Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 087, 40 pages      arXiv:0909.2201
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

James Carlson a, Mark Green b and Phillip Griffiths c
a) Clay Mathematics Institute, United States
b) University of California, Los Angeles, CA, United States
c) The Institute for Advanced Study, Princeton, NJ, United States

Received April 20, 2009, in final form August 31, 2009; Published online September 11, 2009

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds.

Key words: exterior differential systems; variation of Hodge structure, Noether-Lefschetz locus; period domain; integral manifold; Hodge conjecture; Pfaffian system; Chern classes; characteristic cohomology; Cartan-Kähler theorem.

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