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Tropical and non-Archimedean limits of degenerating families of volume forms. (English. French summary) Zbl 1401.32019

The authors study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, they show that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In this way they obtain a measure-theoretic version of a conjecture by M. Kontsevich and Y. Soibelman [Prog. Math. 244, 321–385 (2006; Zbl 1114.14027)] and M. Gross and P. M. H. Wilson [J. Differ. Geom. 55, No. 3, 475–546 (2000; Zbl 1027.32021)], bearing on maximal degenerations of Calabi-Yau manifolds.
Reviewer: Anna Fino (Torino)

MSC:

32Q25 Calabi-Yau theory (complex-analytic aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
32P05 Non-Archimedean analysis
14G22 Rigid analytic geometry
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