Ample tangent bundle on smooth projective stacks

TL;DR

This paper investigates the ampleness of tangent bundles on smooth projective stacks, demonstrating that weighted projective stacks have ample tangent bundles, thus extending classical results from varieties to stacks.

Contribution

It proves that the tangent bundle of weighted projective stacks is ample, providing a new example beyond classical varieties and raising questions about the classification of smooth projective stacks.

Findings

Weighted projective stacks have ample tangent bundles.

Extension of Mori's result from varieties to stacks.

Open question on classification of smooth projective stacks.

Abstract

We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective varieties with ample vector bundle are isomorphic to $\mathbb{P}^N$ for some $N$. Extending our geometric spaces from varieties to projective stacks, we are able to provide a new example. This leaves the open question of knowing if the only smooth projective stacks are the weighted projective stacks.