Automorphisms and deformations of regular semisimple Hessenberg varieties
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| Format: | Preprint |
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2024
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| _version_ | 1866912809986555904 |
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| author | Brosnan, Patrick Escobar, Laura Hong, Jaehyun Lee, Donggun Lee, Eunjeong Mellit, Anton Sommers, Eric |
| author_facet | Brosnan, Patrick Escobar, Laura Hong, Jaehyun Lee, Donggun Lee, Eunjeong Mellit, Anton Sommers, Eric |
| contents | We show that regular semisimple Hessenberg varieties can have moduli. To be precise, suppose $X$ is a regular semisimple Hessenberg variety of codimension $1$ in the flag variety $G/B$, where $G$ is a simple algebraic group of rank $r$ over $\mathbb{C}$ and $B$ is a Borel subgroup. We show that the space~$\mathrm{H}^1(X,TX)$ of first order deformations of $X$ has dimension $r-1$ except in type $A_2$. (In type $A_2$, the Hessenberg varieties in question are all isomorphic to the permutohedral toric surface, and $\dim\mathrm{H}^1(X,TX) = 0$.) Moreover, we show that the Kodaira--Spencer map $\mathfrak{g}\to \mathrm{H}^1(X,TX)$ is onto, that the identity component of the automorphism group of $X$ is a maximal torus of $G$, and that $\mathrm{H}^i(X,TX) = 0$ for $i \geq 2$. Along the way, we prove several theorems of independent interest about the cohomology of homogeneous vector bundles on~$G/B$.
In type $A$, we can give an even more precise statement determining when two codimension $1$ regular semisimple Hessenberg varieties in $G/B$ are isomorphic. We also compute the automorphism groups explicitly in type~$A_{n-1}$ in the terms of stabilizer subgroups of the action of the symmetric group $S_{n}$ on the moduli space $M_{0,n+1}$ of smooth genus $0$ curves with $n + 1$ marked points. Using this, we describe the moduli stack of the regular semisimple Hessenberg varieties $X$ explicitly as a quotient stack of $M_{0,n+1}$.
We prove several analogous results for Hessenberg varieties in generalized flag varieties $G/P$, where $P$ is a parabolic subgroup of $G$. In type $A$, these results are used in the proofs of the results for $G/B$, but they are also of independent interest because the associated moduli stacks are related directly to the action of $S_n$ on $M_{0,n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18313 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automorphisms and deformations of regular semisimple Hessenberg varieties Brosnan, Patrick Escobar, Laura Hong, Jaehyun Lee, Donggun Lee, Eunjeong Mellit, Anton Sommers, Eric Algebraic Geometry 14M15, 14J50, 14J10 We show that regular semisimple Hessenberg varieties can have moduli. To be precise, suppose $X$ is a regular semisimple Hessenberg variety of codimension $1$ in the flag variety $G/B$, where $G$ is a simple algebraic group of rank $r$ over $\mathbb{C}$ and $B$ is a Borel subgroup. We show that the space~$\mathrm{H}^1(X,TX)$ of first order deformations of $X$ has dimension $r-1$ except in type $A_2$. (In type $A_2$, the Hessenberg varieties in question are all isomorphic to the permutohedral toric surface, and $\dim\mathrm{H}^1(X,TX) = 0$.) Moreover, we show that the Kodaira--Spencer map $\mathfrak{g}\to \mathrm{H}^1(X,TX)$ is onto, that the identity component of the automorphism group of $X$ is a maximal torus of $G$, and that $\mathrm{H}^i(X,TX) = 0$ for $i \geq 2$. Along the way, we prove several theorems of independent interest about the cohomology of homogeneous vector bundles on~$G/B$. In type $A$, we can give an even more precise statement determining when two codimension $1$ regular semisimple Hessenberg varieties in $G/B$ are isomorphic. We also compute the automorphism groups explicitly in type~$A_{n-1}$ in the terms of stabilizer subgroups of the action of the symmetric group $S_{n}$ on the moduli space $M_{0,n+1}$ of smooth genus $0$ curves with $n + 1$ marked points. Using this, we describe the moduli stack of the regular semisimple Hessenberg varieties $X$ explicitly as a quotient stack of $M_{0,n+1}$. We prove several analogous results for Hessenberg varieties in generalized flag varieties $G/P$, where $P$ is a parabolic subgroup of $G$. In type $A$, these results are used in the proofs of the results for $G/B$, but they are also of independent interest because the associated moduli stacks are related directly to the action of $S_n$ on $M_{0,n}$. |
| title | Automorphisms and deformations of regular semisimple Hessenberg varieties |
| topic | Algebraic Geometry 14M15, 14J50, 14J10 |
| url | https://arxiv.org/abs/2405.18313 |