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Generic Vanishing for Singular Varieties via Du Bois complexes

Anh Duc Vo

Abstract

We prove appropriate generic vanishing theorems for singular varieties, generalizing the well-known generic vanishing theorem by Green and Lazarsfeld in [GL87] and the generic vanishing theorem of Nakano type in [PS13]. Our theorem explains the counterexample of Hacon and Kovács in [HK15].

Generic Vanishing for Singular Varieties via Du Bois complexes

Abstract

We prove appropriate generic vanishing theorems for singular varieties, generalizing the well-known generic vanishing theorem by Green and Lazarsfeld in [GL87] and the generic vanishing theorem of Nakano type in [PS13]. Our theorem explains the counterexample of Hacon and Kovács in [HK15].

Paper Structure

This paper contains 12 sections, 23 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed Hodge modules
  4. Decomposition theorem and Torsion-freeness theorem for Hodge modules of pure weight
  5. Intersection complex and Du Bois complex
  6. Local cohomological dimension
  7. Integral functors and GV-objects
  8. Generic vanishing of mixed Hodge modules on abelian varieties
  9. Generalization of classical generic vanishing for singular varieties
  10. (Stratified) semismallness defect
  11. Proof of Main Theorems
  12. Non-negativity of Euler characteristic of dual Du Bois complex

Key Result

Theorem 1

Let $a:X\to A$ be a morphism from a projective variety $X$ to an abelian variety $A$. Then

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 1.3
  • Remark 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 2.1: Saito's decomposition theorem
  • Theorem 2.2: Saito_Kollar_conjecture, Saito_Kollar_conjecture*Theorem 3.2
  • Proposition 4.2: MP-lci
  • ...and 36 more