Singular contact varieties

Abstract

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular contact varieties come from the theory of nilpotent orbits: every projectivization of the closure of a nilpotent orbit in a semisimple Lie algebra satisfies our definition after normalization. We show the correspondence between symplectic varieties with the structure of a ${\mathbb{C}}^{\ast }$-bundle and the contact ones along with the existence of the stratification à la Kaledin. In the projective case we demonstrate the equivalence between crepant and contact resolutions of singularities, show the uniruledness and give a full classification of projective contact varieties in dimension 3.

arXiv commentAccepted for publication in manuscripta mathematica. 29 pages, 2nd version containing substantial changes in the presentation of the results and some new ones. In particular, there is now a classification result in dimension 3 and a non-example that satisfies the weaker definition of Campana-Flenner (with nonrational singularities)