Valuative multiplier ideals

TL;DR

This paper develops an algebraic framework for quasi-plurisubharmonic functions using valuation spaces, defining multiplier ideals and proving foundational properties with applications in complex algebraic geometry.

Contribution

It introduces an algebraic analogue of qpsh functions on valuation spaces and establishes their basic properties, expanding tools for complex algebraic geometry.

Findings

Established subadditivity property of multiplier ideals

Proved an approximation theorem for algebraic qpsh functions

Applied the framework to problems in complex algebraic geometry

Abstract

The main goal of this paper is to construct an algebraic analogue of quasi-plurisubharmonic function (qpsh for short) from complex analysis and geometry. We define a notion of qpsh function on a valuation space associated to a quite general scheme. We then define the multiplier ideals of these functions and prove some basic results about them, such as subadditivity property, the approximation theorem. We also treat some applications in complex algebraic geometry.