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arXiv · 1712.09269

Geometry of regular Hessenberg varieties

Hiraku Abe·Naoki Fujita·Haozhi Zeng
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Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. For a regular element $x$ in $\mathfrak{g}$ and a Hessenberg space $H\subseteq \mathfrak{g}$, we consider a regular Hessenberg variety $X(x,H)$ in the flag variety associated with $\mathfrak{g}$. We take a Hessenberg space so that $X(x,H)$ is irreducible, and show that the higher cohomology groups of the structure sheaf of $X(x,H)$ vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.

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Hiraku Abe, Naoki Fujita, Haozhi Zeng. 2017-12-26. Geometry of regular Hessenberg varieties. https://arxiv.org/abs/1712.09269

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