A型偏Flag簇的余切丛上的上同调
Cohomology on Cotangent Bundles of Partial Flag Varieties in Type A
- University of Georgia（佐治亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对A型特殊线性群的偏Flag簇余切丛上的向量丛，通过与Bott--Samelson簇上同调的比较得到高阶上同调消失等结论，进而给出相关函数环的生成元集并证明其仿射化具有终端奇点。
AI中文摘要：
设$G=\mathrm{SL}_n(\mathbf{C})$，$P\subset G$是具有Levi因子$L$的标准抛物子群。对$G$-支配权$\lambda$，考虑从$G/P$上与不可约$L$-模$V_L(\lambda)^*$关联的向量丛拉回至$T^*(G/P)$得到的向量丛。我们将其上同调表示为与仿射Kac-Moody群关联的Bott--Samelson簇上某些线丛上同调的正向极限。该比较得出高阶上同调消失的结论，并说明整体截面在$\mathbf{C}[\mathfrak{g}^*]$上由其零次部分$V_G(\lambda)^*$生成。结合这些结果与早期与A. Slipper合作构造的Braverman--Kazhdan交织子，我们给出$\mathbf{C}[T^*(\mathrm{SL}_n/[P,P])]$的显式生成元集。最后，在与Tom Gannon合作的附录中，我们结合这些结果证明仿射化$\mathrm{Spec}(\mathbf{C}[T^*(\mathrm{SL}_n/[P,P])])$具有终端奇点。
英文摘要：
Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $λ$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(λ)^*$. We express its cohomology as a direct limit of the cohomology of certain line bundles on a Bott--Samelson variety associated to an affine Kac-Moody group. This comparison yields vanishing of higher cohomology and shows that the global sections are generated over $\mathbf C[\mathfrak g^*]$ by their degree-zero part $V_G(λ)^*$. Using these results together with the Braverman--Kazhdan intertwiners constructed in earlier joint work with A. Slipper, we give an explicit generating set for $\mathbf C[T^*(\mathrm{SL}_n/[P,P])]$. Finally, in an appendix joint with Tom Gannon, we combine these results to show the affinization $\mathrm{Spec}(\mathbf{C}[T^*(SL_n/[P,P])])$ has terminal singularities.
