Openness results for uniform K-stability

TL;DR

This paper proves that uniform K-stability of a projective variety with a given polarization persists under small perturbations of the polarization, especially for canonical or anti-canonical cases.

Contribution

It establishes the openness of uniform K-stability for projective varieties when the polarization is canonical or anti-canonical.

Findings

Uniform K-stability is open under polarization perturbations.

Stability persists for polarizations close to the canonical or anti-canonical polarization.

Results apply to projective varieties with specific polarizations.

Abstract

Assume that a projective variety together with a polarization is uniformly K-stable. If the polarization is canonical or anti-canonical, then the projective variety is uniformly K-stable with respects to any polarization sufficiently close to the original polarization.