Bott vanishing using GIT and quantization

TL;DR

This paper introduces a new class of algebraic varieties satisfying Bott vanishing, specifically stable GIT quotients of products of projective lines by PGL_2, expanding known examples beyond toric varieties and del Pezzo surfaces.

Contribution

The paper demonstrates that stable GIT quotients of $(P^1)^n$ by PGL_2 satisfy Bott vanishing, using derived category techniques and quantization theorems, thus broadening the class of known Bott vanishing varieties.

Findings

Stable GIT quotients of $(P^1)^n$ by PGL_2 satisfy Bott vanishing.

Techniques recover Bott vanishing for toric varieties.

Application of Halpern-Leistner's work on derived categories and quantization.

Abstract

A smooth projective variety $Y$ is said to satisfy Bott vanishing if $\Omega_Y^j\otimes L$ 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.