Algebraic Zhou valuations
Shijie Bao, Qi'an Guan, Lin Zhou
TL;DR
This work generalizes Zhou valuations from analytic germs to algebraic Zhou valuations on regular schemes over $\mathbf{Q}$ and links them to Jonsson--Mustaţă conjectures through a valuation-theoretic framework. It introduces a mixed jumping-number theory and algebraic Tian functions to characterize when a valuation is Zhou and when it computes jumping numbers, establishing that algebraic Zhou valuations satisfy $A(v)=1-v(\mathfrak{q})$ and that a valuation computing a jumping number must be a scalar multiple of a Zhou valuation. A core result is that the weak JM conjecture is equivalent to all Zhou valuations being quasi-monomial, with quasi-monomiality guaranteed in dimension $\le 2$ or for $\mathfrak{q}=\mathcal{O}_X$. The paper also proves the denseness of the Zhou-valuations cone, enlarges graded sequences to study stability of Tian functions, and ultimately shows a precise analytic correspondence: algebraic Zhou valuations on $\mathrm{Spec}\,\mathcal{O}_o$ coincide with analytic Zhou valuations related to $|\mathfrak{q}|^2$, thereby connecting algebraic and analytic singularity theories through valuations, mixed jumping numbers, and Tian functions.
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--Mustaţă 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.