On the singularity type of full mass currents in big cohomology classes

By Tam'as Darvas, E. Di Nezza, Chinh H. Lu. Compositio Mathematica, 2016

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Let $X$ be a compact Kähler manifold and $\{\unicode[STIX]{x1D703}\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $\unicode[STIX]{x1D703}$ -plurisubharmonic functions with full mass are the same as those of a current with minimal singularities. Second, given another big and nef class $\{\unicode[STIX]{x1D702}\}$ , we show the inclusion ${\mathcal{E}}(X,\unicode[STIX]{x1D702})\cap \operatorname{PSH}(X,\unicode[STIX]{x1D703})\subset {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ . Third, we characterize big classes whose full mass currents are ‘additive’. Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c47536760830a45556b584a0d69a27df0a68b62a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity (2017, methodology): “speaking, the purpose of this article is threefold. First, we develop the potential theory of non-pluripolar products, by combining techniques of Witt-Nystr¨om [WN17] and previous work of the authors [DDL16]. Second, given φ∈ PSH(X,θ), we introduce and study the spaces E(X,θ,φ) and E1(X,θ,φ), genera”
Plurisubharmonic envelopes and supersolutions (2017, methodology): “ounded density with respect to Lebesgue measure. The latter is a direct consequence of our Theorem A (see Proposition 5.2). An alternative proof of Proposition 5.2 has been given recently in [Ber13], [DDL16] using ideas from the viscosity theory in [EGZ11]. We will also need the domination principle”