Some inequalities for the weighted log canonical thresholds

Let $φ$ be a plurisubharmonic function defined in a neighborhood of the origin in $\mathbb C^n$. For each real number $t>-n$, we associate to $φ$ the weighted log canonical threshold [ c_t(φ):=\sup\Bigl{c\geq 0:|z|^{2t}e^{-2cφ}\in L^1_{\mathrm{loc}} \text{ near }0\Bigr}. ] In this paper, we prove a sharp slope inequality showing that all difference quotients of the function $t\mapsto c_t(φ)$ 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.

💡 Research Summary

The paper introduces a new invariant for plurisubharmonic (psh) functions, called the weighted log canonical threshold (WLCT). For a psh function φ defined near the origin in ℂⁿ and a real parameter t > −n, the WLCT is defined as

 cₜ(φ) = sup { c ≥ 0 | ‖z‖^{2t} e^{−2cφ} ∈ L¹_loc near 0 }.

This definition generalizes the classical log canonical threshold (the case t = 0) and the weighted thresholds previously considered with monomial weights. The main goal of the paper is to relate cₜ(φ) to two classical measures of singularities: the Lelong number ν_φ(0) and the complex Monge‑Ampère mass eₙ(φ) = ∫_{0}(ddᶜφ)ⁿ.

Theorem 1.1 (Weighted Skoda inequality).
If ν_φ(0) > 0 then cₜ(φ) is a finite positive number and satisfies

 1 /