37 research outputs found
Notions of Stein spaces in non-archimedean geometry
Let be a non-archimedean complete valued field and be a -analytic
space in the sense of Berkovich. In this note, we prove the equivalence between
three properties: 1) for every complete valued extension of , every
coherent sheaf on is acyclic; 2) is Stein in the sense of
complex geometry (holomorphically separated, holomorphically convex) and higher
cohomology groups of the structure sheaf vanish (this latter hypothesis is
crucial if, for instance, is compact); 3) admits a suitable exhaustion
by compact analytic domains considered by Liu in his counter-example to the
cohomological criterion for affinoidicity.
When has no boundary the characterization is simpler: in~2) the vanishing
of higher cohomology groups of the structure sheaf is no longer needed, so that
we recover the usual notion of Stein space in complex geometry; in 3) the
domains considered by Liu can be replaced by affinoid domains, which leads us
back to Kiehl's definition of Stein space.
v2: major revision to handle also the case of spaces with boundaryComment: 31 page
Definable sets of Berkovich curves
In this article, we functorially associate definable sets to -analytic
curves, and definable maps to analytic morphisms between them, for a large
class of -analytic curves. Given a -analytic curve , our association
allows us to have definable versions of several usual notions of Berkovich
analytic geometry such as the branch emanating from a point and the residue
curve at a point of type 2. We also characterize the definable subsets of the
definable counterpart of and show that they satisfy a bijective relation
with the radial subsets of . As an application, we recover (and slightly
extend) results of Temkin concerning the radiality of the set of points with a
given prescribed multiplicity with respect to a morphism of -analytic
curves.
In the case of the analytification of an algebraic curve, our construction
can also be seen as an explicit version of Hrushovski and Loeser's theorem on
iso-definability of curves. However, our approach can also be applied to
strictly -affinoid curves and arbitrary morphisms between them, which are
currently not in the scope of their setting.Comment: 53 pages, 1 figure. v2: Section 7.2 on weakly stable fields added and
other minor changes. Final version. To appear in Journal of the Institute of
Mathematics of Jussie
Pushforwards of -adic differential equations
Given a differential equation on a smooth -adic analytic curve, one may
construct a new one by pushing forward by an \'etale morphism. The main result
of the paper provides an explicit formula that relates the radii of convergence
of the solutions of the two differential equations using invariants coming from
the topological behavior of the morphism. We recover as particular cases the
known formulas for Frobenius morphisms and tame morphisms.
As an application, we show that the radii of convergence of the pushforward
of the trivial differential equation at a point coincide with the upper
ramification jumps of the extension of the residue field of the point given by
the morphism. We also derive a general formula computing the Laplacian of the
height of the Newton polygon of a -adic differential equation.Comment: 30 pages; Final version, accepted in American Journal of Mathematic
On the number of connected components of the ramification locus of a morphism of Berkovich curves
Let be a complete, nontrivially valued non-archimedean field. Given a
finite morphism of quasi-smooth -analytic curves that admit finite
triangulations, we provide upper bounds for the number of connected components
of the ramification locus in terms of topological invariants of the source
curve such as its topological genus, the number of points in the boundary and
the number of open ends.Comment: 20 pages, 3 figures; Final version, Accepted in Mathematische Annale