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Asymptotic behaviour of numerical invariants of algebraic varieties

F. L. Zak (2012)

Journal of the European Mathematical Society

We show that if the degree of a nonsingular projective variety is high enough, maximization of any of the most important numerical invariants, such as class, Betti number, and any of the Chern or middle Hodge numbers, leads to the same class of extremal varieties. Moreover, asymptotically (say, for varieties whose total Betti number is big enough) the ratio of any two of these invariants tends to a well-defined constant.

Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Klaus Hulek, Remke Kloosterman (2011)

Annales de l’institut Fourier

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4 -space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Du Bois invariants of isolated complete intersection singularities

Joseph H. M. Steenbrink (1997)

Annales de l'institut Fourier

We define Du Bois invariants for isolated singularities of complex spaces. We relate them to the Hodge numbers of the local and vanishing cohomology groups. Our main results express the Tjurina number of certain Gorenstein singularities in terms of Du Bois invariants and Hodge numbers of the link, and express the Hodge numbers of the Milnor fibre of certain three-dimensional complete intersections in similar terms. We also address the question of the semicontinuity of the Du Bois invariants under...

Families of differential forms on complex spaces

Vincenzo Ancona, Bernard Gaveau (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

On every reduced complex space X we construct a family of complexes of soft sheaves Λ X ; each of them is a resolution of the constant sheaf X and induces the ordinary De Rham complex of differential forms on a dense open analytic subset of X . The construction is functorial (in a suitable sense). Moreover each of the above complexes can fully describe the mixed Hodge structure of Deligne on a compact algebraic variety.

Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré

Michel Raibaut (2012)

Annales de l’institut Fourier

Soit k un corps de caractéristique nulle, P un polynôme de Laurent en d variables, à coefficients dans k et non dégénéré pour son polyèdre de Newton à l’infini. Soit d fonctions non constantes f l à variables séparées et définies sur des variétés lisses. A la manière de Guibert, Loeser et Merle, dans le cas local, nous calculons dans cet article, la fibre de Milnor motivique à l’infini de la composée P ( f ) en termes du polyèdre de Newton à l’infini de P . Pour P égal à la somme x 1 + x 2 nous obtenons une formule...

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...

Mixed Hodge structure of affine hypersurfaces

Hossein Movasati (2007)

Annales de l’institut Fourier

In this article we give an algorithm which produces a basis of the n -th de Rham cohomology of the affine smooth hypersurface f - 1 ( t ) compatible with the mixed Hodge structure, where f is a polynomial in n + 1 variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of f is given in terms of the vanishing of integrals of certain polynomial n -forms in n + 1 over topological n -cycles on...

Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs

Victor V. Batyrev (1999)

Journal of the European Mathematical Society

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log-terminal pairs. There is a natural Kawamata log-terminal pair corresponding to an algebraic variety V having a regular action of a finite group G . In this situation we show that the stringy Euler number of this pair coincides with the physicists’ orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture...

Quelques conséquences locales de la théorie de Hodge

François Loeser (1985)

Annales de l'institut Fourier

Un résultat de positivité de théorie de Hodge nous permet de déterminer certaines pôles de la distribution | f | 2 z pour f une fonction analytique à singularité isolée. Dans le cas des courbes et des singularités quasi-homogènes on détermine l’ensemble exact des pôles. On démontre aussi que si le résidu d’une forme holomorphe est de carré intégrable sur la fibre spéciale, l’intégrale sur la fibre spéciale est limite de celle sur les fibres voisines.

Singularités à l’infini et intégration motivique

Michel Raibaut (2012)

Bulletin de la Société Mathématique de France

Soit k un corps de caractéristique nulle et f une fonction non constante définie sur une variété lisse. Nous définissons dans cet article unefibre de Milnor motivique à l’infiniqui appartient à un anneau de Grothendieck des variétés. Elle est définie en termes d’une compactification choisie, non nécessairement lisse, mais est indépendante de ce choix. Lorsque k est le corps des nombres complexes, en utilisant le morphisme de réalisation de Hodge, elle se réalise en le spectre à l’infini de f . Nous...

Spectrum and multiplier ideals of arbitrary subvarieties

Alexandru Dimca, Philippe Maisonobe, Morihiko Saito (2011)

Annales de l’institut Fourier

We introduce a spectrum for arbitrary subvarieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by using the filtration V of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The key point...

Weights in cohomology and the Eilenberg-Moore spectral sequence

Matthias Franz, Andrzej Weber (2005)

Annales de l’institut Fourier

We show that in the category of complex algebraic varieties, the Eilenberg–Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all spaces involved have pure cohomology. As application, we compute the rational cohomology of an algebraic G -variety X ( G being a connected algebraic group) in terms of its equivariant cohomology provided that H G * ( X ) is pure. This is the case, for example, if X is smooth and has only finitely many orbits. We work in the category...

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