Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929276745416704 |
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| author | Wang, Yingying |
| author_facet | Wang, Yingying |
| contents | We give a description of the cohomology groups of the structure sheaf on smooth compactifications $\overline{X}(w)$ of Deligne--Lusztig varieties $X(w)$ for ${\rm GL}_n$, for all elements $w$ in the Weyl group. As a consequence, we obtain the ${\rm mod}\ p^m$ and integral $p$-adic étale cohomology of $\overline{X}(w)$. Moreover, using our result for $\overline{X}(w)$ and a spectral sequence associated to a stratification of $\overline{X}(w)$, we deduce the ${\rm mod}\ p^m$ and integral $p$-adic étale cohomology with compact support of $X(w)$.
In our proof of the main theorem, in addition to considering the Demazure--Hansen smooth compactifications of $X(w)$, we show that a similar class of constructions provide smooth compactifications of $X(w)$ in the case of ${\rm GL}_n$. Furthermore, we show in the appendix that the Zariski closure of $X(w)$, for any connected reductive group $G$ and any $w$, has pseudo-rational singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02569 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$ Wang, Yingying Algebraic Geometry Representation Theory 20C33, 14G17 We give a description of the cohomology groups of the structure sheaf on smooth compactifications $\overline{X}(w)$ of Deligne--Lusztig varieties $X(w)$ for ${\rm GL}_n$, for all elements $w$ in the Weyl group. As a consequence, we obtain the ${\rm mod}\ p^m$ and integral $p$-adic étale cohomology of $\overline{X}(w)$. Moreover, using our result for $\overline{X}(w)$ and a spectral sequence associated to a stratification of $\overline{X}(w)$, we deduce the ${\rm mod}\ p^m$ and integral $p$-adic étale cohomology with compact support of $X(w)$. In our proof of the main theorem, in addition to considering the Demazure--Hansen smooth compactifications of $X(w)$, we show that a similar class of constructions provide smooth compactifications of $X(w)$ in the case of ${\rm GL}_n$. Furthermore, we show in the appendix that the Zariski closure of $X(w)$, for any connected reductive group $G$ and any $w$, has pseudo-rational singularities. |
| title | Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$ |
| topic | Algebraic Geometry Representation Theory 20C33, 14G17 |
| url | https://arxiv.org/abs/2208.02569 |