## Mixed Hodge modules.(English)Zbl 0727.14004

In his previous paper in Publ. Res. Inst. Math. Sci. 24, No.6, 849-995 (1988; Zbl 0691.14007) the author showed that a polarizable Hodge module with strict support (i.e. such that the underlying perverse sheaf is an intersection complex) is generically a polarizable variation of Hodge structure. In the paper under review the author shows the converse, viz. any polarizable variation of Hodge structure defined on a Zariski open subset can uniquely (and functorially) be extended to a polarizable Hodge module with strict support. Thus the intersection cohomology $$IH^*(X,L)$$ with coefficients in L is endowed with a natural Hodge structure provided that X is compact and bimeromorphic to a compact Kähler manifold $$\tilde X$$ and L is a local system on a Zariski open smooth subset of X and underlies a polarizable variation of Hodge structure. The author also introduces and studies functorial properties of mixed Hodge modules corresponding to perverse mixed complexes. He verifies that (polarizable) mixed Hodge modules on a point can be identified with (polarizable) $${\mathbb{Q}}$$-mixed Hodge structures. He proceeds with showing that polarizable mixed Hodge modules are stable with respect to external product. A special section is devoted to mixed Hodge Modules on algebraic varieties.
Reviewer: F.L.Zak (Moskva)

### MSC:

 14D07 Variation of Hodge structures (algebro-geometric aspects) 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 32J27 Compact Kähler manifolds: generalizations, classification 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 55N33 Intersection homology and cohomology in algebraic topology

Zbl 0691.14007
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### References:

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