3,618 research outputs found
Some model theory of fibrations and algebraic reductions
Let p=tp(a/A) be a stationary type in an arbitrary finite rank stable theory,
and P an A-invariant family of partial types. The following property is
introduced and characterised: whenever c is definable over (A,a) and a is not
algebraic over (A,c) then \tp(c/A) is almost internal to P. The
characterisation involves among other things an apparently new notion of
``descent" for stationary types. Motivation comes partly from results in
Section~2 of [Campana, Oguiso, and Peternell. Non-algebraic hyperk\"ahler
manifolds. Journal of Differential Geometry, 85(3):397--424, 2010] where
structural properties of generalised hyperk\"ahler manifolds are given. The
model-theoretic results obtained here are applied back to the complex analytic
setting to prove that the algebraic reduction of a nonalgebraic (generalised)
hyperk\"ahler manifold does not descend. The results are also applied to the
theory of differentially closed fields, where examples coming from differential
algebraic groups are given.Comment: Substantially revised and augmented. A new section applying the
results to differentially closed fields has been added; title, abstract, and
introduction are new, and several new examples are added. 14 page
F-sets and finite automata
The classical notion of a k-automatic subset of the natural numbers is here
extended to that of an F-automatic subset of an arbitrary finitely generated
abelian group equipped with an arbitrary endomorphism F. This is
applied to the isotrivial positive characteristic Mordell-Lang context where F
is the Frobenius action on a commutative algebraic group G over a finite field,
and is a finitely generated F-invariant subgroup of G. It is shown
that the F-subsets of introduced by the second author and Scanlon are
F-automatic. It follows that when G is semiabelian and X is a closed subvariety
then X intersect is F-automatic. Derksen's notion of a k-normal subset
of the natural numbers is also here extended to the above abstract setting, and
it is shown that F-subsets are F-normal. In particular, the X intersect
appearing in the Mordell-Lang problem are F-normal. This generalises
Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.Comment: The final section is revised following an error discovered by
Christopher Hawthorne; it is no longer claimed that an F-normal subset has a
finite symmetric difference with an F-subset. The main theorems of the paper
remain unchange
- …