Singular contact varieties

TL;DR

This paper generalizes holomorphic contact structures to varieties with rational singularities, explores their properties, and classifies projective contact varieties in dimension three, linking them to nilpotent orbit theory.

Contribution

It introduces a new notion of singular contact varieties, establishes their basic properties, and classifies three-dimensional projective cases, connecting to nilpotent orbit theory.

Findings

Projectivizations of nilpotent orbit closures satisfy the new contact definition.

Equivalence between crepant and contact resolutions in the projective case.

Complete classification of projective contact varieties in dimension 3.

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 $\mathbb{C}^*$-bundle and the contact ones along with the existence of the stratification \`a 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.