On canonical fibrations of algebraic surfaces.
On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3-Surfaces.
-adic cycles
Periodic integrals and tautological systems
We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.
Periods of Mixed Cusp Forms.
Picard-Lefschetz theory and characters of a semisimple Lie group.
Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra.
Polarized Mixed Hodge Structures and the Local Monodromy of a Variation of Hodge Structure.
Problèmes de modules pour les formes différentielles singulières dans le plan complexe.
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...
Quartic del Pezzo surfaces over function fields of curves
We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.
Rational connectivity and sections of families over curves
Real Algebraic Surfaces with Rational or Elliptic Fiberings.
Simple exponential estimate for the number of real zeros of complete abelian integrals
We show that for a generic polynomial and an arbitrary differential 1-form with polynomial coefficients of degree , the number of ovals of the foliation , which yield the zero value of the complete Abelian integral , grows at most as as , where depends only on . The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let , , be a fundamental system of real solutions...
Singularities of nets of quadrics
Some consequences of perversity of vanishing cycles
For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.
Spécialisation du foncteur de Picard
Structure de Hodge mixte sur la cohomologie évanescente
Soit un morphisme propre d’un -schéma intègre dans un germe de courbe algébrique lisse sur . On construit une structure de Hodge mixte sur les cohomologies évanescentes en résolvant les complexes évanescents et par des complexes de Hodge mixtes cohomologiques. Ceci donne une majoration du niveau d’unipotence de l’action de la monodromie.
Sur la cohomologie des faisceaux -adiques entiers sur les corps locaux
On étudie le comportement des faisceaux -adiques entiers sur les schémas de type fini sur un corps local par les six opérations et le foncteur des cycles proches.
Sur la topologie des surfaces affines réglées