Admissible metrics on compact Kähler varieties

Abstract

Let $X$ be a normal compact Kähler variety, and $\mathcal{F}$ a coherent reflexive sheaf on $X$. We investigate the existence of admissible Hermitian metrics on $\mathcal{F}$. If moreover $\mathcal{F}$ is slope stable, we also study the existence of admissible Hermitian-Yang-Mills metrics on it. The existence will hold if one can prove a uniform Sobolev inequality on singular spaces.

The proof of uniform Sobolev inequality in the first version contains a serious gap. We could not fix it, and propose it as a conjecture in the new version. Therefore, the proof on the existence of HYM metrics in the first version is not complete

Cited by in corpus (1)

• Singular cscK metrics on smoothable varieties