We consider manifolds with conic singularities that are isometric to Rn outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author [23] to establish a “very weak” Huygens’ principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr¨odinger equation.

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In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$ . This threshold is defined as the supremum of constants$c$> 0 such that $${e^{-2c\varphi}}$$ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$ to the intermediate multiplicity numbers $${e_j(\varphi)}$$ , defined as the Lelong numbers of $${(dd^c\varphi)^j}$$ at 0 (so that in particular $${e_0(\varphi)=1}$$ ). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$ . This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$ due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.