Complex Geometry and Lie Theory
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In the late 1960s and early 1970s, Phillip Griffiths and his collaborators undertook a study of period mappings and variation of Hodge structure. The motivating problems, which centered on the understanding of algebraic varieties and the algebraic cycles on them, came from algebraic geometry. However, the techiques used were transcendental in nature, drawing heavily on both Lie theory and hermitian differential geometry. Promising approaches were formulated to fundamental questions in the theory of algebraic curves, moduli theory, and the deep interaction between Hodge theory and algebraic cyl...
In the late 1960s and early 1970s, Phillip Griffiths and his collaborators undertook a study of period mappings and variation of Hodge structure. The motivating problems, which centered on the understanding of algebraic varieties and the algebraic cycles on them, came from algebraic geometry. However, the techiques used were transcendental in nature, drawing heavily on both Lie theory and hermitian differential geometry. Promising approaches were formulated to fundamental questions in the theory of algebraic curves, moduli theory, and the deep interaction between Hodge theory and algebraic cyles. Rapid progress on many fronts was made in the 1970s and 1980s, including the discovery of important connections to other fields, including Nevanlinna theory, integrable systems, rational homotopy theory, harmonic mappings, intersection cohomology, and superstring theory. This volume contains thirteen papers presented during the Symposium on Complex Geometry and Lie Theory held in Sundance, Utah in May 1989. The symposium was designed to review twenty years of interaction between these two fields, concentrating on their links with Hodge theory. The organizers felt that the time was right to examine once again the large issues of understanding the moduli and cycle theory of higher-dimensional varieties, which was the starting point of these developments. The breadth of this collection of papers indicates the continuing growth and vitality of this area of research. Several survey papers are included, which should make the book a valuable resource for graduate students and other researchers who wish to learn about the field. With contributions from some of the field's top researchers, this volume testifies to the breadth and vitality of this area of research.
Table of contents:
Enrico Arbarello and Corrado De Concini, Abelian varieties, infinite-dimensional Lie algebras, and the heat equation; Robert L Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory; Herbert Clemens, The quartic double solid revisited; Robert Friedman, On threefolds with trivial canonical bundle; H Garland and G J Zuckerman, Representations of Heisenberg systems and vertex operators; Phillip A Griffiths, Some aspects of exterior differential systems; Richard M Hain, Algebraic cycles and extensions of variations of mixed Hodge structure; E Looijenga and M Rapoport, Weights in the local cohomology of Baily-Borel compactification; C A M Peters, Curvature for period domains; Masa-Hiko Saito and Steven Zucker, On the Tortelli problem for fiber spaces; Morihiko Saito, Hodge conjecture and mixed motives. I; Rob Schrauwen, Joseph Steenbrink, and Jan Stevens, Spectral pairs and the topology of curve singularities; Carlos T Simpson, The ubiquity of variations of Hodge structure.
Table of contents:
Enrico Arbarello and Corrado De Concini, Abelian varieties, infinite-dimensional Lie algebras, and the heat equation; Robert L Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory; Herbert Clemens, The quartic double solid revisited; Robert Friedman, On threefolds with trivial canonical bundle; H Garland and G J Zuckerman, Representations of Heisenberg systems and vertex operators; Phillip A Griffiths, Some aspects of exterior differential systems; Richard M Hain, Algebraic cycles and extensions of variations of mixed Hodge structure; E Looijenga and M Rapoport, Weights in the local cohomology of Baily-Borel compactification; C A M Peters, Curvature for period domains; Masa-Hiko Saito and Steven Zucker, On the Tortelli problem for fiber spaces; Morihiko Saito, Hodge conjecture and mixed motives. I; Rob Schrauwen, Joseph Steenbrink, and Jan Stevens, Spectral pairs and the topology of curve singularities; Carlos T Simpson, The ubiquity of variations of Hodge structure.