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

Abstract

In this article we establish a very general non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments combine the proof of the classical non-vanishing theorem due to V. Shokurov, as well as many ideas from Y.-T. Siu’s analytic proof of the finite generation of the canonical ring. An important tool is the notion of relative critical exponent of two closed positive currents with respect to a measure.