Abstract
In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of \(\mathfrak{gl }_n(\mathbb C )\) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology.
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Precup, M. Affine pavings of Hessenberg varieties for semisimple groups. Sel. Math. New Ser. 19, 903–922 (2013). https://doi.org/10.1007/s00029-012-0109-z
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DOI: https://doi.org/10.1007/s00029-012-0109-z