187,833 research outputs found
Some Definability Results in Abstract Kummer Theory
Let be a semiabelian variety over an algebraically closed field, and let
be an irreducible subvariety not contained in a coset of a proper algebraic
subgroup of . We show that the number of irreducible components of
is bounded uniformly in , and moreover that the bound is
uniform in families .
We prove this by purely Galois-theoretic methods. This proof applies in the
more general context of divisible abelian groups of finite Morley rank. In this
latter context, we deduce a definability result under the assumption of the
Definable Multiplicity Property (DMP). We give sufficient conditions for finite
Morley rank groups to have the DMP, and hence give examples where our
definability result holds.Comment: 21 pages; minor notational fixe
Inner geometry of complex surfaces: a valuative approach
Given a complex analytic germ in , the standard
Hermitian metric of induces a natural arc-length metric on , called the inner metric. We study the inner metric structure of the germ
of an isolated complex surface singularity by means of an infinite
family of numerical analytic invariants, called inner rates. Our main result is
a formula for the Laplacian of the inner rate function on a space of
valuations, the non-archimedean link of . We deduce in particular that
the global data consisting of the topology of , together with the
configuration of a generic hyperplane section and of the polar curve of a
generic plane projection of , completely determine all the inner rates
on , and hence the local metric structure of the germ. Several other
applications of our formula are discussed in the paper.Comment: Proposition 5.3 strengthened, exposition improved, some typos
corrected, references updated. 42 pages and 10 figures. To appear in Geometry
& Topolog
The weakness of the pigeonhole principle under hyperarithmetical reductions
The infinite pigeonhole principle for 2-partitions ()
asserts the existence, for every set , of an infinite subset of or of
its complement. In this paper, we study the infinite pigeonhole principle from
a computability-theoretic viewpoint. We prove in particular that
admits strong cone avoidance for arithmetical and
hyperarithmetical reductions. We also prove the existence, for every
set, of an infinite low subset of it or its complement. This
answers a question of Wang. For this, we design a new notion of forcing which
generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page
Zariski Geometries
We characterize the Zariski topologies over an algebraically closed field in
terms of general dimension-theoretic properties. Some applications are given to
complex manifold and to strongly minimal sets.Comment: 9 page
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