Infinitesimal Methods in Hodge Theory

Abstract

LECTURE 1: Kahler manifolds, the Hodge Theorem, Lefschetz decomposition, Hodge index theorem, degeneration of the Hodge-De Rham spectral sequence, Hodge structures.

LECTURE 2: Poincaré dual class of a homology cycle, cycle class of an algebraic subvariety, Griffiths intermediate Jacobian, Abel-Jacobi map, logarithmic differential forms, introduction to Deligne cohomology, infinitesimal Abel-Jacobi map.

LECTURE 3: Kodaira-Spencer class, the period map and its derivative, infinitesimal period relation, Yukawa coupling, second fundamental form of the period map for curves.

LECTURE 4: Poincaré residue representation of the Hodge groups of a hypersurface, pseudo-Jacobi ideal, derivative of the period map for hypersurfaces, generalized Macaulay’s theorem, infinitesimal Torelli for projective hypersurfaces and hypersurfaces of high degree, Hodge class of a complete intersection curve.

LECTURE 5: Mixed Hodge structure of a quasi-projective variety, Gysin sequence and Lefschetz duality, examples of extension classes using mixed Hodge structures.

LECTURE 6: Normal functions, normal function associated to a family of cycles, infinitesimal condition for normal functions, Griffiths infinitesimal invariant, normal function associated to a Deligne class.

LECTURE 7: Macaulay’s lower bound on the growth of ideals, Gotzmann persistence theorem, Koszul vanishing theorem, explicit Noether-Lefschetz theorem, Donagi symmetrizer lemma, generic Torelli theorem for projective hypersurfaces, image of the Abel-Jacobi map for a general 3-fold of degree $\ge 6$, Nori’s vanishing lemma.

LECTURE 8: Nori connectedness theorem and its consequences.