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arXiv 2212.07930math.AG
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奇异切触簇

Singular contact varieties

Robert Śmiech

备注：
Accepted 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)
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AI总结：

本研究将光滑流形的全纯切触结构推广到有理奇点簇，证明其基本性质，给出幂零轨道闭包射影化等实例，建立辛簇与切触簇的对应及Kaledin型分层，还证明射影情形下 crepant 与切触解消等价、簇的单直纹性并完成三维分类。

摘要：

在这篇注记中，我们将流形（光滑簇）上全纯切触结构的概念推广到具有有理奇点的簇，并证明了这类对象的基本性质。奇异切触簇的自然例子来源于幂零轨道理论：半单李代数中幂零轨道闭包的每个射影化在正规化后都满足我们的定义。我们证明了具有$\mathbb{C}^*$-丛结构的辛簇与切触簇之间的对应关系，以及Kaledin型分层的存在性。在射影情形下，我们证明了奇点的crepant解消与切触解消之间的等价性，证明了簇的单直纹性，并给出了三维射影切触簇的完整分类。

英文摘要：

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 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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