The openness conjecture for projective manifolds

TL;DR

This paper proves the openness conjecture for positively curved singular metrics on line bundles over projective varieties, confirming a key hypothesis in complex geometry and its implications for plurisubharmonic functions.

Contribution

It provides a proof of the openness conjecture for singular metrics on line bundles over projective varieties, extending the understanding of plurisubharmonic functions with isolated singularities.

Findings

Proof of the openness conjecture for positively curved singular metrics

Confirmation of the conjecture for plurisubharmonic functions with isolated singularities

Implications for complex geometry and pluripotential theory

Abstract

We give a proof of the openness conjecture of Demailly and Koll\'ar for positively curved singular metrics on ample line bundles over projective varieties. As a corollary it follows that the openness conjecture for plurisubharmonic functions with isolated sigularities holds.