge structures are related to pure Hodge structures. A mixed Hodge module on a smooth variety X consists of a regular holonomic D-module with two filtrations W and F , a filtered perverse sheaf (L,W ) and an isomorphism α, as above, compatible with W . Given a mixed Hodge module M, W the graded objects Grk M = WkM/Wk−1M ∈ HM(X, k). Any Hodge module can W be regarded as a mixed Hodge module where all but one Grk are zero. Theorem 6.3 (Saito). The category MHM(X) of mixed Hodge modules is abelian, W and the functors Grk : MHM(X) → HM(X, k) are exact. When X is a point, MHM(X) is just the category of polarizable mixed Hodge structures. We have forgetful map MHM(X) → P erv(X). This extends to a map on derived categories b b D MHM(X) → Dc(X) b b thanks to a theorem of Beilinson [B] that D P erv(X) is equivalent to Dc(X). Theorem 6.4 (Saito). Given a morphism f : Y → X, there is an operation b b f∗ : D MHM(Y ) → D MHM(X) compatible with the derived direct image Rf∗ : b b Dc(Y ) → Dc(X). As a corollary, we get the refined decomposition theorem promised above. Corollary 6.5. Let f : Y → X be a morphism of complex projective varieties. If b M is a Hodge module, then f∗M decomposes in D MHM(X) as a direct sum of translates of Hodge modules.

7. The Kashiwara-Malgrange filtration Although we gave a working definition of Hodge modules, for some applications, one needs to understand what’s happening under the hood. The original definition 12 DONU ARAPURA in [S1] is by induction on the dimension of support. We only give the rough idea, and refer the original papers and more detailed surveys [S3, SS] etc. for more in- formation. The base step of the induction goes as follows:

A Hodge module on a point is the same thing as polarizable Hodge structure. A Hodge module with zero dimensional support is a finite sum of modules of the form i+M, for inclusions of points i : x → X.

The next axiom says very roughly that a Hodge module restricts to a Hodge module on a hypersurface defined by a function. The precise formulation is rather delicate. We only explain the D-module aspects in broad outline. Suppose that, we have a smooth divisor E = f −1(0) for some regular function f : X → C. The assumptions about E are not too restrictive, because they can be achieved by shrinking X and replacing E by the inclusion of the graph in X × C. Let t be ∗ the coordinate on C, and let T ∈ π1(C ) be a generator. Let M be the regular holonomic module on X. We have a bunch of local systems associated to the perverse sheaf DR(M), and T will act on these. We say that M is quasi-unipotent if the eigenvalues of these actions are all roots of unity. For example, it is a theorem of Borel [Sc] that M is quasi-unipotent if it comes from a polarized variation of Hodge structure. Let us assume that M satisfies this quasi-unipotency condition, then M carries a decreasing filtration V αM, called the Kashiwara-Malgrange or simply the V -filtration, indexed by Q with discrete jumps and the following properties (1) tV αM ⊆ V α+1M with equality for α > 0 α α−1 (2) ∂tV M ⊆ V M α α >α (3) t∂t − α is nilpotent on GrV M = V M/V M. When M carries a filtration F , one can impose additional compatibility conditions between V and F sometimes called specializability. We won’t explain the precise conditions, but will point out two useful consequences Theorem 7.1. Suppose that (M,F ) is specializiable .

(1) If none of the simple factors of M are supported on E, then F•M is deter- mined by its restriction to X − E. The formula

X >0 FpM = FiDX (j∗Fp−i ∩ V M) i≥0 is similar to (H). • (2) If M is supported on E, then GrF M is annihilated by f, so that it becomes OE-module. A crude formulation of the inductive axiom is:

α A Hodge module (M,F ) is specializable, and GrV M with filtration induced F (but shifted) is a summand of the filtered D-module associated to a mixed Hodge module supported on E.

To state things more precisely would involve introducing vanishing cycle and nearby α functors of which GrV M is a constituent. But this would be too a long story. Finally, we add that although we focussed on the case where X is smooth, Saito defines Hodge modules on singular varieties as well. When X is embeddable into a HODGE MODULES 13 smooth variety Y , one can take HM(X, n) to consists of the subcategory of Hodge modules on Y supported on X. This is independent of Y .

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Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A. E-mail address: [email protected]