Log canonical thresholds and Monge-Ampère masses (Q681636)

The log canonical threshold \(c(\varphi)\) of a plurisubharmonic function \(\varphi\) on an open subset of \(\mathbb C^n\) is the supremum over all positive \(c\) such that \(e^{-2c\varphi}\) is \(L^1\) on a neighbourhood of \(0\in\mathbb C^n\). Let \(\Omega\) be a bounded hyperconvex domain. For \(\varphi\in\mathcal{E}(\Omega)\), the largest subset of the space of negative plurisubharmonic functions on which the Monge-Ampère operator is well-defined, a lower bound is given for \(c(\varphi)\) in terms of the log canonical threshold \(c_{n-1}(\varphi)\) of the restriction of \(\varphi\) to \((n-1)\)-dimensional subspaces through the origin and the Lelong number \(e_{n}(\varphi)\) of \((dd^c\varphi)^n\) at \(0\). Together with the restriction formula of \textit{Q. Guan} and \textit{X. Zhou} [``Multiplier ideal sheaves, jumping numbers, and the restriction formula'', Preprint, \url{arXiv:1504.04209}] this gives the upper bound \(c(\varphi)\leq \frac n{n-1}c_{n-1}(\varphi)\).