Tropical and non-Archimedean limits of degenerating families of volume forms

By S. Boucksom, Mattias Jonsson. 2016

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We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich--Soibelman and Gross--Wilson, bearing on maximal degenerations of Calabi--Yau manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/128f83cd7a52fbf5bce010e3ef670f603ee591f4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, background): “)n; Theorem 3.6 is therefore a consequence of the following result. Lemma 3.10. We have R X log h MA(˚s) e2 s ref i (ddc˚s)n= O(logs) as s!1. Let us rst prove an estimate of independent interest. See [BJ16a] for more precise results. Lemma 3.11. Let Xbe an snc test conguration for Xand a smooth metr”
Algebraicity of the metric tangent cones and equivariant K-stability (2018, methodology): “w the upper boundedness of G(˜ϕt) can be seen in two ways. For one way, one can resolve the singularity of {r ≤1}∩(X|D) and estimate the integral using the method as in [Li17a, Proof of Lemma 3.7] or [BJ17]. The other approximation approach is the following. Recall that r2 is the radius function ass”