Abstract Aspects of Mixed Hodge Structures
We continue to study the more formal aspects of Hodge theory, this time for the case of mixed Hodge structures. In §3.1 the basic definitions are given; the Deligne splittings are introduced which make it possible to prove strictness of morphisms of mixed
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We continue to study the more formal aspects of Hodge theory, this time for the case of mixed Hodge structures. In § 3.1 the basic definitions are given; the Deligne splittings are introduced which make it possible to prove strictness of morphisms of mixed Hodge structures and to show that the category of mixed Hodge structures is abelian. The complexes which come up in constructions for mixed Hodge structures have two filtrations and any one of these defines a priori different natural filtrations on the terms of the spectral sequence for the other filtration. We compare these in § 3.2. This study reveals (§ 3.3) that certain abstract properties built in the definition of a mixed Hodge complex of sheaves guarantee that their hypercohomology groups carry a mixed Hodge structure. If one can interpret a geometric object as a hypercohomology group of a complex underlying a mixed Hodge complex of sheaves, this object carries a mixed Hodge structure. This is the technique which will be employed in subsequent chapters. Given a morphism of mixed Hodge structures, there is no canonical way to put the structure on the cone of the morphism. However, as we show in § 3.4, for a morphism of mixed Hodge complexes of sheaves the mixed cone is a canonical mixed Hodge structure on the cone of the underlying morphism of complexes of sheaves. It depends explicitly on the comparison morphisms, but this is built in in the definitions. The mixed cone construction will often be used later. As an example of its geometric significance we explain how to put a mixed Hodge structure on relative cohomology of a pair of compact smooth K¨ ahler manifolds. In § 3.5 we return to the categorical study of mixed Hodge structures. We first study extensions of two mixed Hodge structures and after that the higher Extgroups. The category of mixed Hodge structures is abelian, but it does not have enough injectives; we use Verdier’s direct approach (§ A.2.2) to the derived category. The higher Ext-groups turn out to be zero if R = Z or if R is a field. This is related to Beilinson’s construction of absolute Hodge cohomology as we shall briefly indicate.
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3 Abstract Aspects of Mixed Hodge Structures
3.1 Introduction to Mixed Hodge Structures: Formal Aspects We let R be a noetherian subring of C such that R ⊗ Q is a field and we let VR be a finite type R-module. Definition 3.1. An R-mixed Hodge structure on VR consists of two filtrations, an increasing filtration on VR ⊗R (R ⊗ Q), the weight filtration W• and a decreasing filtration F • on VC = V ⊗R C, the Hodge filtration which has the additional property that it induces a pure (R ⊗ Q)-Hodge structure of weight k on each graded piece GrW k (VR ⊗Z Q) = Wk /Wk−1 . We say that the R-mixed Hodge structure is graded-polarizable if the GrW k (VR ⊗Z Q) are pure, polarizable (R ⊗ Q)-Hodge structures. The mixed Hodge structure on V defines a class in the Grothendieck group (see Def. A.4.3) of pure R-Hodge structures X [V ] := GrW ∈ K0 (hsR ). (III–1) k k∈Z
The Hodge numbers of these pure Hodge structures hp
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