Complex Geometry and Lie Theory

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2 Complex Geometry and Lie Theory

3 Proceedings of Symposia in PURE MATHEMATICS Volume 53 Complex Geometry and Lie Theory James A. Carlson C. Herbert Clemens David R. Morrison Editors sffl^^fw American Mathematical Society Sll ^ TT TT TT TT 3 Ijm * \.ilmm. Jn Providence, Rhode Island

4 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 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 ; v. 53) Proceedings of a symposium held at Sundance, Utah, May 26-30, ISBN Geometry, Differential Congresses. 2. Functions of complex variables Congresses. 3. Lie algebras Congresses. I. Carlson, James A., II. Clemens, C. Herbert (Charles Herbert), III. Morrison, David R., IV. Series. QA641.C '6 dc20 CIP COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright 1991 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was typeset using ^A^S-TgX, the American Mathematical Society's TgX macro system

5 Contents Preface vii Abelian Varieties, Infinite-Dimensional Lie Algebras, and the Heat Equation ENRICO ARBARELLO AND CORRADO DE 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

6 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.

7 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 inwardlooking. 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 higherdimensional 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 vii

8 viii 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

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