Reflexive differential forms on singular spaces -- Geometry and Cohomology

TL;DR

This paper advances the understanding of reflexive differential forms on singular spaces by extending key theorems, analyzing their cohomology, and exploring their geometric and topological properties.

Contribution

It generalizes the extension theorem for reflexive forms to complex spaces and investigates their existence, vanishing properties, and topological aspects on singular varieties.

Findings

Extension theorem for reflexive forms on complex spaces

Non-existence of reflexive pluri-differentials on certain singular varieties

Failure of Kodaira-Akizuki-Nakano vanishing in general cases

Abstract

Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira-Akizuki-Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.