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arXiv · 2212.07930

Singular contact varieties

Robert Śmiech
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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.

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Robert Śmiech. 2022-12-15. Singular contact varieties. https://arxiv.org/abs/2212.07930

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