Some inequalities for the weighted log canonical thresholds

2025/11/27 by Nguyen Xuan Hong, Hong, Nguyen Xuan
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2511.22373

openalex publication_date 2025/11/27 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

Let φ be a plurisubharmonic function defined in a neighborhood of the origin in \mathbb Cⁿ. For each real number t>-n, we associate to φ the weighted log canonical threshold c_t(φ):=sup\c≥ 0:‖z‖^2te^-2cφ∈ L¹_loc near 0\. In this paper, we prove a sharp slope inequality showing that all difference quotients of the function t↦ c_t(φ) are uniformly controlled by the Lelong number ν_φ(0). Moreover, we derive explicit lower bounds for the growth of c_t(φ) in terms of the complex Monge-Ampère mass of φ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory.

Citations

• Singularity invariants of plurisubharmonic functions and complex spaces
• Log canonical thresholds and Monge-Ampere masses
• Extension of L² holomorphic functions
• Extremal cases for the log canonical threshold
• The openness conjecture and complex Brunn-Minkowski inequalities
• The weighted log canonical threshold
• A sharp lower bound for the log canonical threshold
• Partial pluricomplex energy and integrability exponents of plurisubharmonic functions
• Monge-Ampere measures on pluripolar sets
• The pluri-fine topology is locally connected

Related

• A log canonical threshold test
• Extremal cases for the log canonical threshold
• The weighted log canonical threshold
• Log canonical thresholds and coregularity
• ACC for log canonical thresholds