Abstract:
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.
Keywords: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.
Received:April 20, 2009; in final form August 31, 2009; Published online September 11, 2009
Citation:
Mark Green, James Carlson, Phillip Griffiths, “Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results”, SIGMA, 5 (2009), 087, 40 pp.
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\by Mark Green, James Carlson, Phillip Griffiths
\paper Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results
\jour SIGMA
\yr 2009
\vol 5
\papernumber 087
\totalpages 40
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Brevik J., Nollet S., “Developments in Noether-Lefschetz Theory”, Hodge Theory, Complex Geometry, and Representation Theory, Contemporary Mathematics, 608, eds. Doran R., Friedman G., Nollet S., Amer Mathematical Soc, 2014, 21–50
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