88 research outputs found
An algorithm for computing the integral closure
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
Let be a non-archimedean local field, a smooth and proper -scheme,
and fix a pluricanonical form on . For every finite extension of ,
the pluricanonical form induces a measure on the -analytic manifold
. We prove that, when runs through all finite tame extensions of
, suitable normalizations of the pushforwards of these measures to the
Berkovich analytification of 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 . We also prove a similar result for all
finite extensions under the assumption that 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
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
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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