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On the Hodge theory of toroidal embeddings and corresponding vanishings

Chuanhao Wei

Abstract

In this paper, we establish Deligne's logarithmic comparison theorem and the $E_1$-degeneration of the corresponding Hodge-de Rham spectral sequence, in the setting of toroidal embeddings. Along the way, we prove Kawamata-Viehweg Vanishing and Bott Vanishing for toroidal varieties and toric varieties respectively.

On the Hodge theory of toroidal embeddings and corresponding vanishings

Abstract

In this paper, we establish Deligne's logarithmic comparison theorem and the -degeneration of the corresponding Hodge-de Rham spectral sequence, in the setting of toroidal embeddings. Along the way, we prove Kawamata-Viehweg Vanishing and Bott Vanishing for toroidal varieties and toric varieties respectively.

Paper Structure

This paper contains 20 sections, 49 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic forms of toric triples
  4. Twistor de Rham complexes of a toric triple
  5. Saito's mixed Hodge module
  6. Reformulation of the smooth case
  7. Simplicial case
  8. Resolution of fan triples
  9. Sorting functions, and sortedness
  10. Efficient and convex subdivisions
  11. Extension of fans
  12. Order of fan triples and sequential star resolution
  13. Log-simplicial model for a general toric triple
  14. Proof of main theorems
  15. Resolution of a toroidal triple/quadruple
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume that $X$ is smooth, with $D=B+C$ being a normal crossing divisor. We have the following quasi-isomorphisms in the derived category of complexes of constructable sheaves on $X$: where and $n=\dim X$.

Theorems & Definitions (105)

  • Theorem 1.1: Deligne's Logarithmic Comparison Theorem
  • Theorem 1.2: Hodge-de Rham degeneration
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Definition 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 95 more