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Beauville-Bogomolov decomposition for klt varieties

Henri Guenancia

TL;DR

This work presents a largely self-contained account of the singular Beauville–Bogomolov decomposition for compact Kähler varieties with klt singularities and zero first Chern class. Building on a singular Calabi–Yau metric and Bochner principle, it identifies a finite quasi-étale cover admitting a product decomposition into a torus and factors that are irreducible Calabi–Yau or irreducible holomorphic symplectic in the singular sense. The approach blends metric (holonomy and Ricci-flat Kähler metrics), topological (finite local fundamental groups and Albanese geometry), and algebro-geometric (stability and algebraic integrability) methods, with a crucial shift from the smooth to singular setting via quasi-étale covers and deformation arguments. Key new ingredients include the identification of flat directions via 1-forms on quasi-étale covers, the finiteness of holonomy components, and a splitting theorem for algebraically integrable foliations, enabling the passage from tangent-sheaf decompositions to product decompositions. The results unify and extend projective and Kähler theories, offering a robust template for understanding the structure of singular varieties with trivial canonical class and their fundamental groups, and linking to applications in primitive symplectic theory and orbifold Chern-class characterizations.

Abstract

These lecture notes present a mostly self-contained proof of the singular version of Beauville-Bogomolov decomposition theorem for compact Kähler varieties with log terminal singularities and zero first Chern class.

Beauville-Bogomolov decomposition for klt varieties

TL;DR

This work presents a largely self-contained account of the singular Beauville–Bogomolov decomposition for compact Kähler varieties with klt singularities and zero first Chern class. Building on a singular Calabi–Yau metric and Bochner principle, it identifies a finite quasi-étale cover admitting a product decomposition into a torus and factors that are irreducible Calabi–Yau or irreducible holomorphic symplectic in the singular sense. The approach blends metric (holonomy and Ricci-flat Kähler metrics), topological (finite local fundamental groups and Albanese geometry), and algebro-geometric (stability and algebraic integrability) methods, with a crucial shift from the smooth to singular setting via quasi-étale covers and deformation arguments. Key new ingredients include the identification of flat directions via 1-forms on quasi-étale covers, the finiteness of holonomy components, and a splitting theorem for algebraically integrable foliations, enabling the passage from tangent-sheaf decompositions to product decompositions. The results unify and extend projective and Kähler theories, offering a robust template for understanding the structure of singular varieties with trivial canonical class and their fundamental groups, and linking to applications in primitive symplectic theory and orbifold Chern-class characterizations.

Abstract

These lecture notes present a mostly self-contained proof of the singular version of Beauville-Bogomolov decomposition theorem for compact Kähler varieties with log terminal singularities and zero first Chern class.

Paper Structure

This paper contains 41 sections, 38 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. The decomposition theorem
  3. Skeleton of the proof
  4. The singular version
  5. Brief outline of the proof
  6. Preliminary material
  7. Varieties
  8. Sheaves
  9. Holonomy
  10. Klt singularities: topological, analytic and metric aspects
  11. Definition and examples
  12. Topological properties of klt singularities
  13. Analytic properties of klt singularities
  14. Kähler-Einstein metrics on klt varieties
  15. Polystability, Bochner principle and Albanese map
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $X$ be a compact Kähler manifold such that $c_1(X)=0\in H^2(X,{\mathbb{C}})$. There exists a finite étale cover $f:X'\to X$ and a product decomposition where $T$ is a complex torus and for any index $i\in I$ (resp. $j\in J$), the manifold $Y_i$ (resp. $Z_j$) is irreducible Calabi--Yau (resp. irreducible holomorphic symplectic).

Theorems & Definitions (80)

  • Theorem 1.1: Beauville-Bogomolov decomposition theorem
  • Conjecture 1.2
  • Theorem 1.3
  • Definition 2.1: Quasi-étale covers
  • Definition 2.2: Kähler variety
  • Definition 2.3: Reflexive differentials
  • Definition 2.5: Foliations
  • Definition 2.6: Stable sheaves
  • Example 2.7
  • Definition 2.8: Kähler holonomy
  • ...and 70 more