Bott Vanishing Using GIT and Quantization

Abstract

A smooth projective variety Y is said to satisfy Bott vanishing if ${\mathrm{\Omega }}_{\mathit{Y}}^{\mathit{j}}\otimes \mathit{L}$ 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 ${({\mathbb{P}}^1)}^{\mathit{n}}$ by the action of ${\mathit{PGL}}_2$ 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.