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Hamiltonian stability and subanalytic geometry

Laurent Niederman (2006)

Annales de l’institut Fourier

In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Using theorems of real subanalytic geometry, we derive a geometric criterion for steepness: a numerical function h which is real analytic around a...

Harmonic metrics and connections with irregular singularities

Claude Sabbah (1999)

Annales de l'institut Fourier

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L 2 complex.

Hermitian (a,b)-modules and Saito's "higher residue pairings"

Piotr P. Karwasz (2013)

Annales Polonici Mathematici

Following the work of Daniel Barlet [Pitman Res. Notes Math. Ser. 366 (1997), 19-59] and Ridha Belgrade [J. Algebra 245 (2001), 193-224], the aim of this article is to study the existence of (a,b)-hermitian forms on regular (a,b)-modules. We show that every regular (a,b)-module E with a non-degenerate bilinear form can be written in a unique way as a direct sum of (a,b)-modules E i that admit either an (a,b)-hermitian or an (a,b)-anti-hermitian form or both; all three cases are possible, and we give...

Hodge numbers attached to a polynomial map

R. García López, A. Némethi (1999)

Annales de l'institut Fourier

We attach a limit mixed Hodge structure to any polynomial map f : n . The equivariant Hodge numbers of this mixed Hodge structure are invariants of f which reflect its asymptotic behaviour. We compute them for a generic class of polynomials in terms of equivariant Hodge numbers attached to isolated hypersurface singularities and equivariant Hodge numbers of cyclic coverings of projective space branched along a hypersurface. We show how these invariants allow to determine topological invariants of f such...

Hodge–type structures as link invariants

Maciej Borodzik, András Némethi (2013)

Annales de l’institut Fourier

Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in a three–sphere. We call them H–numbers. They contain the same amount of information as the (non degenerate part of the) real Seifert matrix. We study their basic properties, and we express the Tristram–Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce...

Holomorphic bijections of algebraic sets

Sławomir Cynk, Kamil Rusek (1997)

Annales Polonici Mathematici

We prove that every holomorphic bijection of a quasi-projective algebraic set onto itself is a biholomorphism. This solves the problem posed in [CR].

Holomorphic extensions of formal objects

Javier Ribón (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We are interested on families of formal power series in ( , 0 ) parameterized by n ( f ^ = m = 0 P m ( x 1 , , x n ) x m ). If every P m is a polynomial whose degree is bounded by a linear function ( d e g P m A m + B for some A > 0 and B 0 ) then the family is either convergent or the series f ^ ( c 1 , , c n , x ) { x } for all ( c 1 , , c n ) n except a pluri-polar set. Generalizations of these results are provided for formal objects associated to germs of diffeomorphism (formal power series, formal meromorphic functions, etc.). We are interested on describing the nature of the set of parameters where...

Holomorphic foliations by curves on 3 with non-isolated singularities

Gilcione Nonato Costa (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Let be a holomorphic foliation by curves on 3 . We treat the case where the set Sing ( ) consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of , the multiplicity of along the curves and the degree and genus of the curves.

Homogeneous polynomials with isomorphic Milnor algebras

Imran Ahmed (2010)

Czechoslovak Mathematical Journal

We recall first Mather's Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. We show that two homogeneous polynomials having isomorphic Milnor algebras are right-equivalent.

Homology for irregular connections

Spencer Bloch, Hélène Esnault (2004)

Journal de Théorie des Nombres de Bordeaux

Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration defines a perfect pairing between de Rham cohomology with values in the connection and homology with values in the dual connection.

Hyperbolic Cauchy problem and Leray's residue formula

Susumu Tanabé (2000)

Annales Polonici Mathematici

We give an algebraic description of (wave) fronts that appear in strictly hyperbolic Cauchy problems. A concrete form of a defining function of the wave front issued from the initial algebraic variety is obtained with the aid of Gauss-Manin systems satisfied by Leray's residues.

Hyperplane arrangements and Milnor fibrations

Alexander I. Suciu (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.

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