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.
Funding source: DFG
Award Identifier / Grant number: “Classification of Algebraic Surfaces and Compact Complex Manifolds”
Funding source: Baden-Württemberg-Stiftung
Award Identifier / Grant number: “Eliteprogramm für Postdoktorandinnen und Postdoktoranden”
Daniel Greb wants to thank János Kollár for interesting discussions concerning the contents of this paper. The authors also want to thank Sándor Kovács and Luca Migliorini for numerous discussions and hints. They also thank the referee for reading the paper extremely carefully, and for several helpful suggestions which led to a partial generalisation of Proposition 4.5, and to a substantial simplification of the argument in Appendix A.1.
© 2014 by De Gruyter
