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arXiv · 1807.00276

Log-concavity of volume and complex Monge-Amp\`ere equations with prescribed singularity

Tamás Darvas·Eleonora Di Nezza·Chinh H. Lu
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Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the K\"ahler case as well. As an application we confirm a conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi concerning log-concavity of the volume of closed positive $(1,1)$-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn-Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and $P$-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

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Tamás Darvas, Eleonora Di Nezza, Chinh H. Lu. 2018-07-01. Log-concavity of volume and complex Monge-Amp\`ere equations with prescribed singularity. https://arxiv.org/abs/1807.00276

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