Duality and Local Cohomology in Hodge Theory

Scott M Hiatt, Purdue University

Abstract

A Hodge module on an algebraic variety may be viewed as a variation of Hodge structure with singularities. Given an irreducible variety X, for any polarized variation of Hodge structure H on a smooth open subvariety U ⊂ X, there exists a unique Hodge module M ∈ HMX(X) that extends H. Conversely, for any Hodge module M ∈ HMX(X) with strict support on X, there exists a polarized variation of Hodge structure H on a smooth open subset U ⊂ X such that M:V ∼= H. In this thesis, we first study the singularities of a Hodge module M ∈ HMX(X) by using Morihiko Saito’s theory of S-sheaves and duality. Then using local cohomology and the theory of mixed Hodge modules, we study the Hodge structure of Hi (X, DR(M)) when X is a projective variety. Finally, we consider a variation of Hodge structure H on U as a Hodge module N ∈ HM(U) on U, and study the local cohomology of the complex GrF p DR(j!N ) ∈ Db coh(OX), where j : U ,→ X is the natural map.

Degree

Ph.D.

Advisors

Arapura, Purdue University.

Subject Area

Mathematics

COinS