论文标题
deligne- lusztig品种$ {\ rm gl} _n $的同时和几何形状
Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$
论文作者
论文摘要
我们对平滑的压缩$ \ OVILLINE {x}（w）$ deligne-lusztig品种的$ x（w）$ for $ {\ rm gl} _n $的$ x（w）$，对于Weyl Group的所有元素$ W $。结果，我们获得了$ \ p $ p $ p $ p $ p $ - adicétale的共同体的$ {\ rm mod} \ p^m $。此外，使用$ \ OVILLINE {X}（W）$的结果以及与$ \ Overline {X}（W）$分层相关的光谱序列，我们推断出$ {\ rm mod} \ rm mod} \ p^m $和Integral $ p $ p $ - adiC-adiC-adiCétale共同体，并具有综合支持$ x（w）$ x（w）$。 为了证明主要定理的证明，除了考虑$ x（w）$的汉森光滑压实外，我们还表明，在$ {\ rm gl} _n $的情况下，类似类似的构造提供了$ x（w）$的平滑压缩。此外，我们在附录中表明，对于任何连接的还原组$ g $和任何$ w $，zariski关闭$ x（w）$都具有伪理性的颗粒。
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.
