Hodge-theoretic aspects of the Decomposition Theorem

TL;DR

This paper explores how the decomposition isomorphism in the Decomposition Theorem can be chosen to preserve pure Hodge structures, linking algebraic geometry, Hodge theory, and cohomological methods.

Contribution

It introduces a new cohomological characterization of the decomposition isomorphism that ensures Hodge structure compatibility, advancing understanding of the theorem's geometric and Hodge-theoretic aspects.

Findings

Decomposition isomorphism can be chosen to yield pure Hodge structures.

Corollaries include results on intersection forms and Hodge cycles.

Provides new insights into the structure of intersection cohomology.

Abstract

Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomology, projectors and Hodge cycles, and induced morphisms in intersection cohomology.