Log canonical thresholds and Monge-Ampère masses. (English) Zbl 1441.14017
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 Q. Guan and X. Zhou [“Multiplier ideal sheaves, jumping numbers, and the restriction formula”, Preprint, arXiv:1504.04209] this gives the upper bound \(c(\varphi)\leq \frac n{n-1}c_{n-1}(\varphi)\).
Reviewer: Jan Stevens (Göteborg)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 32U05 | Plurisubharmonic functions and generalizations |
| 32U25 | Lelong numbers |
| 32S10 | Invariants of analytic local rings |
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