Singular spaces with trivial canonical class

TL;DR

This paper extends the Beauville-Bogomolov Decomposition Theorem to singular projective varieties with trivial canonical class, proposing a structure theory based on two classes of canonical varieties.

Contribution

It proves a decomposition theorem for tangent sheaves of singular varieties with trivial canonical class, advancing the understanding of their structure in the context of minimal model theory.

Findings

Decomposition theorem for tangent sheaf of singular varieties

Identification of two fundamental classes of canonical varieties

Progress towards a structure theory for varieties with Kodaira dimension zero

Abstract

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact K\"ahler manifold with numerically trivial canonical bundle admits an \'etale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation. Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class. In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau-- and irreducible holomorphic-symplectic manifolds, respectively.