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Complex Geometry and Lie Theory

http://dx.doi.org/10.1090/pspum/053

Proceedings of Symposia in

PURE MATHEMATICS

Volume 53

Complex Geometry and Lie Theory James A. Carlson C. Herbert Clemens David R. Morrison Editors

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PROCEEDINGS OF A SYMPOSIUM ON COMPLEX GEOMETRY AND LIE THEORY

HELD AT SUNDANCE, UTAH MAY 26-30, 1989

with the support of the National Science Foundation Grant DMS-8815065.

1991 Mathematics Subject Classification. Primary 32-06; Secondary 14-06, 22-06, 58A14.

Library of Congress Cataloging-in-Publication Data Complex geometry and Lie theory / [edited by] James A. Carlson, C. Herbert Clemens, and David R. Morrison.

p. cm.—(Proceedings of symposia in pure mathematics, ISSN 0082-0717; v. 53) Proceedings of a symposium held at Sundance, Utah, May 26-30, 1989. ISBN 0-8218-1492-3 1. Geometry, Differential—Congresses. 2. Functions of complex variables—Congresses.

3. Lie algebras—Congresses. I. Carlson, James A., 1946- . II. Clemens, C. Herbert (Charles Herbert), 1939- . III. Morrison, David R., 1955- . IV. Series. QA641.C6143 1991 91-25148 516.3'6—dc20 CIP

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Contents

Preface vii

Abelian Varieties, Infinite-Dimensional Lie Algebras, and the Heat Equation ENRICO ARBARELLO AND CORRADO D E CONCINI 1

Two Exotic Holonomies in Dimension Four, Path Geometries, and Twistor Theory ROBERT L. BRYANT 33

The Quartic Double Solid Revisited HERBERT CLEMENS 89

On Threefolds with Trivial Canonical Bundle ROBERT FRIEDMAN 103

Representations of Heisenberg Systems and Vertex Operators H. GARLAND AND G. J. ZUCKERMAN 135

Some Aspects of Exterior Differential Systems PHILLIP A. GRIFFITHS 151

Algebraic Cycles and Extensions of Variations of Mixed Hodge Structure RICHARD M. HAIN 175

Weights in the Local Cohomology of a Baily-Borel Compactihcation E. LOOIJENGA AND M. RAPOPORT 223

Curvature for Period Domains C. A. M. PETERS 261

On the Torelli Problem for Fiber Spaces

MASA-HIKO SAITO AND STEVEN ZUCKER 269

vi CONTENTS

Hodge Conjecture and Mixed Motives. I MORIHIKO SAITO 283

Spectral Pairs and the Topology of Curve Singularities ROB SCHRAUWEN, JOSEPH STEENBRINK, AND JAN STEVENS 305

The Ubiquity of Variations of Hodge Structure CARLOS T. SIMPSON 329

Preface

On May 26-29, 1989, the American Mathematical Society sponsored a symposium on complex geometry and Lie theory at the Sundance Center, Sundance, Utah. The symposium was designed to review twenty years of interaction between these two fields, concentrating on their links with Hodge theory.

This interaction began with the study of period mappings and variation of Hodge structure undertaken by Phillip Griffiths and his coworkers in the late 1960s and early 1970s. The motivating problems—How can the moduli of algebraic varieties and the algebraic cycles on them be understood?—came from algebraic geometry. But the techniques used were transcendental in nature, and drew heavily on both Lie theory and hermitian differential geometry. Promising approaches were formulated to fundamental questions in the theory of algebraic curves, the theory of moduli, and the deep interaction between Hodge theory and algebraic cycles.

Progress in these areas was made rapidly on many fronts during the 1970s and 1980s. Advances were made in the theory of moduli (through the proofs of several Torelli-type theorems), and in the study of algebraic cycles (where many intriguing new phenomena in higher codimension were uncovered). Old problems were settled: the cubic threefold was shown to be nonrational, Petri's conjecture on quadrics through the canonical curve was proved, and solutions were found for the Schottky problem of determining which abelian varieties are Jacobians of curves. Internal issues were resolved: the asymptotic behavior of the period mappings was established, and connections were found with the geometry of singular fibers in a degenerating family. And numerous connections with other fields were found, ranging from Nevanlinna theory to integrable systems to rational homotopy theory to harmonic mappings to intersection cohomology to superstring theory in theoretical physics.

Entire new fields of research have grown up out of a single beginning. There is the danger that these fields may concentrate too much on their own internal dynamics and questions, and become too technical and inward-looking. The symposium organizers felt that it was time to look once again at the larger issue—the understanding of the moduli and cycle-theory of higher-dimensional varieties—which was a common starting point of these developments. It was our hope that this would allow the many researchers who got their start in a common vision to reflect on the broader implications of

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their contributions to the fulfillment of that vision. In this way both they and the next generation of researchers can gain new energy and insight, and capitalize more fully on the gains made by their mathematical cousins.

We are fortunate to be able to publish here contributions from thirteen of the twenty-five symposium lecturers. The breadth of this collection of papers indicates the continuing growth and vitality of this area of research. Most of the contributions are written versions of lectures given at the symposium; one or two represent substantially different work of the author, not reported on at the symposium. The papers published here are in final form, and will not be published elsewhere.

The symposium was supported by the National Science Foundation, Duke University, and the University of Utah. To them, and to the able staff at the American Mathematical Society, we express our deep appreciation.

James A. Carlson C. Herbert Clemens David R. Morrison