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Hodge modules and Kähler morphisms

Mads Bach Villadsen

Abstract

We prove the decomposition theorem for Hodge modules with integral structure along proper Kähler morphisms, partially generalizing M. Saito's theorem for projective morphisms. Our proof relies on compactifications of period maps of generically defined variations of Hodge structure, as well as the theorems of Cattani-Kaplan-Schmid on \(L^2\)-cohomology of variations of Hodge structure for the hard Lefschetz theorem.

Hodge modules and Kähler morphisms

Abstract

We prove the decomposition theorem for Hodge modules with integral structure along proper Kähler morphisms, partially generalizing M. Saito's theorem for projective morphisms. Our proof relies on compactifications of period maps of generically defined variations of Hodge structure, as well as the theorems of Cattani-Kaplan-Schmid on -cohomology of variations of Hodge structure for the hard Lefschetz theorem.
Paper Structure (17 sections, 15 theorems, 51 equations)

This paper contains 17 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Related work
  4. Preliminaries
  5. Hodge structures, filtrations and polarizations
  6. Kähler forms and Chern classes
  7. Sesquilinear pairings and triples of $\widetilde{\mathscr{D}}$-modules
  8. Complex Hodge modules
  9. A remark on polarizations in terms of perverse sheaves
  10. Hodge modules with integral structure
  11. A compactification theorem by Sommese
  12. The decomposition theorem
  13. Behaviour under alterations
  14. Normal crossings case
  15. Reduction to normal crossings case
  16. ...and 2 more sections

Key Result

theorem 1.1

Let $f\colon X\to Y$ be a proper morphism of complex manifolds, and suppose $l\in H^2(X,\mathbb{R}(1))$ is a relative Kähler class. Then for any $M\in \operatorname{HM}_{(\mathbb{Z})}(X,n)^p$ a polarizable pure $\mathbb{C}$-Hodge module of weight $n$ which admits an integral structure, we have the f

Theorems & Definitions (41)

  • theorem 1.1
  • corollary 1
  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • remark 1
  • lemma 1
  • lemma 2
  • ...and 31 more