Algebraic Zhou valuations

TL;DR

This paper extends Zhou valuations from complex domains to general schemes, establishing their algebraic properties, characterizations, and connections to analytic counterparts, and relates them to a conjecture involving quasi-monomial valuations.

Contribution

It generalizes Zhou valuations to algebraic schemes, introduces mixed jumping numbers and Tian functions, and links algebraic and analytic Zhou valuations.

Findings

Algebraic Zhou valuations are characterized via Tian functions.

An algebraic version of the Jonsson--Mustață conjecture is formulated.

Correspondence between algebraic and analytic Zhou valuations is established.

Abstract

In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Musta\c{t}\u{a} conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.