88 research outputs found

    An algorithm for computing the integral closure

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    We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field

    Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

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    Let KK be a non-archimedean local field, XX a smooth and proper KK-scheme, and fix a pluricanonical form on XX. For every finite extension KK' of KK, the pluricanonical form induces a measure on the KK'-analytic manifold X(K)X(K'). We prove that, when KK' runs through all finite tame extensions of KK, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of XX converge to a Lebesgue-type measure on the temperate part of the Kontsevich--Soibelman skeleton, assuming the existence of a strict normal crossings model for XX. We also prove a similar result for all finite extensions KK' under the assumption that XX has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi--Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields

    The determination of integral closures and geometric applications

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    We express explicitly the integral closures of some ring extensions; this is done for all Bring-Jerrard extensions of any degree as well as for all general extensions of degree < 6; so far such an explicit expression is known only for degree < 4 extensions. As a geometric application we present explicitly the structure sheaf of every Bring-Jerrard covering space in terms of coefficients of the equation defining the covering; in particular, we show that a degree-3 morphism f : Y --> X is quasi-etale if and only if the first Chern class of the sheaf f_*(O_Y) is trivial (details in Theorem 5.3). We also try to get a geometric Galoisness criterion for an arbitrary degree-n finite morphism; this is successfully done when n = 3 and less satifactorily done when n = 5.Comment: Advances in Mathematics, to appear (no changes, just add this info

    Essential skeletons of pairs and the geometric P=W conjecture

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    We construct weight functions on the Berkovich analytification of a variety over a trivially-valued field of characteristic zero, and this leads to the definition of the Kontsevich-Soibelman skeletons and the essential skeletons of pairs. We prove that the weight functions determine a metric on the pluricanonical bundles which coincides with Temkin's canonical metric in the smooth case. The weight functions are defined in terms of log discrepancies, which makes the Kontsevich-Soibelman and essential skeletons computable: this allows us to relate the essential skeleton to its discretely-valued counterpart, and explicitly describe the closure of the Kontsevich-Soibelman skeletons. As a result, we employ these techniques to compute the dual boundary complexes of certain character varieties: this provides the first evidence for the geometric P=W conjecture in the compact case, and the first application of Berkovich geometry in non-abelian Hodge theory.Comment: Sections 1.6-1.7 rewritten and minor changes in Sections 6-
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