Toric plurisubharmonic functions and analytic adjoint ideal sheaves

TL;DR

This paper explores the properties of toric plurisubharmonic functions, describes their multiplier ideal sheaves, generalizes the analytic adjoint ideal sheaves, and applies these concepts to extension theorems in complex geometry.

Contribution

It provides a detailed description of multiplier ideal sheaves for toric psh functions and extends the notion of adjoint ideal sheaves to the analytic setting, with applications to extension theorems.

Findings

Characterization of multiplier ideal sheaves for toric psh functions

Generalization of adjoint ideal sheaves to the analytic context

Proof of a generalized adjunction exact sequence

Abstract

In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald's theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa-Takegoshi-Manivel theorem, we prove the existence of the so-called generalized adjunction exact sequence, which enables us to recover a weak version of the global extension theorem of Manivel, for compact K\"ahler manifolds.