Some inequalities for the weighted log canonical thresholds

Abstract

Let $\ddot{I}†$ be a plurisubharmonic function defined in a neighborhood of the origin in ${\mathbb{C}}^n$. For each real number $t>-n$, we associate to $\ddot{I}†$ 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 $t\mapsto c_t(\ddot{I}†)$ are uniformly controlled by the Lelong number $\hat{I}{½}_{\ddot{I}}†(0)$. Moreover, we derive explicit lower bounds for the growth of $c_t(\ddot{I}†)$ in terms of the complex Monge-Ampère mass of $\ddot{I}†$ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory.