Positivity of the cotangent sheaf of singular Calabi-Yau varieties

TL;DR

This paper proves that the tangent and cotangent sheaves of certain singular Calabi-Yau varieties are not pseudoeffective, extending previous results and providing examples of low-dimensional Calabi-Yau varieties with specific singularities.

Contribution

It generalizes the non-pseudoeffectivity of tangent and cotangent sheaves to singular Calabi-Yau and holomorphic symplectic varieties, and offers examples with codimension 2 singularities.

Findings

Tangent and reflexivized cotangent sheaves are not pseudoeffective for these varieties.

Provides explicit examples of low-dimensional Calabi-Yau varieties with codimension 2 singularities.

Abstract

We prove that the tangent and the reflexivized cotangent sheaves of any normal projective klt Calabi-Yau or irreducible holomorphic symplectic variety are not pseudoeffective, generalizing results of A. H\"oring and T. Peternell arXiv:1710.06183v2 [math.AG]. We provide examples of Calabi-Yau varieties of small dimension with singularities in codimension 2.