Ko{\l}odziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property

TL;DR

This paper proves a conjecture related to Morse-type integrals on compact Hermitian manifolds with bounded mass, leading to solutions for other longstanding conjectures in complex geometry.

Contribution

It confirms Ko{ }odziej-Tosatti's conjecture and resolves Demailly-Pfcan's and Tosatti-Weinkove's conjectures under the bounded mass property assumption.

Findings

Confirmed Ko{ }odziej-Tosatti's conjecture.

Provided positive answers to Demailly-Pfcan's and Tosatti-Weinkove's conjectures.

Established results for Morse-type integrals on Hermitian manifolds.

Abstract

In this note, we show a conjecture of Ko{\l}odziej-Tosatti about Morse-type integrals in nef $(1,1)$ classes on compact Hermitian manifold with bounded mass property. As a consequence, we give positive answers to Demailly-P\u{a}un's conjecture and Tosatti-Weinkove's conjecture when compact Hermitian manifold with bounded mass property.