La conjecture de Milnor
Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type
We classify the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = −2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that...
Landau-Ginzburg models in real mirror symmetry
In recent years, mirror symmetry for open strings has exhibited some new connections between symplectic and enumerative geometry (A-model) and complex algebraic geometry (B-model) that in a sense lie between classical and homological mirror symmetry. I review the rôle played in this story by matrix factorizations and the Calabi-Yau/Landau-Ginzburg correspondence.
L'arithmétique du groupe de Chow des zéro-cycles
Lawson homology for quasiprojective varieties
L'équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques
Local-global principle for certain biquadratic normic bundles
Let X be a proper smooth variety having an affine open subset defined by the normic equation over a number field k. We prove that: (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of X; (2) the analogue for rational points is also valid assuming Schinzel’s hypothesis.