Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four

Patrick Graf

Abstract

Let $X$ be a compact Kähler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that $X$ admits a Beauville-Bogomolov decomposition: a finite quasi-étale cover of $X$ splits as a product of a complex torus and singular Calabi-Yau and irreducible holomorphic symplectic varieties. We also prove that $X$ has small projective deformations and the fundamental group of $X$ is projective. To obtain these results, we propose and study a new version of the Lipman-Zariski conjecture.

A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four

Abstract

Let be a compact Kähler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that admits a Beauville-Bogomolov decomposition: a finite quasi-étale cover of splits as a product of a complex torus and singular Calabi-Yau and irreducible holomorphic symplectic varieties. We also prove that has small projective deformations and the fundamental group of is projective. To obtain these results, we propose and study a new version of the Lipman-Zariski conjecture.

Paper Structure

This paper contains 11 sections, 10 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Notation and basic facts
  3. The Lipman--Zariski Conjecture for direct summands
  4. Proof of \ref{['directlz']}
  5. Proof of \ref{['directlz-cor']}
  6. Proof of \ref{['main bound']}
  7. Proof of main results
  8. Torus covers revisited
  9. Proof of \ref{['btt']}
  10. Proof of \ref{['kod']}
  11. Proof of Corollaries \ref{['pi1']} and \ref{['bb']}

Key Result

Theorem 1.1

Let $X$ be a normal compact Kähler space of dimension $\le 4$, with klt singularities and such that $\mathrm{c}_{1}(X) = 0 \in \mathrm{H}^{2} \!\left( X, \mathbb R \right)$. Assume that $\dim {X}_{\mathrm{sg}} \le 1$. Then the semiuniversal locally trivial deformation space $\mathop{\mathrm{Def^{lt}

Theorems & Definitions (31)

  • Theorem 1.1: Singular BTT theorem in dimension four
  • Corollary 1.2: Kodaira problem in dimension four
  • Corollary 1.3: Fundamental groups
  • Corollary 1.4: BB decomposition
  • Theorem 1.6: Bounding the flat factor
  • Remark
  • Theorem 1.8: \ref{['directlz ques']} for klt singularities
  • Definition 2.1: Torsion-free differentials
  • Definition 2.2: Quasi-étale covers
  • Definition 2.3
  • ...and 21 more