433 research outputs found
Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang
We prove that a real analytic subset of a torus group that is contained in
its image under an expanding endomorphism is a finite union of translates of
closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and
Wang for real analytic varieties. Our proof uses real analytic geometry,
topological dynamics and Fourier analysis.Comment: 25 page
Connectivity and a Problem of Formal Geometry
Let be a product of weighted
projective spaces, and let be the diagonal of . We prove
an algebraization result for formal-rational functions on certain closed
subvarieties of along the intersection .Comment: 9 pages, to appear in the Proceedings volume "Experimental and
Theoretical Methods in Algebra, Geometry and Topology", series Springer
Proceedings in Mathematics & Statistic
Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures
We study the basic properties of Higgs sheaves over compact K\"ahler
manifolds and we establish some results concerning the notion of semistability;
in particular, we show that any extension of semistable Higgs sheaves with
equal slopes is semistable. Then, we use the flattening theorem to construct a
regularization of any torsion-free Higgs sheaf and we show that it is in fact a
Higgs bundle. Using this, we prove that any Hermitian metric on a
regularization of a torsion-free Higgs sheaf induces an admissible structure on
the Higgs sheaf. Finally, using admissible structures we proved some properties
of semistable Higgs sheaves.Comment: 18 pages; some typos correcte
Bergman kernel and complex singularity exponent
We give a precise estimate of the Bergman kernel for the model domain defined
by where
is a holomorphic map from to ,
in terms of the complex singularity exponent of .Comment: to appear in Science in China, a special issue dedicated to Professor
Zhong Tongde's 80th birthda
Exact Lagrangian submanifolds in simply-connected cotangent bundles
We consider exact Lagrangian submanifolds in cotangent bundles. Under certain
additional restrictions (triviality of the fundamental group of the cotangent
bundle, and of the Maslov class and second Stiefel-Whitney class of the
Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically
indistinguishable from the zero-section. This implies strong restrictions on
their topology. An essentially equivalent result was recently proved
independently by Nadler, using a different approach.Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the
spectral sequence considerably expanded. Other minor change
Analytic curves in algebraic varieties over number fields
We establish algebraicity criteria for formal germs of curves in algebraic
varieties over number fields and apply them to derive a rationality criterion
for formal germs of functions, which extends the classical rationality theorems
of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to
arbitrary algebraic curves over a number field.
The formulation and the proof of these criteria involve some basic notions in
Arakelov geometry, combined with complex and rigid analytic geometry (notably,
potential theory over complex and -adic curves). We also discuss geometric
analogues, pertaining to the algebraic geometry of projective surfaces, of
these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor
of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200
Techniques for the study of singularities with applications to resolution of 2-dimensional schemes
We give an overview of invariants of algebraic singularities over perfect
fields. We then show how they lead to a synthetic proof of embedded resolution
of singularities of 2-dimensional schemes.Comment: 26 pages; minor changes have been adde
Formal properties in small codimension
In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces
Singular open book structures from real mappings
We prove extensions of Milnor's theorem for germs with nonisolated
singularity and use them to find new classes of genuine real analytic mappings
with positive dimensional singular locus \Sing \psi \subset
\psi^{-1}(0), for which the Milnor fibration exists and yields an open book
structure with singular binding.Comment: more remark
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