Some inequalities for the weighted log canonical thresholds

Abstract

Let $\varphi$ be a plurisubharmonic function defined in a neighborhood of the origin in $\mathbb C^n$. For each real number $t>-n$, we associate to $\varphi$ the weighted log canonical threshold \[ c_t(\varphi):=\sup\Bigl\{c\geq 0:\|z\|^{2t}e^{-2c\varphi}\in L^1_{\mathrm{loc}} \text{ near }0\Bigr\}. \] In this paper, we prove a sharp slope inequality showing that all difference quotients of the function $t\mapsto c_t(\varphi)$ are uniformly controlled by the Lelong number $\nu_\varphi(0)$. Moreover, we derive explicit lower bounds for the growth of $c_t(\varphi)$ in terms of the complex Monge-Amp\`ere mass of $\varphi$ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory.