Relative critical exponents, non-vanishing and metrics with minimal singularities

TL;DR

This paper investigates metrics with minimal singularities on adjoint bundles, establishing non-vanishing results and exploring their properties using advanced analytic techniques and the concept of relative critical exponents.

Contribution

It introduces new properties of metrics with minimal singularities and employs the notion of relative critical exponent to advance understanding in complex algebraic geometry.

Findings

Proves a non-vanishing statement for metrics with minimal singularities

Establishes several properties of these metrics

Utilizes the concept of relative critical exponent as a key technical tool

Abstract

In this article we prove a non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments involve many ideas from Y.-T. Siu's analytic proof of the finite generation of the canonical ring. An important technical tool is the notion of relative critical exponent of two closed positive currents with respect to a measure.