A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

2025/11/03 by Itay Glazer, Glazer, Itay, Yotam I. Hendel +1
Mathematics · #11S40 #11S80 (Secondary) #14B05 (Primary) 14G20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO) #math.AG #math.LO #msc:11S40 #msc:11S80 #msc:14B05 #msc:14G20

paper · pdf · doi:10.48550/arxiv.2511.01270

11 pages. Revised version. To appear in Proceedings of the AMS

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

Given a local field F of positive characteristic, an F-analytic manifold X and an analytic function f:X→ F, the F-analytic log-canonical threshold lct_F(f;x₀) is the supremum over the values s≥0 such that |f|_F^-s is integrable near x₀∈ X. We show that lct_F(f;x₀)>0. Moreover, if f is a regular function on a smooth algebraic F-variety, we obtain an effective lower bound lct_F(f;x₀)>C, where C>0 is explicit and depends only on the complexity class of X and f.

Citations

• Integrability and singularities of Harish-Chandra characters
• Integrability of pushforward measures by analytic maps
• Anti-concentration of polynomials: dimension-free covariance bounds and decay of Fourier coefficients
• Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero
• IMPANGA lecture notes on log canonical thresholds
• Log canonical thresholds, F-pure thresholds, and non-standard extensions
• Singularity invariants related to Milnor fibers: survey
• Which powers of holomorphic functions are integrable?
• On real log canonical thresholds
• Integration in valued fields
• On F-pure thresholds

Related

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• A sharp lower bound for the log canonical threshold
• ACC for log canonical thresholds for complex analytic spaces
• Log canonical thresholds, F-pure thresholds, and non-standard extensions
• ACC for foliated log canonical thresholds