Tame maximal weights, relative types and valuations

TL;DR

This paper introduces Zhou weights and Zhou valuations, establishing their properties and applications in measuring singularities of plurisubharmonic functions, with implications for complex analysis and algebraic geometry.

Contribution

It develops a new class of tame maximal weights called Zhou weights and characterizes their relative types as valuations, linking singularity measures with tropical algebra.

Findings

Zhou weights are shown to satisfy tropical multiplicativity and additivity.

Relative types to Zhou weights characterize division relations in holomorphic germs.

Global Zhou weights generalize pluricomplex Green functions with continuity and approximation properties.

Abstract

In this article, we obtain a class of tame maximal weights (Zhou weights). Using Tian functions (the function of jumping numbers with respect to the exponents of a holomorphic function or the multiples of a plurisubharmonic function) as a main tool, we establish an expression of relative types (Zhou numbers) to these tame maximal weights in integral form, which shows that the relative types satisfy tropical multiplicativity and tropical additivity. Thus, the relative types to Zhou weights are valuations (Zhou valuations) on the ring of germs of holomorphic functions. We use Tian functions and Zhou numbers to measure the singularities of plurisubharmonic functions, involving jumping numbers and multiplier ideal sheaves. Especially, the relative types to Zhou weights characterize the division relations of the ring of germs of holomorphic functions. Finally, we consider a global version of Zhou weights on domains in $\mathbb{C}^n$, which is a generalization of the pluricomplex Green functions, and we obtain some properties of them, including continuity and some approximation results.