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A geometric application of Nori’s connectivity theorem

Claire Voisin (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general K -trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens’ conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang’s conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict.

de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities

Matthew Satriano (2012)

Annales de l’institut Fourier

We prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic p (as defined by Abramovich, Olsson, and Vistoli) which lift mod p 2 degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are étale locally the quotient of a smooth scheme by a finite linearly reductive group scheme.

Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich

Raphaël Krikorian (2003/2004)

Séminaire Bourbaki

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension 2 , il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à 2 , et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

Finiteness results for Teichmüller curves

Martin Möller (2008)

Annales de l’institut Fourier

We show that for each genus there are only finitely many algebraically primitive Teichmüller curves C , such that (i) C lies in the hyperelliptic locus and (ii) C is generated by an abelian differential with two zeros of order g - 1 . We prove moreover that for these Teichmüller curves the trace field of the affine group is not only totally real but cyclotomic.

Hodge metrics and the curvature of higher direct images

Christophe Mourougane, Shigeharu Takayama (2008)

Annales scientifiques de l'École Normale Supérieure

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct image bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.

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 : ℂ n → ℂ . 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-gaussian maps

Elisabetta Colombo, Gian Pietro Pirola, Alfonso Tortora (2001)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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...

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