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Naive boundary strata and nilpotent orbits

Matt Kerr, Gregory Pearlstein (2014)

Annales de l’institut Fourier

We give a Hodge-theoretic parametrization of certain real Lie group orbits in the compact dual of a Mumford-Tate domain, and characterize the orbits which contain a naive limit Hodge filtration. A series of examples are worked out for the groups S U ( 2 , 1 ) , S p 4 , and G 2 .

Non-commutative Hodge structures

Claude Sabbah (2011)

Annales de l’institut Fourier

This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.

On the genus of reducible surfaces and degenerations of surfaces

Alberto Calabri, Ciro Ciliberto, Flaminio Flamini, Rick Miranda (2007)

Annales de l’institut Fourier

We deal with a reducible projective surface X with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the ω -genus p ω ( X ) of X , i.e. the dimension of the vector space of global sections of the dualizing sheaf ω X . Then we prove that, when X is smoothable, i.e. when X is the central fibre of a flat family π : 𝒳 Δ parametrized by a disc, with smooth general fibre, then the ω -genus of the fibres of π is constant.

Positive sheaves of differentials coming from coarse moduli spaces

Kelly Jabbusch, Stefan Kebekus (2011)

Annales de l’institut Fourier

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base Y , and suppose the family is non-isotrivial. If Y is a smooth compactification of Y , such that D : = Y Y is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along D . Viehweg and Zuo have shown that for some m > 0 , the m th symmetric power of this sheaf admits many sections. More precisely, the m th symmetric power contains an invertible...

Quantum Cohomology and Periods

Hiroshi Iritani (2011)

Annales de l’institut Fourier

In a previous paper, the author introduced an integral structure in quantum cohomology defined by the K -theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation of toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of...

Représentations linéaires des groupes kählériens et de leurs analogues projectifs

Fréderic Campana, Benoît Claudon, Philippe Eyssidieux (2014)

Journal de l’École polytechnique — Mathématiques

Dans cette note nous établissons le résultat suivant, annoncé dans [CCE13] : si G GL n ( ) est l’image d’une représentation linéaire d’un groupe kählérien π 1 ( X ) , il admet un sous-groupe d’indice fini qui est l’image d’une représentation linéaire du groupe fondamental d’une variété projective complexe lisse X ' .Il s’agit donc de la solution (à indice fini près) pour les représentations linéaires d’une question usuelle demandant si le groupe fondamental d’une variété kählérienne compacte est aussi celui d’une variété...

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