Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures.
Mirror symmetry in dimension 3
Mixed Hodge structures on log deformations
Monodromy and weight filtration for smoothings of isolated singularities
Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part
Naive boundary strata and nilpotent orbits
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 , , and .
Non-commutative Hodge structures
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 Grothendieck's generalized Hodge conjecture for a family of threefolds with trivial canonical bundle.
On Hodge structures and non-representability of Chow groups
On the de Rham-Witt complex attached to a semi-stable family
On the genus of reducible surfaces and degenerations of surfaces
We deal with a reducible projective surface with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the -genus of , i.e. the dimension of the vector space of global sections of the dualizing sheaf . Then we prove that, when is smoothable, i.e. when is the central fibre of a flat family parametrized by a disc, with smooth general fibre, then the -genus of the fibres of is constant.
On the Hodge spectral sequence for some classes of non-complete algebraic manifolds.
On the period map for abelian covers of projective varieties
On the variational Torelli problem for complete intersections
Periods of integrals on algebraic manifolds, III (Some global differential-geometric properties of the period mapping)
Positive sheaves of differentials coming from coarse moduli spaces
Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base , and suppose the family is non-isotrivial. If is a smooth compactification of , such that is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along . Viehweg and Zuo have shown that for some , the symmetric power of this sheaf admits many sections. More precisely, the symmetric power contains an invertible...
Quantum Cohomology and Periods
In a previous paper, the author introduced an integral structure in quantum cohomology defined by the -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
Dans cette note nous établissons le résultat suivant, annoncé dans [CCE13] : si est l’image d’une représentation linéaire d’un groupe kählérien , 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 .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...
Rigidity for variations of Hodge structure and Arakelov-type finiteness theorems
Semiregularity, obstructions and deformations of Hodge classes