Extending Hodge Bundles for Abelian Variations

A. Némethi^{1}, Department of mathematics, Ohio State University, Columbus, OH 43210, U.S.A.

J.H.M. Steenbrink, Department of mathematics, Catholic University, Toernooiveld, 6525 ED Nijmegen, The Netherlands.

Abstract. We consider a local variation of Hodge structure with abelian monodromy group. We show the existence of a canonical limit mixed Hodge structure, and we prove the nilpotent orbit theorem in this context.

1 Introduction.

In [Schmid], the asymptotic behaviour of a (polarized) variation of Hodge structure over a product of discs and punctured discs has been considered. An essential feature of such a space is, that its fundamental group is abelian, and hence the monodromy group of any local system over it is abelian. In this paper we consider still a local situation: a variation of Hodge structure over B \ D where B is a sufficiently small polydisc and D is a divisor in B. In the case considered by Schmid D has normal crossing singularities at 0. Instead of supposing this, we consider variations of Hodge structure with abelian monodromy group, i.e. for which the monodromy representation factors via H_{1}(B \ D). Each irreducible component of D gives rise to a generator of H_{1}(B \ D) and hence of the monodromy group. Under the assumption that these generators are unipotent we show the existence of a canonical limit mixed Hodge structure (W, F) of the variation. Moreover we prove that, if {N_{i}}_{i=1}^{r} are the logarithms of the local monodromies around the irreducible components of D, then the filtration W agrees with the weight filtration of any element of the positive real cone spanned by N_{i}’s, and

θ(z_{1}, …, z_{r}) = exp(∑_{i=1}^{r} z_{i}N_{i}) · F

1Partially supported by NSF-grant No. DMS-9203482, and by the Netherlands Organisation for the Advancement of Scientific Research N.W.O.

is a nilpotent orbit.

The unipotence assumption is not essential: the pull-back to a finite ramified covering of B will have this property. This introduces some singularities which however are not a serious obstruction. The fact that B is smooth is never used in an essential way in the paper: the main point is that B \ D is smooth, a property preserved under the base change. For simplicity of exposition, we have written the proof for the smooth case. For the general case, see section 4.

One way to interpret our result is the following. Given a variation of Hodge structure on B \ D one may restrict to analytic arcs in B intersecting D only at 0. Each such arc gives a limit mixed Hodge structure (more precisely: a one-variable nilpotent orbit). Our result shows, that this limit mixed Hodge structure is independent of the choice of the analytic arc.

The authors thank the MSRI at Berkeley for its hospitality during May 1993.

2 Specialization.

(2.1) For the definition of the category of mixed Hodge modules MHM(X)^{p} (where X can be any separated and reduced complex analytic space), and for their basic functorial properties, see [Saito 1990]. Notice that the forgetful functor

Rat : MHM(X)^{p} → Perv(Q_{X})

which assigns to each mixed Hodge module on X the underlying perverse sheaf with Q–coefficients, commutes with the various direct and inverse image functors.

Let X be a complex manifold, D ⊂ X a divisor with simple normal crossings such that each component of D is smooth, and V a polarized variation of Hodge structure on X \ D whose local monodromy transformations around the irreducible components of D are unipotent. There are various ways to extend V to a mixed Hodge module on X (see [Saito 1990]), e.g. as j_{∗}V, Rj_{∗}V or IC(V). Here j : X \ D → X is the inclusion map and IC stands for ‘Intersection Complex’. Let M an arbitrary extension of V to a mixed Hodge module on X.

In [Saito 1991] section 3 the Verdier specialization of a mixed Hodge module M on X along a closed subvariety Y of X is considered (see also [Verdier]). This construction produces a mixed Hodge module Sp_{Y} M on the normal cone C_{Y} X of Y in X. We will make this explicit for the case of the extension M and Y an irreducible component of D.

Let V := V ⊗_{C} O_{X\D}. An integrable connection ∇ on V is defined by ∇(v ⊗ f) = v ⊗ df for v and f local sections of V and O_{X\D} respectively. By [Deligne 1970] Proposition 5.2, there exists a unique holomorphic subbundle Ṽ of j_{∗}V such that ∇ extends to a connection with logarithmic poles along D:

∇ : Ṽ → Ω_{X}^{1}(log D) ⊗ Ṽ

and such that the residue of ∇ along each component of D is nilpotent.

(2.2) Let Y be an irreducible component of D which is projective. The following construction produces a smooth compactification of C_{Y} X. As Y is smooth, C_{Y} X is just the total space of the normal bundle of Y in X, and a natural compactification of C_{Y} X is the projective space bundle P(N_{Y/X} ⊕ O_{Y}) =: P_{Y} X. Let π : P_{Y} X → Y denote the projection. We let Y_{0} and Y_{∞} denote the zero and infinity sections of P_{Y} X respectively. Let D′ denote the union of the components of D which are different from Y and let C = D′ ∩Y. Then C and π^{−1}(C) ∪ Y_{0} ∪ Y_{∞} are divisors with normal crossings on Y and P_{Y} X respectively. Let p : B_{Y} X → X × C denote the blowing-up of X × C with centre Y × {0}. Then p^{−1}(Y × {0}) = P_{Y} X, and p^{−1}(X × {0}) = P_{Y} X ∪ X′, where X′ is the strict transform of X × {0} in B_{Y} X; it is mapped isomorphically onto X × {0} by p, as Y is a divisor in X. Moreover P_{Y} X \ X′ ≅ C_{Y} X.

[figure: P_{Y} X; Y × {0}; B_{Y} X; X × C; C; X; ↪; ↪; f; p_{1}; p]

Let f : B_{Y} X → C denote the composition of p with the projection of X × C onto its second factor. Then Sp_{Y} M is defined as the restriction to C_{Y} X of ψ_{f}p^{∗}p_{1}^{∗}M, where p_{1} is the projection of X × C onto its first factor. (We neglect the shifts of complexes.) Let U = P_{Y} X \(π^{−1}(C)∪Y_{0} ∪Y_{∞}).

Theorem 1. Let V be a polarized variation of Hodge structure on X \ D whose local monodromy transformations around the irreducible components of D are unipotent. Let M be any extension of V to a mixed Hodge module on X. Then the restriction of Sp_{Y} M to U does not depend on the choice of extension, and it is an admissible variation of mixed Hodge structure in the sense of [Kashiwara].

If the canonical extension Ṽ is trivial, then the canonical extension to the space P_{Y} X of the restriction Rat(Sp_{Y} M|U) is also trivial. Proof. Let U_{B} = B_{Y} X \ [p^{−1}(p_{1}^{−1}(D)) ∪ X′]. Then

(ψ_{f}p^{∗}p_{1}^{∗}M)|U = ψ_{f}((p^{∗}p_{1}^{∗}M)|U_{B}) = ψ_{f}p^{∗}p_{1}^{∗}V.

In particular, this mixed Hodge module is independent of the extension M. Since the restriction f|U_{B} : U_{B} → C has the space U = (f|U_{B})^{−1}(0) as a smooth fiber, the above identities shows also that Rat(Sp_{Y} M)|U is a local system.

Consider the canonical extension (Ṽ, ∇) of V. Then the canonical extension of the local system V′ = p_{1}^{∗}V, defined on (X \ D) × C, is exactly p_{1}^{∗}(Ṽ, ∇) = (Ṽ⊗_{O_{X}} O_{X×C}, ∇⊗1). Consider now the local system V′′ = p^{∗}p_{1}^{∗}V on B_{Y} X \ p^{−1}(D × C). Its canonical extension is Ṽ′ := p^{∗}p_{1}^{∗}Ṽ. We denote its connection by:

∇_{B} : Ṽ′ → Ω_{B_{Y} X}^{1}(log p^{−1}(D × C)) ⊗ Ṽ′.

This connection, and the map f : B_{Y} X → C induce the relative connection:

∇_{B/C} : Ṽ′ → Ω_{B_{Y} X/C}^{1}(log p^{−1}(D × C)) ⊗ Ṽ′.

Here ∇_{B/C} = ∇_{B} modulo f^{∗}Ω_{C}^{1}(log{0}).

Tensoring this connection with O_{P_{Y} X} and using the identity

Ω_{B_{Y} X/C}^{1}(log p^{−1}(D × C)) ⊗ O_{P_{Y} X} = Ω_{P_{Y} X}^{1}(log π^{−1}(C) + Y_{0} + Y_{∞}),

we obtain a connection on P_{Y} X:

∇_{P_{Y} X} : Ṽ′′ → Ω_{P_{Y} X}^{1}(log π^{−1}(C) + Y_{0} + Y_{∞}) ⊗ Ṽ′′,

where Ṽ′′ = Ṽ′ ⊗_{O_{B_{Y} X}} O_{P_{Y} X}. Then it can be verified that the logarithmic connection (∇_{P_{Y} X}, Ṽ′′) is, in fact, the canonical extension of Rat(Sp_{Y} M|U) on P_{Y} X.

Therefore, the restriction (Sp_{Y} M)|U is a mixed Hodge module with Rat(Sp_{Y} M)|U a local system. In particular, it underlies a good variation of mixed Hodge structure. Its canonical extension is (p^{∗}p_{1}^{∗}Ṽ) ⊗ O_{P_{Y} X}, in particular it is trivial if the canonical extension Ṽ is trivial. □

(2.3) Let us consider the particular case that X has dimension one and D = Y is just a point. Then C_{Y} X = A^{1} ⊂ P^{1} = P_{Y} X and U ≅ C^{∗}. The variation of Hodge structure V has an limit nilpotent orbit, which is a variation of mixed Hodge structure on C^{∗}. This is exactly the restriction of Sp_{Y} M to U.

The construction of (2.2) is functorial in the following sense. If T is a complex submanifold of X which is transverse to D, then Sp_{Y ∩T} (M|T) is the restriction of Sp_{Y} M to C_{Y ∩T} T.

It follows from this that the restriction of Sp_{Y} M to a fibre of π : P_{Y} X → Y is a nilpotent orbit. In particular, Gr_{W} Sp_{Y} M is locally constant on the copy of C^{∗} contained in each fibre. This also follows from the fact that any polarized variation of Hodge structure on C^{∗} is locally constant, because it has a cyclic monodromy group, which moreover must be semi-simple (by [Deligne 1972]) and generated by a quasi-unipotent transformation, hence it is finite.

3 Abelian variations

(3.1) Let Δ be a hypersurface in an open neighborhood of 0 in C^{n} with 0 ∈ Δ. Let B denote an open ball with centre 0 in C^{n} such that all spheres with centre 0 which are contained in B are transverse to Δ (i.e. to all strata of a Whitney stratification of Δ). We write Δ instead of Δ∩B in the sequel.

Let V be a polarized variation of Hodge structure on B \ Δ whose underlying local system is associated with a representation ρ of π_{1}(B \ Δ) whose image is an abelian group: in short: V is an abelian variation. If V would live on a quasi-projective variety, this fact would imply that V has finite monodromy group, but in the local situation this need not be the case. If Δ has normal crossings, π_{1}(B \Δ) is itself abelian and every polarized variation of Hodge structure on B \ Δ is abelian.

Let D be a classifying space for the Hodge structures occurring in V and let Γ ⊂ Aut(D) be the corresponding discrete subgroup of isometries of the integral structure. The choice of a base point in B \ Δ determines a period mapping φ : B \ Δ → D/Γ.

Let Δ = Δ_{1} ∪ … ∪ Δ_{r} be the irreducible decomposition of Δ and let f_{j}, j = 1, …, r be a generator for the ideal of Δ_{j} in O(B). We obtain the holomorphic map:

f = (f_{1}, …, f_{r}) : B → C^{r}

with f^{−1}((C^{∗})^{r}) = B \ Δ. Let

(B \ Δ)^{ab} := {(s, z) ∈ (B \ Δ) × C^{r}| f_{j}(s) = exp(2πiz_{j}) for j = 1, …, r}.

A choice of a base point in (B\Δ)^{ab} (over the base point of B\Δ) determines a single-valued holomorphic lifting of φ:

φ̃ : (B \ Δ)^{ab} → D

which satisfies φ̃(s, z + e_{j}) = T_{j}φ̃(s, z), if e_{j} is the j^{th} standard basis vector in C^{r}. Here T_{j} ∈ Γ is the image under ρ of a loop in B \ Δ which winds once around Δ_{j} . Recall that T_{j} is quasi-unipotent. As T_{1}, …, T_{r} commute, their semisimple parts generate a finite abelian group G.

From now on we assume that all T_{j}’s are in fact unipotent, so G = {1}. Let Ď denote the compact dual of D. Let N_{j} = log T_{j} and define a mapping

ψ̃ : (B \ Δ)^{ab} → Ď

by

ψ̃(s, z) = exp(− ∑_{j} z_{j}N_{j})φ̃(s, z).

Then ψ̃(s, z + e_{j}) = ψ̃(s, z) for j = 1, …, r so ψ̃ descends to a mapping ψ : B \ Δ → Ď.

As usual, let σ be the monodromy cone:

σ = {∑_{j} λ_{j}N_{j} : λ_{j} ∈ R; λ_{j} > 0}.

Let σ_{C} denotes its C–span, and σ_{O_{B}} = σ_{C} ⊗ O_{B}.

Notice that (B\Δ)^{ab} can be defined independently of the map f, in fact, it is the topological covering corresponding to the epimorphism π_{1}(B \ Δ, ∗) → H_{1}(B \ Δ, Z). Moreover, the monodromies T_{j} (j = 1, …, r) (hence the N_{j}’s, too) are independent of the choice of f_{j}’s. On the other hand, in our realization (B \Δ)^{ab} ⊂ (B \Δ)×C^{r}, the map f̃: (B \Δ)^{ab} → C^{r}, induced by the projection on the second factor, depends on the choice of the generators f_{j}. In particular, the coordinates {z_{j} = f̃_{j}}_{j} depend on this choice. This implies the dependence on f of ψ̃ and ψ too.

Let f′_{j} be another generator of the ideal of Δ_{j}, then f′_{j} = f_{j} · g_{j}, where g_{j} ∈ O_{B}^{∗} is an invertible function on B, and

ψ_{f′}(s) = exp(− ∑_{j} \frac{log g_{j}(s)}{2πi}N_{j}) · ψ_{f}(s).

Let V = V ⊗ O_{B\Δ} as in section 2, and let j : B \ Δ → B denote the inclusion. For each multivalued horizontal section u of V the expression v_{f} = exp(− ∑_{j} z_{j}N_{j})u defines a single valued section of V. Let V_{0} denote the subspace of Γ(B, j_{∗}V) consisting of such sections. Again as above, the section v_{f} depends on the choice of f_{j}’s in the following way:

v_{f′} = exp(− ∑_{j} \frac{log g_{j}(s)}{2πi}N_{j})v_{f}.

This shows that the vector space V_{0} also depends on this choice: (V_{0})_{f′} = exp(A)(V_{0})_{f}, where A ∈ σ_{O_{B}}. But this last fact shows that the subsheaf V_{B} of j_{∗}V generated by V_{0} is independent of the choice of f_{j}’s. Obviously, by its very definition, V_{B} is a free O_{B}–module.

If Δ has normal crossings, then V_{B} is the canonical extension Ṽ. Moreover, this identification is compatible with the Hodge filtration in the following sense.

Consider X and D as in section 2, and let V be a polarized variation of Hodge structure on X \ D. Let {F^{p}} denotes the Hodge filtration of V = V ⊗ O_{X\D}. By [Schmid], the sheaves F̃^{p} := j_{∗}F^{p} ∩ Ṽ are holomorphic subbundles of Ṽ. We call (Ṽ, F̃^{∗}) the canonical extension of the variation of Hodge structure V.

On the other hand, {F^{p}} induces a filtration {F_{B}^{∗}} of V_{B}, defined by F_{B}^{∗} := j_{∗}F^{p} ∩ V_{B}. If Δ has normal crossings, then (Ṽ, F̃^{∗}) = (V_{B}, F_{B}^{∗}). In general, the restriction of V_{B} to the complement of the singular locus of Δ coincides with the canonical extension of V to this set.

The isomorphism V_{B} ≅ V_{0} ⊗ O_{B} gives a (non–canonical) trivialization of V_{B} and hence of V. This enables one to give an alternative description of the modified period map ψ. The space Ď can be considered as a space of certain flags on V_{0} (defined by the Hodge-Riemann bilinear relations). In these terms, the map ψ associates to b ∈ B \Δ the flag {F^{p}(b)} of V(b) ≅ V_{0}.

The following generalizes [Schmid], Theorem (4.12):

Theorem 2. Let B, Δ be as above. Suppose V is an abelian variation on B \ Δ with unipotent monodromy. Then the mapping ψ has a holomorphic extension to all of B.

First we fix some notations, which will be useful in the following. Let τ : X → B be an embedded resolution of Δ and let D = τ^{−1}(Δ), which is a divisor with normal crossings on X. We assume that each irreducible component of D is smooth. Let d ∈ D and let W be a small neighbourhood of d. Choose local coordinates (x_{i})_{i=1}^{n} in W such that f_{j} ∘ τ|_{W} = a_{j}(x) · ∏_{i} x_{i}^{m_{ji}}, with a_{j} ∈ O^{∗}(W). Let S_{i} be the logarithm of the monodromy of the variation τ^{∗}V|_{W \D} around the divisor {x_{i} = 0}. Let x_{i} = exp(2πit_{i}). Then ∑_{j} m_{ji}N_{j} = S_{i}. Actually, these relations are provided by the map τ_{d} : H_{1}(W \ D, Z) → H_{1}(B \ Δ, Z), τ(S_{i}) = ∑_{j} m_{ji}N_{j}. Let σ_{d} be the local monodromy cone {∑_{i=1}^{n} λ_{i}S_{i} : λ_{i} > 0, i = 1, …, n} of the variation on W \ D. Then its image by τ_{d} is Γ_{d} = {∑_{i=1}^{n} λ_{i} ∑_{j=1}^{r} m_{ji}N_{j}; λ_{i} > 0} ⊂ σ. In this way, we obtain a finite number of different subsets {Γ_{d}}_{d∈D} of σ. Denote by Γ their union ∪_{d∈D}Γ_{d} ⊂ σ.

Since ∑_{j} m_{ji}N_{j} = S_{i}, one has:

ψ ∘ τ|_{W \D} = exp(− ∑_{j} \frac{log a_{j}}{2πi} N_{j}) · exp(− ∑_{i} t_{i}S_{i})φ̃_{d}.

Here ψ_{d} = exp(− ∑_{i} t_{i}S_{i})φ̃_{d} is exactly Schmid’s modified period map, which can be extended to W.

Proof of theorem 2. We will show that the sheaves F_{B}^{p} are locally free direct summands of V_{B}. Then the extended map ψ_{B} is just given by ψ_{B}(b) = the flag {F_{B}^{p}(b)} on V_{B}(b) ≅ V_{0}. By Riemann’s extension theorem it suffices to show that ψ has a continuous extension to all of B. Let τ and D as above. As τ is a closed surjective mapping, it suffices to show that ψ (or more precisely: its lifting to X \ D) has a continuous extension to X which is constant on the fibres of τ.

As D has normal crossings and the local monodromies of V around the components of D are unipotent, ψ has a holomorphic extension ψ_{X} to all of X by [Schmid]. Indeed, let d ∈ D and choose W and local coordinates (x_{i})_{i=1}^{n} in W as above. Then

ψ ∘ τ|_{W \D} = exp(− ∑_{j} \frac{log a_{j}}{2πi} N_{j}) · ψ_{d}.

Since ψ_{d} can be extended to W, and the first factor is holomorphic, we obtain that ψ ∘ τ|_{W \D} can be extended holomorphically to all of W.

It remains to be shown that ψ_{X} is constant on the fibres of τ. As τ has connected fibres, it suffices to show that ψ_{X} is constant on each irreducible component of a fibre of τ.

A similar argument as above shows that τ^{∗}V_{B}|_{W} differs from the restriction of the canonical extension Ṽ_{X} of V = τ^{∗}V ⊗O_{X\D} to W by the action of exp(− ∑_{j} \frac{log a_{j}}{2πi} N_{j}) ∈ GL(n, O_{W}). Therefore Ṽ_{X} = τ^{∗}V_{B} = V_{0} ⊗ O_{X} is trivial.

Let b ∈ B and let Y be an irreducible component of τ^{−1}(b). If Y has codimension one in X, it is an irreducible component of D. In that case we can apply the construction of section 2 to Y. As Ṽ_{X} = V_{0}⊗O_{X}, the admissible variation of mixed Hodge structure which we obtain from Theorem 1 also has a trivial canonical extension. By [Hain–Zucker] Prop. (4.6), the extended Hodge bundles are also trivial in a way compatible to the trivialization of the canonical extension. The restriction of the extended Hodge bundles on P_{Y} X to the zero section Y_{0} coincides with the restriction to Y of the extended Hodge bundles from X. Hence these are also trivial, and ψ_{X} is constant on Y.

If τ^{−1}(b) is not everywhere of codimension one in X, we can use a further blowing-up g : Z → X to achieve that (τg)^{−1}(b) has everywhere codimension one in Z. Then the extension ψ_{Z} of ψ to Z satisfies ψ_{X}g = ψ_{Z} and is constant on (τg)^{−1}(b), so ψ_{X} is constant on τ^{−1}(b). □

Notice that the limiting Hodge filtration ψ_{B}(0) depends on the choice of the generators f_{j}. Similarly as in the classical case, when Δ has normal crossings, it is well defined only up to the action of exp σ_{C}.

We want to define a weight filtration W on V_{0} such that W and F = ψ_{B}(0) define a mixed Hodge structure on V_{0}: the limit mixed Hodge structure of the abelian variation V.

A limit mixed Hodge structure should not change if one replaces B by some smooth subvariety which is not contained in Δ. In particular, for a curve S on B with S ∩ Δ = {0} the restriction of V to S \ {0} has a limit mixed Hodge structure, and what we have to show is that the weight filtration of that limit mixed Hodge structure does not depend on the choice of the curve S.

Consider again an embedded resolution τ : (X, D) → (B, Δ) of Δ. For each irreducible component Y of D let γ_{Y} ∈ H_{1}(B \ Δ) be the homology class of the oriented boundary of an analytic disc S_{Y} in X which intersects Y transversely in one point. We let T_{Y} denote the image of γ_{Y} under the monodromy representation and N_{Y} = log T_{Y} (remark that T_{Y} is unipotent as the T_{i}’s are unipotent and commute with each other). If Y is the strict transform of a component Δ_{i} of Δ, then T_{Y} = T_{i} and N_{Y} = N_{i} . In general, N_{Y} = ∑_{i} n_{i}(Y)N_{i}, where n_{i}(Y) is the intersection multiplicity of τ(S_{Y}) and Δ_{i} at their unique point of intersection. We consider N_{Y} as an endomorphism of V_{0}.

Consider the logarithmic connection ∇_{X} : Ṽ_{X} → Ω_{X}^{1}(log D) ⊗ Ṽ_{X}. It induces the endomorphism Res_{Y} (∇_{X}) of Ṽ_{X} ⊗ O_{Y}, and Γ(Y, Ṽ_{X} ⊗ O_{Y}) ≅ V_{0}. Under this identification, N_{Y} = −2πi · Res_{Y} (∇_{X})) (see [Deligne 1970] Cor. 5.6 p. 96).

Let k be the weight of V. Each nilpotent endomorphism N of V_{0} defines a unique increasing filtration W = W (N, k) on V_{0} with the properties that N(W_{i}) ⊂ W_{i−2} for all i and the induced map N^{j} : Gr_{k+j}^{W} → Gr_{k−j}^{W} is an isomorphism for all j.

We need a relative version of this. Suppose that V_{0} already carries an increasing filtration W and N(W_{i}) ⊂ W_{i} for all i. Then there exists at most one increasing filtration M(N, W) of V_{0} such that N(M_{i}) ⊂ M_{i−2} and the filtration induced by M on Gr_{k}^{W} is precisely W (Gr_{k}^{W} N, k) (see [Steenbrink-Zucker] section 2). In case N(W_{i}) ⊂ W_{i−1}, this M(N, W) exists if and only if the stronger condition N(W_{i}) ⊂ W_{i−2} holds, and then M = W.

By the normal crossing case, any S ∈ σ_{d} defines the same weight filtration W (S, k) = W (σ_{d}) (by [Cattani–Kaplan]), and (W (σ_{d}), ψ_{d}(0), S) is a polarized mixed Hodge structure for any d ∈ D and S ∈ σ_{d}.

Proposition 1. Suppose that Y is a compact irreducible component of D and let d ∈ Y. Then W (N_{Y} , k) = W (σ_{d}). Proof. By Theorem 1, the restriction of the mixed Hodge module Sp_{Y} M to the Zariski-open subset U of P = P_{Y} X is an admissible variation of mixed Hodge structure. We have seen in the proof of Theorem 2 that it has a trivial canonical extension Ṽ_{P} = O_{P} ⊗ V_{0}. Therefore by [Hain–Zucker], the weight filtration W on Ṽ_{P} comes from a filtration on V_{0} (which is exactly W (N_{Y} , k)). Also, any N ∈ σ_{d} acts on Ṽ_{P} via its action on V_{0} . As U is quasi-projective, the monodromy group of the variation of Hodge structure Gr_{i}^{W} Ṽ_{P} is semisimple. As it is also abelian and unipotent, it must be trivial. We conclude that

(∗) N(W (N_{Y} , k)_{i}) ⊂ W (N_{Y} , k)_{i−1}.

By [Cattani-Kaplan], the relative weight filtration M(N, W (N_{Y} , k)) exists and is equal to W (N, k). This existence together with (*) implies that M(N, W (N_{Y} , k)) = W (N_{Y} , k). □

Corollary. Suppose that Y and Z are compact components of D with Y ∩ Z ≠ ∅. Then the filtrations W (N_{Y} , k) and W (N_{Z}, k) of V_{0} are equal. Proof. In that case, both are equal to W (σ_{d}), where d ∈ Y ∩ Z. □

Theorem 3.

1) Any N ∈ σ defines the same weight filtration W (σ) of V_{0}.

2) (W (σ), ψ_{B}(0)) is a mixed Hodge structure which is polarized by any N ∈ σ in the sense of [Cattani–Kaplan]. In particular,

θ(z_{1}, …, z_{r}) = exp(∑_{j=1}^{r} z_{j}N_{j}) · ψ_{B}(0) is a nilpotent orbit.

Proof. First notice that 1)-2) imply the fact that θ is a nilpotent orbit by [Cattani–Kaplan–Schmid] (4.66).

Consider a resolution τ : X → B as above. We keep the notations introduced before the proof of Theorem 2.

Since D is connected, the space Γ is connected too. Actually, its (topological) closure in σ̄ = {∑_{j} λ_{j}N_{j} : λ_{j} ≥ 0} is a finite simplicial complex, therefore it is path–connected. Proposition 1 implies that any N ∈ Γ defines the same weight filtration of V_{0}. This will be denoted by W (Γ). Actually this agrees with any “local” weight filtrations W (σ_{d}).

Consider the strict transform Z_{j} of Δ_{j} and a point d_{j} ∈ Z_{j} ∩ D. If in local coordinates Z_{j} is given by {x_{i_{0}} = 0}, then N_{j} = S_{i_{0}}. Since S_{i_{0}}W_{l}(σ_{d}) ⊂ W_{l−2}(σ_{d}) (use, for example, 2.11 in [Cattani–Kaplan]), we obtain that N_{j}W_{l}(Γ) ⊂ W_{l−2}(Γ) for any j and l. This shows that

(3.1) NW_{∗}(Γ) ⊂ W_{∗−2}(Γ) for any N ∈ σ̄.

A similar relation is true for the Hodge filtration. Let F_{B}^{∗} (respective F_{d}^{∗}) be the Hodge filtration given by ψ_{B}(0) (respective by ψ_{d}(0)). Then by (2.1), F_{B}^{p} = exp(A_{d})F_{d}^{p}, where A_{d} = − ∑_{j} \frac{log a_{j}(0)}{2πi}N_{j}. Since S_{i_{0}}F_{d}^{p} ⊂ F_{d}^{p−1}, we obtain that N_{j}F_{B}^{p} ⊂ F_{B}^{p−1} for any j and p; therefore

(3.2) NF_{B}^{∗} ⊂ F_{B}^{∗−1} for any N ∈ σ̄.

By the local theory, any M ∈ σ_{d} polarizes the mixed Hodge structure (W (σ_{d}), ψ_{d}(0)) = (W (Γ), exp(A_{d}) · F_{B}), where A_{d} ∈ σ_{C} is defined above. But thanks to (3.1), exp(A_{d}) acts trivially on Gr^{W (Γ)}, therefore:

(3.3) (W (Γ), F_{B}, M) is a polarized mixed Hodge structure for any M ∈ Γ.

Notice that in general the cone Γ can be a very thin subset of σ. Its closure Γ̄ in σ_{C} is a simplicial complex, of dimension at most n . However, the next fact shows that, in some sense, Γ generates σ. Let d_{j} as above, then S_{i_{0}} ∈ σ̄_{d}, therefore N_{j} ∈ Γ̄_{d} ⊂ Γ̄. Hence:

(3.4) the convex closure c(Γ) of Γ in σ is exactly σ.

Now the result follows from the following Convexity Theorem:

Theorem 4. Let a ⊂ g_{0} be an abelian subalgebra of the Lie algebra g_{0} of G_{R}, consisting of nilpotent elements. Assume that there exist a path– connected set Γ ⊂ a and a Hodge filtration F ∈ Ď of weight k such that

a) any N ∈ Γ defines the same weight filtration W (Γ) = W (N, k); and

b) (W (Γ), F) is a mixed Hodge structure polarized by any N ∈ Γ.

Then these properties remain true for the convex closure c(Γ) of Γ in a, i.e. any N ∈ c(Γ) polarizes the mixed Hodge structure (W (Γ), F). Proof. We can assume that Γ is a cone (i.e. R_{+} · Γ = Γ), and spans a over R. Then

(4.1) NW_{∗}(Γ) ⊂ W_{∗−2}(Γ) for any N ∈ a.

By [Cattani–Kaplan] (2.20), there exists F_{0} ∈ Ď such that

(4.2) \begin{matrix}a) (W (Γ), F_{0}) is a mixed Hodge structure split over R; \\ b) any M ∈ Γ polarizes (W (Γ), F_{0}); and \\ c) Gr^{W (Γ)}F = Gr^{W (Γ)}F_{0} \end{matrix}

Now since for any M ∈ Γ one has M(F_{0}^{p}) ⊂ F_{0}^{p−1}, we have:

(4.3) N(F_{0}^{p}) ⊂ F_{0}^{p−1} for any N ∈ c(Γ).

Let U_{Γ} = {N ∈ c(Γ) : W (N, k) = W (Γ)}. Then (4.1) implies that N^{l} : Gr_{k+l}^{W (Γ)} → Gr_{k−l}^{W (Γ)} is well–defined for any N ∈ a, and

U_{Γ} = {N ∈ c(Γ) : N^{l} : Gr_{k+l}^{W (Γ)} → Gr_{k−l}^{W (Γ)} is an isomorphism for all l}.

This is an open condition, hence U_{Γ} is a Zariski open subcone of c(Γ) in the following sense: there is a real hypersurface X (provided by the product of the determinants of the N^{l}’s) such that U_{Γ} = (a − X) ∩ c(Γ). By condition a), Γ ⊂ U_{Γ}.

As a first step, we wish to prove that (W (Γ), F_{0}) is polarized by any N ∈ U_{Γ}^{0}, where U_{Γ}^{0} is the connected component of U_{Γ} which contains Γ.

Fix N ∈ U_{Γ}^{0}. Then W (N, k) = W (Γ) and N(F_{0}^{p}) ⊂ F_{0}^{p−1} by (4.3), therefore (W (Γ), F_{0}, N) is a polarized mixed Hodge structure if and only if the Hodge structure induced by F_{0} on the primitive subspace ker N^{l+1} of Gr_{k+l}^{W (Γ)} is polarized by the bilinear form S_{l}(N) = S(·, N^{l} ·) for all l ≥ 0.

Let I^{p,q} = F_{0}^{p} ∩ F̄_{0}^{q} ∩ W_{p+q}(Γ) be the generalized Hodge decomposition of (W (Γ), F_{0}) (see for example [Cattani–Kaplan]). Then I^{p,q} projects isomorphically onto H^{p,q}(Gr_{p+q}^{W (Γ)}). (4.1) and (4.3) imply that for any N ∈ c(Γ) one has N(I^{p,q}) ⊂ I^{p−1,q−1}.

Fix (r, s) and (r′, s′) so that r + s − k = r′ + s′ − k = l ≥ 0. Using the facts that S(F_{0}^{p}, F_{0}^{q}) = S(F̄_{0}^{p}, F̄_{0}^{q}) = 0 if p + q ≥ k + 1, we obtain that for any N ∈ c(Γ), S(I^{r,s}, N^{l}I^{r′,s′}) = 0 if r + r′ − l ≥ k + 1 or s + s′ − l ≥ k + 1. But if (r, s) ≠ (s′, r′) then at least one of the above inequalities are satisfied. Therefore, the first Hodge–Riemann bilinear relation for S_{l}(N) is automatically satisfied for any N ∈ c(Γ).

If N ∈ U_{Γ}^{0}, then S_{l}(N) is non–degenerate (because N^{l} : I^{p,q} → I^{p−l,q−l} , p+ q − k = l ≥ 0, is an isomorphism). Consider the Lefschetz–decomposition

I^{p,q} = ⊕_{j≥0}N^{j}(I_{0}^{p+j,q+j}(N)) with I_{0}^{p+j,q+j}(N) = I^{p,q} ∩ ker N^{r+s−k+1}.

This direct sum decomposition is orthogonal relative to the form S_{l}(N). Since for at least one N ∈ U_{Γ}^{0} the form i^{p−q}S_{l}(N) is positive definite on I_{0}^{p,q}(N), for the whole connected component U_{Γ}^{0} this is so. Hence, we have proven:

(4.4) (W (Γ), F_{0}, N) is a polarized MHS for any N ∈ U_{Γ}^{0}.

Our goal is the inclusion c(Γ) ⊂ U_{Γ}^{0}. If dim_{R}a = r ≤ 2 then this is obviously true. Assume in the sequel that r ≥ 3.

Notice that c(Γ) can be generated in the following way. Set Γ_{0} = Γ. Suppose that Γ_{i} (i ≥ 0) has already been constructed. Then define Γ_{i+1} by the set {tx + (1 − t)y : t ∈ [0, 1], x ∈ Γ_{i}, y ∈ Γ_{i}}. Then Γ_{i} ⊂ Γ_{i+1} and c(Γ) = ∪_{i}Γ_{i}. Notice that if Γ_{i} is path–connected then Γ_{i+1} is so.

If S_{1}, S_{2}, S_{3} ∈ c(Γ), then let σ(S_{1}, S_{2}, S_{3}) be the open cone {N ∈ c(Γ) : N = ∑_{i=1}^{3} λ_{i}S_{i} for some λ_{i} > 0} generated by S_{1}, S_{2} and S_{3}; and let σ̄(S_{1}, S_{2}, S_{3}) be its closure in c(Γ).

Assume that c(Γ) ⊄ U_{Γ}^{0}. Then there exists i_{0} ≥ 0 such that Γ_{i_{0}} ⊂ U_{Γ}^{0}, but Γ_{i_{0}+1} ⊄ U_{Γ}^{0}; i.e. there exist x_{1}, x_{2} ∈ U_{Γ}^{0} such that the segment [x_{1}, x_{2}] = {tx_{1} + (1 − t)x_{2} : t ∈ [0, 1]} is not in U_{Γ}^{0}. Since U_{Γ}^{0} is open and path connected, we can take a sequence of points y_{1}, …, y_{s} in such a way that y_{1} = x_{1}, y_{s} = x_{2}, and the segments [y_{i}, y_{i+1}] (i = 1, …, s − 1) are in U_{Γ}^{0}. Let s be minimal with this property. Since [x_{1}, x_{2}] ⊄ U_{Γ}^{0} one has s ≥ 3.

We wish to construct three points S_{1}, S_{2} and S_{3} in U_{Γ}^{0} (called “bad triple”) such that σ(S_{1}, S_{2}, S_{3}) ⊂ U_{Γ}^{0} but σ̄(S_{1}, S_{2}, S_{3}) ⊄ U_{Γ}^{0}. Consider the triples (y_{1}, y_{2}, (1 − t)y_{2} + ty_{3}) for t ∈ (0, 1]. For t small, the closed cone generated by them is in U_{Γ}^{0}. If for any t the closed cone generated by them were in U_{Γ}^{0} then the segment [y_{1}, y_{3}] would be in U_{Γ}^{0} which contradicts the minimality of y_{i}’s or the fact that [x_{1}, x_{2}] is not in U_{Γ}^{0}. Therefore, a “bad triple” exists.

Now, consider a bad triple S_{1}, S_{2}, S_{3}. Since σ_{S} = σ(S_{1}, S_{2}, S_{3}) ∈ U_{Γ}^{0}, (4.4) together with [Cattani–Kaplan–Schmid] Prop. (4.66) implies that θ(z_{1}, z_{2}, z_{3}) = exp(∑_{i=1}^{3} z_{i}S_{i}) · F_{0} is a nilpotent orbit. Now by [Cattani–Kaplan] (3.3), for any face τ of σ_{S} there is a filtration W (τ) such that for any N ∈ τ one has W (N, k) = W (τ). Since S_{i} ∈ U_{Γ}^{0}, any face has a non–empty intersection with U_{Γ}^{0}; in particular, any element N ∈ σ̄_{S} defines the same weight filtration W (Γ). Therefore σ̄_{S} ⊂ U_{Γ}^{0} which contradicts with the “badness” of the triple S_{1}, S_{2}, S_{3}. The contradiction comes from the assumtion c(Γ) ⊄ U_{Γ}^{0}.

At this moment we know that U_{Γ}^{0} contains c(Γ). In particular, any N ∈ c(Γ) defines the same weight filtration W (Γ) and any N ∈ c(Γ) polarizes (W (Γ), F_{0}) (cf. 4.4). Since the graded Hodge structures induced by F and F_{0} on Gr^{W (Γ)} agree (see 4.2.c), any N ∈ c(Γ) polarizes (W (Γ), F) too. This completes the proof. □ Corollary. Let ϕ be a symmetric trilinear form on a real vector space V of dimension n. Let V^{+} be the set of elements v ∈ V such that ϕ(v, v, v) > 0 and the quadratic form ϕ(v, x, x) is nondegenerate with signature (1, n − 1). Then the connected components of V^{+} are convex cones. Proof. Consider the graded R-algebra

A = A_{0} ⊕ A_{1} ⊕ A_{2} ⊕ A_{3}

with A_{0} = A_{3} = R, A_{1} = V and A_{2} = V^{∨} and the following multiplication: i) 1 ∈ A_{0} is the unit element; ii) if x, y ∈ A_{1} then xy is the linear form z → ϕ(x, y, z); iii) xα = α(x) for x ∈ A_{1} and α ∈ A_{2};

and equipped with the mixed Hodge structure such that A_{i} ⊗ C is of type (3 − i, 3 − i). Consider V as a subalgebra of the endomorphism algebra of A. Then Theorem 4 applies to V. □ Corollary. Let H^{∗} be the real cohomology of a compact Kähler manifold X purely of dimension n. In H^{1,1} ⊂ H^{2}(X, R) let U be the set of elements u such that for each l, the cup product with u^{l} induces an isomorphism H^{n−l} → H^{n+l}. Then, the connected component of U containing the Kähler class is a convex cone. Proof. Consider the mixed Hodge structure ⨁ H^{i}(X)(i) and let N be the cup product with u ∈ U. □ Remarks. Let a ⊂ g_{0} be an abelian subalgebra consisting of nilpotent elements.

1) Assume that there exists a Hodge filtration F ∈ Ď of weight k such that (W (N, k), F, N) is a polarized mixed Hodge structure for every N in some open subset of a. Then by [Cattani–Kaplan] (2.16) there exists a non– empty Zariski open set U of a such that any N ∈ U defines the same weight filtration W. Moreover, the assumption also implies that N(F^{p}) ⊂ F^{p−1} for any p and N ∈ a. If U^{0} is a connected component of U and at least one N ∈ U^{0} polarizes (W, F), then any N ∈ U^{0} does (see the proof of Theorem 4).

2) Fix a filtration W. Assume that U = {N ∈ a : W (N, k) = W } is a non–empty Zariski open set of a. Let U^{0} be a connected component of U. Assume that there exists a Hodge filtration F of weight k such that any N ∈ U^{0} polarizes (W, F) (or equivalently, for any N ∈ U^{0} one has N(F^{p}) ⊂ F^{p−1} and at least one N ∈ U^{0} polarizes (W, F)). Then by our last theorem U^{0} is convex.

In this case, any boundary point P of U^{0} defines a weight filtration W (P, k) with the property that for any N ∈ U^{0} one has M(N, W (P, k)) = W. (Indeed, any P ∈ ∂U^{0} and N ∈ U^{0} can be considered as a boundary point P ∈ σ̄ \ σ, respectively interior point N ∈ σ for a cone σ well–chosen in U^{0}; then apply [Cattani–Kaplan] (3.3).)

4 Dropping the unipotence assumption

Dropping the smoothness of B.

Let (B′, 0) be a reduced analytic space–germ and (Δ′, 0) ⊂ (B′, 0) a Cartier divisor. We denote a small representative of the pair (B′, Δ′, 0) by (B′, Δ′). Assume that B′ \ Δ′ is smooth and connected.

Let V′ be a polarized variation of Hodge structure on B′ \ Δ′ whose underlying local system is associated with an abelian unipotent representation ρ′. Assume that the following holds: (A) Assumption.

There exist w_{j} ∈ O(B′) (j = 1, …, r) such that:

a) Δ′ = ∪_{j=1}^{r}Z(w_{j}),

b) the map w = (w_{1}, …, w_{r}) : B′ \ Δ′ → (C^{∗})^{r} induces a surjective map w_{∗} : π_{1}(B′ \ Δ′, ∗) → Z^{r}, and

c) the representation ρ′ can be factorized through w_{∗}; i.e. there exists the following commutative diagram:

[figure: π_{1}(B′ \ Δ′, ∗); AutV_{∗}; Z^{r}; ρ′; w_{∗}; ρ^{c}]

Consider a system (B′, Δ′, V′) which satisfies the assumption (A). Construct the topological covering:

(B′ \Δ′)^{c} = {(s, z′) ∈ (B′ \Δ′)×C^{r} : w_{j}(s) = exp(2πi·z′_{j}) for j = 1, …, r}.

Assumption (b) implies that (B′ \ Δ′)^{c} is connected. In fact, it corresponds to the subgroup ker w_{∗} ⊂ π_{1}(B′ \ Δ′, ∗). The condition (c) is equivalent to the inclusion ker w_{∗} ⊂ ker ρ′.

Let e_{j} be the j^{th} standard basis vector in Z^{r}. Let T′_{j} = ρ^{c}(e_{j}). Then T′_{j} is unipotent and T′_{1}, …, T′_{r} commute. Let N′_{j} = log T′_{j}. These nilpotent elements define the monodromy cone σ′.

Fix compatible base points in B′ \Δ′ and (B′ \Δ′)^{c}. These determine the period map φ′ : B′ \ Δ′ → D/Γ of V′, and its lifting φ̃′ : (B′ \ Δ′)^{c} → D. By the definition of T′_{j} one has: φ̃′(s, z′ + e_{j}) = T′_{j} · φ̃′(s, z′). Similarly as in the smooth case, define ψ̃′ : (B′\Δ′)^{c} → Ď by ψ̃′(s, z′) = exp(− ∑_{j} z′_{j}N′_{j})φ̃′(s, z′). Then ψ̃′(s, z′ + e_{j}) = ψ̃′(s, z′) for any j = 1, …, r. Therefore, ψ̃′ descends to a mapping ψ′ : B′ \ Δ′ → Ď. Define V′_{0} similarly as V_{0} in section 3.

Theorem 5. Let B′, Δ′ and V′ be as above. Assume that they satisfy the assumtion (A) and B′ \ Δ′ is connected and smooth. Then:

a) the mapping ψ′ has a holomorphic extension to all of B′,

b) any N ∈ σ′ defines the same weight filtration W (σ′) of V′_{0},

c) (W (σ′), ψ′(0)) is a mixed Hodge structure which is polarized by any N ∈ σ′. In particular

θ(z′_{1}, …, z′_{r}) = exp(∑_{j=1}^{r} z′_{j}N′_{j}) · ψ′(0) is a nilpotent orbit.

The proof is the same as one of Theorem 2 and 3.

Dropping the unipotence assumption.

Let (B, Δ) be a reduced analytic space–germ and (Δ, 0) ⊂ (B, 0) a Cartier divisor. Let (B, Δ) be a representative of (B, Δ, 0). Assume that B \ Δ is smooth and connected.

Let V be a polarized variation of Hodge structure on B \ Δ whose underlying local system is associated with an abelian representation ρ. Assume that there exist {f_{j}}_{j=1}^{r} ∈ O(B) such that the system (B, Δ, V, {f_{j}}_{j=1}^{r}) satisfies Assumption A, in particular, ρ = ρ^{c} ∘ f_{∗} for a map ρ^{c} : Z^{r} → Aut V_{∗}.

Construct (B \ Δ)^{c} as above. After choosing the base points, we have the period map φ : B \ Δ → D/Γ and its lifting φ̃ : (B \ Δ)^{c} → D which satisfies φ̃(s, z + e_{j}) = T_{j} · φ̃(s, z). Assume that T_{j} is quasi–unipotent for all j = 1, …, r. Let T_{j} = T_{j,s} · T_{j,u} be its decomposition in its semisimple and unipotent part. Let m_{j} be the order of T_{j,s} and N_{j} = log T_{j,u}. Define the modified period map ψ̃ : (B \ Δ)^{c} → Ď by ψ̃(s, z) = exp(− ∑_{j} z_{j}N_{j})φ̃(s, z). Then

(∗)  ψ̃(s, z + m_{j}e_{j}) = ψ̃(s, z).

Therefore, ψ̃ descends to a finite covering of B \ Δ. Its explicit construction follows. Construct the space:

B′ = {(s, w) ∈ B × C^{r} : f_{j}(s) = w_{j}^{m_{j}} for j = 1, …, r}

and Δ′ = {(s, w) ∈ B′ : w_{1} ⋯ w_{r} = 0}. The irreducible decomposition of Δ′ is ∪_{j=1}^{r}Δ′_{j}, where Δ′_{j} = Z(w_{j}). Then B′ \ Δ′ is smooth and connected, in fact it is an (m_{1}, …, m_{r})–fold topological covering of B \Δ by the projection q_{1}(s, w) = s.

Consider k : Z^{r} → Z^{r} defined by k(e_{j}) = m_{j}e_{j} for any j. Then f_{∗} ∘ q_{1∗} = k ∘ w_{∗}. Since q_{1∗} and k are one-to-one, and their cokernels are isomorphic, by the five–lemma w_{∗} is onto. Moreover, since ρ = ρ^{c} ∘ f_{∗}, the representation of the lifting V′ of V to B′ \ Δ′ is ρ′ = ρ^{′c} ∘ w_{∗}, where ρ^{′c} = ρ^{c} ∘ k. Since T′_{j} = ρ^{′c}(e_{j}) = ρ^{c}(m_{j}e_{j}) = T_{j,u}^{m_{j}} is unipotent, the representation ρ^{′c} (hence, ρ′ too) is unipotent. Therefore, the system (B′, Δ′, V′, {w_{j}}_{j}) satisfies the conditions of Theorem 5.

Notice that (B′ \ Δ′)^{c} can be identified with

{(s, z′) ∈ (B \ Δ) × C^{r} : f_{j}(s) = exp(2πi · m_{j}z′_{j}) for j = 1, …, r}.

The projection q_{2} : (B′\Δ′)^{c} → B′\Δ′ is given by q_{2}(s, z′) = (s, exp(2πiz′_{1}), …). Then (B′ \ Δ′)^{c} and (B \ Δ)^{c} are isomorphic by the isomorphism q_{3} : (B \ Δ)^{c} → (B′ \ Δ′)^{c}, q_{3}(s, z) = (s; z_{1}/m_{1}, …, z_{r}/m_{r}).

The condition (∗) implies that ψ̃ descends to a mapping ψ : B′ \ Δ′ → Ď. On the other hand ψ̃′(s, z′) = exp(− ∑_{j} z′_{j}N′_{j})φ̃′(s, z′) = exp(− ∑_{j} z′_{j}m_{j}N_{j}) · φ̃(s; m_{1}z′_{1} …, m_{r}z′_{r}). Therefore ψ̃′ ∘ q_{3} = ψ̃, in particular ψ = ψ′.

These maps are summarized in the diagram:

[figure: D/Γ; D; B \ Δ; B′ \ Δ′; (B′ \ Δ′)^{c}; (B \ Δ)^{c}; Ď; φ; φ′; φ̃′; φ̃; ψ = ψ′; ψ̃′; ψ̃; q_{1}; q_{2}; q_{3}; ∼]

The covering q_{1} is regular. The transformation group G = ∏_{j} Z_{m_{j}} acts by (k̂_{1}, …, k̂_{r}) ∗ (s; w_{1}, …, w_{r}) = (s; exp(2πik_{1}/m_{1})w_{1}, …). Therefore:

(∗∗) ψ((k̂_{1}, …, k̂_{r}) ∗ (s, w)) = T_{1,s}^{k_{1}} ⋯ T_{r,s}^{k_{r}} · ψ(s, w)

for any (s, w) ∈ B′ \ Δ′.

By Theorem 5, ψ = ψ′ can be extended to B′. On the other hand, the action of G can be extended (by the same formula) to B′ too. By continuity, the relation (**) remains true for any (s, w) ∈ B′. In particular T_{1,s}^{k_{1}} … T_{r,s}^{k_{r}} · ψ(0, 0) = ψ(0, 0) for any (k_{1}, …, k_{r}). Therefore one has:

Theorem 6. Let B, Δ be as above and let V be an abelian variation of Hodge structure on B \ Δ whose monodromies T_{j} are quasi-unipotent. Let T_{j} = T_{j,s} ·T_{j,u} be the Jordan decomposition, m_{j} = ord(T_{j,s}) and N_{j} = log T_{j,u}. Define the monodromy cone σ and V_{0} as in section 3. Then:

a) the modified period map ψ : B′ \ Δ′ → Ď has a holomorphic extension to all of B′,

b) ψ(0) is a fixed point of the action of ∏_{j} Z_{m_{j}} via the semisimple monodromies, i.e.: T_{j,s} · ψ(0) = ψ(0) for any j = 1, …, r,

c) any N ∈ σ defines the same weight filtration W (σ) of V_{0},

d) (W (σ), ψ(0)) is a mixed Hodge structure which is polarized by any N ∈ σ ; in particular

θ(z_{1}, …, z_{r}) = exp(∑_{j=1}^{r} z_{j}N_{j}) · ψ(0) is a nilpotent orbit.

Consider the map φ̃_{∞} : (B \ Δ)^{c} → Ď given by φ̃_{∞}(s, z) = θ(z). By Theorem 6, if B is sufficiently small, φ̃_{∞} take its values in D. Moreover, T_{j,s} · φ̃_{∞} = φ̃_{∞} for any j and φ̃_{∞}(s, z + e_{j}) = T_{j,u} · φ̃_{∞}(s, z) = T_{j} · φ̃_{∞}(s, z). Therefore, φ̃_{∞} descends to a map φ_{∞} : B \ Δ → D/Γ. The map φ_{∞} defines a polarized variation V_{∞} of Hodge structures on B \ Δ with the same underlying representation as V (and the same stalk), but with the property T_{j,s}φ̃_{∞} = φ̃_{∞}. Actually, this is the nilpotent orbit of V.

References:

Cattani, E. and A. Kaplan: Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure. Invent. math. 67(1982), 101-115.

Cattani, E., Kaplan, A. and Schmid, W.: Degeneration of Hodge structures, Annals of Math., 123, (1986) 457-535.

Deligne, P.: Équations différentielles à points singuliers réguliers. Lecture Notes in Math. 163, Springer 1970.

Deligne, P.: Théorie de Hodge II, Publ. Math. I.H.E.S., 40, (1972) 5-57. Hain, R.M. and S. Zucker: A guide to unipotent variations of mixed

Hodge structure. In: Hodge Theory, Proceedings, Sant Cugat, Spain, 1985. Lecture Notes in Math. 1246, Springer 1987. pp. 92-106.

Kashiwara, M.: A study of variation of mixed Hodge structure. Publ. RIMS Kyoto Univ. 22 (1986), 991-1024.

Saito, M.: Mixed Hodge modules. Publ. RIMS Kyoto Univ. 26 (1990) 221-333.

Saito, M.: On Steenbrink’s Conjecture. Math. Ann. 289 (1991) 703-716. Schmid, W.: Variation of Hodge structure: the singularities of the period

mapping. Invent. math. 22 (1973) 211-319.

Steenbrink, J.H.M. and S. Zucker: Variation of mixed Hodge structure I. Invent. math. 80 (1985) 489-542.

Verdier, J.-L.: Spécialisation de faisceaux et monodromie modérée. Astérisque 101-102 (1983) 332-364.