Canonical volume forms on compact K\"{a}hler manifolds

TL;DR

This paper constructs a canonical singular hermitian metric with semipositive curvature on the canonical line bundle of a compact Kähler manifold, advancing understanding of geometric structures in complex geometry.

Contribution

It introduces a canonical metric with semipositive curvature on the canonical bundle for compact Kähler manifolds with pseudoeffective canonical bundle, providing new tools for complex geometric analysis.

Findings

Existence of a canonical singular hermitian metric with semipositive curvature.

Application to the study of the geometry of Kähler manifolds.

Enhancement of techniques for analyzing pseudoeffective line bundles.

Abstract

We construct a canonical singular hermitian metric with semipositive curvature current on the canonical line bundle of a compact K\"{a}hler manifold with pseudoeffective canonical bundle.