Log canonical thresholds at infinity

TL;DR

This paper introduces a global version of log canonical thresholds for plurisubharmonic functions in complex space, providing explicit formulas in the toric case and a new approximation method with applications to polynomial maps.

Contribution

It develops a global framework for log canonical thresholds, derives explicit formulas in special cases, and introduces a novel polynomial approximation technique for plurisubharmonic functions.

Findings

Explicit formulas for thresholds in the toric case

A new polynomial approximation with controlled singularities

Applications to tame polynomial maps

Abstract

The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in L^p(\mathbb{C}^n)$. Explicit formulas are obtained in the toric case. By considering Bergman functions of corresponding weighted Hilbert spaces, a new polynomial approximation of plurisubharmonic functions of logarithmic growth with control over its singularities and behavior at infinity (a global version of Demailly's approximation theorem) is established. Some application to tame polynomial maps are given.