利用GIT与量子化证明Bott消失
Bott vanishing using GIT and quantization
AI总结：
本文利用Halpern-Leistner的GIT导出范畴与量子化定理，证明了$(\mathbb{P}^1)^n$在$PGL_2$作用下的稳定GIT商满足Bott消失，并用类似方法重新得到环面簇情形。
摘要：
一个光滑射影簇$Y$被称为满足Bott消失，如果对于每个$j$和每个丰富线丛$L$，$Ω_Y^j\otimes L$没有高阶上同调。已知满足这一性质的例子很少，其中包括环面簇，以及Totaro最近证明的五次del Pezzo曲面。本文提出一类新的满足Bott消失的簇，即在特征零的代数闭域上，$(\mathbb{P}^1)^n$在$PGL_2$作用下的稳定GIT商。为此，我们利用Halpern-Leistner关于GIT商导出范畴的工作及其版本的量子化定理。我们还看到，使用类似的技术，可以重新得到环面情形的Bott消失。
英文摘要：
A smooth projective variety $Y$ is said to satisfy Bott vanishing if $Ω_Y^j\otimes L$ has no higher cohomology for every $j$ and every ample line bundle $L$. Few examples are known to satisfy this property. Among them are toric varieties, as well as the quintic del Pezzo surface, recently shown by Totaro. Here we present a new class of varieties satisfying Bott vanishing, namely stable GIT quotients of $(\mathbb{P}^1)^n$ by the action of $PGL_2$, over an algebraically closed field of characteristic zero. For this, we use the work done by Halpern-Leistner on the derived category of a GIT quotient, and his version of the quantization theorem. We also see that, using similar techniques, we can recover Bott vanishing for the toric case.
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