VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK

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1 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK TUDOR PĂDURARIU Contents 1. Introduction 1 2. Pure and mixed Hodge structures 2 3. Variations of Hodge structures over the punctured unit disk 4 4. Nearby cycles and the relative de Rham complex 7 References 9 1. Introduction Suppose we have a family of complex algebraic varieties X B parametrized by a complex manifold B, such that the generic fiber of this family is smooth. Then there exists an open subset B B which parametrizes the points with smooth preimage. The purpose of these notes is to explain how one can obtain information singular fibers via a study of the variation of Hodge structures on the smooth locus B. To be more precise, recall that last time we have defined variation of (polarized) Hodge structures. One would prefer to have such a vhs for the whole family X, but this is not possible because the singular fibers do not have a pure Hodge structure on the singular cohomology groups. Instead, by work of Deligne [1], they carry mixed Hodge structures, which can be thought as extensions of pure Hodge structures. However, let s observe that the singular cohomology (up to torsion) H n (X t, Z) has the same rank for any t B, so we might hope that we can associate to any singular point a Hodge structure with the same underlying abelian group, but which varies in the family X B. This Hodge structure can be taught as the limit of the pure (polarized) Hodge structures near the singular fiber. Schmid [3] proves that one can actually define such a limit as a mixed Hodge structure, and that this structure is compatible with the mixed Hodge structure on the singular fiber coming from Deligne s theorem. The plan for today is the following: in Section 2 we will recall the definition of pure polarized Hodge structures and we will give a brief overview of Deligne s theory of mixed Hodge structures, inclusing some of the ingredients that go into his proof. In Section 3 we recall the definition of variations and Hodge structure and explain how one might try to extend a variation on B to the whole space B. We will end 1

2 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 2 by stating Schmid s theorem. In section 4 we discuss a proof of the geometric case of Schmid s theorem due to Steenbrink [2, Chapter 11], and we end by stating some geometric applications of the main result, such as the local invariant cycle theorem. 2. Pure and mixed Hodge structures Throughout the notes, the varieties will be defined over C. We will also abuse notation and denote by H n (X, Z) (and call singular cohomology) what is actually the singular cohomology of X modulo torsion. Let s recall that for X a compact Kahler manifold, the singular cohomology (up to torsion) H := H n (X, Z) has a Hodge structure of weight n, meaning that there exists a decreasing filtration of H C = H n (X, C) 0 = F 0 H n (X, Z) F 1 H n (X, Z) F n H n (X, C) = H n (X, C) with F p H n F q H n = 0 and F p H n F q H n = H for p + q = n + 1. Such a filtration can be defined via the de Rham complex, and it satisfies the two above properties as a consequence of Hodge theory. Define H p,q := F p H n F q H n. Let s further recall that a polarized Hodge structure of weight n consists of a nondegenerate integer bilinear pairing Q on H C with Q(F p, F n p+1 ) = 0 and Q(Cφ, φ) > 0 for φ 0, where C is the Weil operator. The cohomology H n (X, C) has such a bilinear form Q(φ, ψ) = φ ψ ω n k, where ω is the Kahler form on X and k is the dimension of the form φ. An important example of Kahler manifolds are the complex projective algebraic varieties. A natural question is whether this structure can be generalized to other classes of spaces, for example to all algebraic varieties. One cannot define a pure Hodge structure, in general, but one can show that the singular cohomology H n (X, Z) is made up of pure Hodge structures, in a sense that will be made precise later. Before stating the general theorems, let s discuss an example which shows the need to extend the category of Hodge structures, if we want to associate a HS to every variety. Let X be a curve of genus g, and let U be the complement of n + 1 points in X. sequence: We can compute H n (U) using the relative cohomology long exact and thus that, for i = 1, H i (X) H i (U) H i+1 (X, U) H i+1 (X), 0 H 1 (X)/H 1 (X, U) H 1 (U) ker (H 2 (X, U) H 2 (X)) 0.

3 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 3 One can put a Hodge structure of weight i on H i (X, U) = H i 2 (X U)( 1) such that all the maps in the above long exact sequence are maps of Hodge structures. The short exact sequence exhibits H 1 (U) as an extension of a weight 1 piece H 1 (X)/H 1 (X, U) = Z g and a weight 2 piece ker (H 2 (X, U) H 2 (X)) = Z n. In general, we can imagine how a similar argument might be used to show that for any complex variety X, the singular cohomology H n (X, Z) is an extension of Hodge structures of various weights: first, as above, we might try to compactify X to a projective variety Y : this can be done such that the locus of points added is a divisor with strict normal crossing inside Y. Then, once we have a Hodge structure on Y, we can use the Hodge structure on the divisor Y X to put, via the relative cohomology sequence, a Hodge structure on X. Now, the question is how one can put a HS on Y, for Y projective, but not necessarily smooth. One can resolve Y by a smooth projective variety Z such that the exceptional locus is once again a divisor with smooth normal crossing, and one can patch together the HS on Z and on the exceptional locus to a HS on Y. One can also show this HS on H n (X, Z) is independent on the choices we have made, such as the compactification Y or the resolution of singularities Z Y. Definition 2.1. A mixed Hodge structure on an abelian group H consists of a finite decreasing (Hodge) filtration F p H C H C and a finite increasing weight filtration W i H Q H Q such that the induced filtration on gr W n H := W n C/W n 1 C given by F p gr W n H := (F p W n C + W n 1 C)/(W n 1 C) is a pure Hodge structure of weight n. Theorem 2.2. (Deligne) Let X be a complex variety. Then there exists a mixed Hodge structure on H n (X, Z) which recovers the standard pure Hodge structure for X smooth projective and which is functorial with respect to morphisms of varieties. We will give a sketch of proof for the above theorem in the case of a smooth, but not necesarily proper variety U. A similar idea was employed by Steenbrink to give an alternative proof of Schmid s theorem, which we will discuss in section 4. Let s begin by choosing a smooth compactification X of U such that D = X U is a divisor with smooth normal crossings inside X, meaning that all its irreducible components are smooth and locally D looks like the intersection of some finite number of transversal hyperplanes. Denote the inclusion by j : D X. A holomorphic form ω on U is said to have logarithmic singularities along D if ω and dω have at most a pole of order one along D, that is, if fω and fdω are elements of Ω (X), where f is a local equation for D. Observe that Ω X (log D) j Ω U. It can be more explictly given as follows: if D is a divisor with strict normal crossings

4 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 4 and locally around p X the divisor is given by an equation z 1 z k = 0, then Ω 1 X(log D) p = O X,p dz 1 z 1 O X,p dz k z k O X,p dz k+1 O X,p dz n, and, further, Ω p X (log D) p = p Ω 1 X(log D) p. Theorem 2.3. Let U be a complex smooth variety, X a projective smooth variety containing U, with D = X U a divisor with strict normal crossings. Then: (1) H k (U, C) = H k (X, Ω X (log D)). (2) Define the weight filtration W m Ω p X (log D) by 0, if m < 0, Ωp X (log D), for m p, and by Ω m p X Ωm X (log D) for 0 m p. Then this filtration induces in cohomology a filtration W m H k (U, C) = image (H k (X, W m k Ω Ẋ (log D)) Hk (U, C)) which can be defined over Q and which, together with the Hodge filtration, induces a mixed Hodge structure on H k (U, C). 3. Variations of Hodge structures over the punctured unit disk In the introduction, we said that Schmid s theorem gives a description families of smooth varieties near a degenerating point where it becomes singular. We will restrict, for simplicity, to the case where the smooth family X is over D, the unit disk with the origin removed. How can we capture the behaviour over the origin? First, we can extend our family to X D, where X 0, the fiber over 0, is a divisor with strict normal crossings inside X. Over D, we have a variation of pure Hodge structures, where the underlying abelian group is Z r = H n (X t, Z), for t 0. We have seen in the previous section that there exists a mixed Hodge structure on H n (X 0, Z). However, it is natural to ask whether this family of Hodge structures over D can be extended over 0 is some way. For this, we will need to put a Hodge structure on Z r which captures the behaviour of the family near 0. Schmid s theorem says that one can define a limiting Hodge structure on Z r which will be a mixed Hodge structure compatible with the mixed Hodge structure coming from Deligne s theorem on H n (X 0, Z). Before delving into the statement and the ingredients of Schmid s theorem, let s discuss an example of a variation of Hodge structure. Look at the family X D, where X := {y 2 = x(x 2)(x t)} P 2 D, for t D. Then all the fibers over t 0 are smooth, while the fiber over 0 is a nodal cubic. We blow up the node over t = 0 so that the exceptional curve will be a strict normal crossing divisor inside the whole family. We look at the variation of Hodge structures determined by the H 1 (X t, Z), for t D. Schmid s theorem will

5 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 5 put a mixed Hodge structure on the underlying abelian group Z 2. We know that H 1 (X 0, Z) = Z and that the map sp : H 1 (X 0, Z) H 1 (X t, Z) is a morphism of mixed Hodge structures, where the first one has the Deligne mhs, while the second one has the Schmid mhs determined by X 0. Similar to example 1, one can show that the mixed Hodge structure on H 1 (X 0, Z) is all supported in weight 0, so the weight 0 graded piece of H 1 (X t, Z) will need to be at least Z, the image of sp. However, we will see that the Schmid weight structure is determined by the monodromy operator acting on H 1 (X t, Z) = Z 2. This can be easily computed in our above example, and is given by T (a) = a and T (b) = 2a + b, where a and b are some generators of H 1 (X t, Z). The weight filtration will need to satisfy a Poincare duality type property centered at 1, the index of the cohomology group, and this will determine that the weight 2 piece is at least Z, and thus that the weight filtration is given by 0 W 0 = image of sp = Z W 1 W 2 = H 1 (X t, Z) = Z 2. Finally, observe that the image of the specialization map can be directly read from the monodromy. Indeed, the image of the map sp : H 1 (X 0, Z) H 1 (X t, Z) is exactly spanned by the cycles in H 1 (X t, Z) invariant under the monodromy T. Let s go back to the statement of Schmid s theorem. Schmid starts with a variation of (polarized) Hodge structure of weight n not necessarily coming from geometry. Let V be a holomorphic vector bundle with a flat connection : V V Ω 1 D, decreasing filtration F V by holomorphic subbundles, and a hermitian pairing Q : V V CD such that (1) Q is O D linear in both arguments, (2) dq(u, v) = Q( (u), v) + Q(u, (v)), (3) F V satisfies Griffiths transversality (F p V) Ω 1 F p 1 V, (4) for all t D, (F. V t, Q t ) is a polarized Hodge structure of weight n on the complex vector space V t. We can also put a metric on each fiber, using the decomposition H = p+q=n H p,q, h 2 = ( 1) p Q(h p,q, h p,q ). p+q=n

6 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 6 Back to Schmid s theorem, assume that we are given a vhs over D, which we will call V. The question is whether it can be extended to a vhs over D. First, we need to extend V to a holomorphic vector bundle on D. For this we need the following theorem: Theorem 3.1. Let V be a vhs over D. Then the monodromy operator T is quasiunipotent. More precisely, if l = max {p q V p,q t 0} and if T = T s T u is the Jordan decomposition of T with T u unipotent and T s semisimple, then and T s has a finite order. (T u I) l+1 = 0 We will recall in the next paragraph the definition of monodromy. Assuming the proposition, for each eigenvalue λ of T s, choose β with real part in [0, 1), so that T = e 2πR e 2πiN, with R semisimple with eigenvalues as above. Let s now discuss in more detail the monodromy T is defined. Along the way, we will also explain how to extend the holomorphic vector bundle V over D to a holomorphic vector bundle over D. From V, define V the corresponding local system, where V is the locally constant sheaf of flat sections of V. Further, if p : H D is the exponential map, define V = H 0 (H, p 1 V ). The monodromy T : V V is induced by the deck transformation z z + 1 of the exponential map. Now for any nonzero v V, s v : H V defined by satisfies s v (z) = e 2πizR e 2πizN v s v (z + 1) = T 1 s v (z), and therefore drops to a nowhere vanishing section s v H 0 (D, V). We thus have a found a trivialization of V on D, given by s v H 0 (D, V), for v V, and therefore a canonical way to extend V to a trivial holomorphic vector bundle V n on D. This means that we have successfully extended the vector bundle V to a holomorphic vector bundle V n. The next natural question is what happens with the connection. For this, we compute ds v = d(e 2πi(R N) v) = 2πidz e 2πiz(R N) (R N)v = 2πidz s (R N)v.

7 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 7 Taking into consideration that t = e 2πiz we find out that This implies that the connection extends to a connection (s v ) = dt t s (R N)v. : V V Ω 1 D n : V n V n Ω 1 D (log 0), a connection with logarithmic poles at the origin whose residue is R N End(V ). To make a short recapitulation, we have started with a variation of phs V on D. By Borel s theorem, it is quasiunipotent. We have extended V to a holomorphic vector bundle V n on D and the connection to n : V n V n Ω 1 D (log 0). Assume further that T s = 1. In particular, R = 0. This happens, for example, if the vhs comes from geometry and the divisor X 0 has multiplicities one along its irreducible components. Further, V V0 n is an isomorphism. Next, we need to investigate the other pieces of information from the vhs package, namely the Hodge subbundles. Theorem 3.2. (Schmid) The Hodge bundles F p V over D extend to holomorphic subbundles F p,n V n. In particular, we obtain a filtration F p,n 0 of V n 0 = V. Recall also that N acts on V. Lemma 3.3. Given a nilpotent endomorphism N of a finite dimensional vector space V, the weight filtration of N centered at k is the unique increasing filtration W = W (N, k) of V such that N(W i ) W i 2, for i 2, and such that N l : gr w k+l V grw k l V is an isomorphism for l 0, where gr w i V := W i 1/W i. As a consequence of the construction of the Hodge bundles F p,n and of the above lemma, one obtains, with some work, the following result: Theorem 3.4. The two filtrations F V and W V are part of a polarized mixed Hodge structure on V. 4. Nearby cycles and the relative de Rham complex In this section, we will focus on the geometric case and explain a proof of Schmid s theorem in this particular case, due to Steenbrink [2, Chapter 11]. Assume that we are given a smooth proper family X D. Extend it to a proper family X D

8 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 8 such that the fiber over 0 is a divisor with strict normal crossings X 0. Assume further that the multiplicities of the components of the divisor X 0 are all 1. This implies that T s = I. Denote the maps X k X i X j X 0, where the map k is the exponential map lying over H D. The space X is homotopic to any fiber X t, for t 0. One can think of X as the generic fiber of X without actually realizing it as the fiber over a point t D. Also, the whole space X can be contracted to the special fiber X 0. One can define the specialization map sp : H k (X 0 ) H k (X ) via sp : H k (X 0 ) = H k (X ) H k (X t ) = H k (X ). We can now define the vanishing cycle functor φ f : D b (X ) D b (X 0 ) by φ f (F ) = j (ik) (ik) F, where all the functors appearing in the definition of φ f are derived. Theorem 4.1. In the above setting, we have that H k (X ) = H k (φ f C X ). Using this theorem, we can compute the smooth fiber via a certain complex on X 0, but this does not seem in any case easier than computing directly the singular cohomology of X. One reason for using this theorem is that we can show that φ f is related to a de Rham type complex and we know, for example from Deligne s proof of the existence of mixed Hodge structures, that we have chances of defining Hodge and weight filtrations once we are in such a setting. More precisely, define where f : X D. Ω X /D (log E) = Ω X(log E)/f Ω 1 D(log 0) Ω 1 X (log E), Theorem 4.2. The complex φ f C X is quasiisomorphic to Ω X /D(log E). One can define a Hodge filtration by truncating the de Rham complex, and a weight filtration based on the order of the pole, which induce a mixed Hodge structure on H k (φ f C X ). In particular, this result puts a mixed Hodge structure on H k (X t ) capturing the behaviour of the family near the singular fiber X 0, giving an alternative proof to the geometric case of Schmid s theorem. There are two main observations that need to be made. First, the specialization map sp : H (X 0 ) H k (X )

9 VARIATIONS OF HODGE STRUCTURES AFTER SCHMID AND STEENBRINK 9 is a morphism of mixed Hodge structures. The second is that, because the above construction gives the same mhs as the one in Schmid s theorem, the weight filtration can be read from the nilpotent operator N = log(t ). Theorem 4.3. The weight filtration W on H k (X ) coincides with the weight filtration of N = log(t ) at k, and is an isomorphism of Hodge structures. N r : gr w k+r Hk (X ) gr w k r Hk (X )( r) Theorem 4.4. (local invariant cycle theorem) For a family X D as above, the sequence H k (X 0 ) sp H k (X ) T I H k (X ) is exact, that is, the image of the specialization map, which are cycles on the special fiber which extend to the generic fiber, are exactly the cycles on the generic fiber fixed by the monodromy. References [1] Deligne, Pierre. Theorie de Hodge: II. Publications Mathematiques de l IHES 40 (1971): [2] Peters, Chris AM, and Joseph HM Steenbrink. Mixed hodge structures. Vol. 52. Springer Science and Business Media, [3] Schmid, Wilfried. Variation of Hodge structure: the singularities of the period mapping. Inventiones mathematicae 22.3 (1973): Department of Mathematics, Massachusetts Institute of Techonology, 182 Memorial Drive, Cambridge, MA address: tpad@mit.edu