Algebraic Zhou valuations

2025/05/26 by Bao, Shijie, Guan, Qi'an, Zhou, Lin
#12J20 #14B05 #14F18 #32U05 #32U35 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.19451

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.

Citations

• Equivalence of the sharp effectiveness results of strong openness property
• Zhou valuations and jumping numbers
• Tame maximal weights, relative types and valuations
• Concavity property of minimal L² integrals with Lebesgue measurable gain IV: product of open Riemann surfaces
• Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity
• Boundary points, Minimal L² integrals and Concavity property
• A Minimizing Valuation is Quasi-monomial
• Decreasing equisingular approximations with analytic singularities
• A sharp effectiveness result of Demailly's strong openness conjecture
• Extension of L² holomorphic functions
• The openness conjecture and complex Brunn-Minkowski inequalities
• Modules of square integrable holomorphic germs
• The weighted log canonical threshold
• The openness conjecture for plurisubharmonic functions
• An algebraic approach to the openness conjecture of Demailly and Kollar
• Valuations and asymptotic invariants for sequences of ideals
• Functorial desingularization over Q: boundaries and the embedded case
• Valuations and plurisubharmonic singularities
• Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
• Valuations and multiplier ideals
• Jumping Coefficients of Multiplier Ideals
• The valuative tree
• Uniform Approximation of Abhyankar Valuation Ideals in Smooth Function Fields
• A Subadditivity Property of Multiplier Ideals
• A subadditivity property of multiplier ideals.
• Singularities of Pairs
• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

Related

• Zhou valuations and jumping numbers
• The existence of valuative interpolation
• Tame maximal weights, relative types and valuations
• Valuation spaces and multiplier ideals on singular varieties
• Valuation theoretic methods in the birational geometry of algebraic varieties