Log canonical thresholds at infinity
Carles Bivià-Ausina, Alexander Rashkovskii
TL;DR
The paper develops a global theory of log canonical thresholds at infinity for plurisubharmonic functions with logarithmic growth on ${\mathbb C}^n$, introducing ${\operatorname{c_\infty}}(u)$ and its variant ${\hat{c}_\infty}(u)$. It leverages logarithmic indicators and Newton polyhedra to obtain explicit, toric-case formulas, and constructs a global Demailly-type analytic approximation via Bergman kernels that controls singularities and infinity behavior. It also analyzes multipliers at infinity, toric and monomial maps, and the relation between ${\operatorname{c_\infty}}(u)$ and Monge-Ampère masses, providing both general bounds and sharp results under lower-set conditions. The results yield practical tools for describing the integrability of $e^{-cu}$ at infinity and for understanding asymptotic singularity behavior in global pluripotential theory with 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.