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Infinitesimal Methods in Hodge Theory

C.I.M.E. Notes - June 1993
  • Mark L. Green
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1594)

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.

Mathematics Subject Classification (1991):

14C25 14C30 14D07 19E15 32I25 

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Copyright information

© Springer-Verlag Berlin/Heidelberg 1994

Authors and Affiliations

  • Mark L. Green
    • 1
  1. 1.Department of MathematicsU.C.L.ALos AngelesUSA

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