Lecture Notes in Mathematics

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1 Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 1246 Hodge Theory Proceedings of the U.S.-Spain Workshop held in Sant Cugat (Barcelona), Spain June 24-30, 1985 Edited by E. Cattani, F. Guillen, A. Kaplan and F. Puerta Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

2 Editors Eduardo Cattani Aroldo Kaplan Dep.artmentof Mathematics, University of Massachusetts Amherst, Massachusetts 01003, USA Francisco Guillen Fernando Puerta Departament de Maternatiques ETSEIB Universitat Politecnica de Catalunya Avenida Diagonal 647, Barcelona, Spain Mathematics Subject Classification (1980): Primary: 14C30, 32J25 Secondary: 14B05, 14B15, 14005, 14F40, 14J 15, 32C38, 32G 13, 32G20 ISBN Springer-Verlag Berlin Heidelberg New York ISBN Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in dats banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyriqht Law. Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/

3 Introduction This volume contains most of the papers presented at the 1985 U.S.-Spain workshop on Hodge Theory, which took place in June of that year in St. Cugat del Valles (Barcelona), under the auspices of the Joint Committee for Scientific and Technological Cooperation between the two countries. The methods of harmonic theory introduced by Riemann in the study of algebraic curves and extended by Hodge to higher dimensions, yield important consequences for the structure of smooth projective varieties; prototypical examples are the decomposition of the complex cohomology groups of such varieties according to Hodge type, and the Lefschetz decomposition in terms of primitive cohomology classes. Since Hodge, these ideas have undergone several generalizations of great interest in algebraic geometry. This volume focuses on those concerning singular and open varieties, families, and the rational homotopy of varieties, which have been developed during the past 20 years. The structure present in the local system of cohomology defined by a family of smooth varieties was encoded by Griffiths in the notion of a variation of (polarized) Hodge s t ruct.ur-e l. Some of the related questions explored in these proceedings include: a geometric realization of certain maximal variations of Hodge structure; an asymptotic description of a variation degenerating at a divisor with normal crossings; an explicit construction, for some particular cases, of the equivalence 1 cf. the survey papers: P. Griffiths: "Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems", Bull. Amer. Math. Soc. 2Q, (1970). P. Griffiths and W. Schmid: "Recent developments in Hodge Theory: a discussion of techniques and results", Proc. of the International Colloquium on Discrete Subgroups of Lie Groups and Applications to Moduli, Bombay 1973, Oxford University Press, 1975,

4 IV between unipotent variations of mixed Hodge structure and mixed Hodge theoretic representations of the fundamental group, and a study of the higher Albanese manifolds that play a role in the classification of unipotent variatiorts of mixed Hodge structure. In "Theorie de Hodge, I, II and 111,,2, Pierre Deligne introduced the notion of a mixed Hodge structure and proved the existence and functoriality of such a structure in the (ordinary) cohomology groups of singular or open varieties over C. Two very fruitful approaches to the study of mixed Hodge theory are presented in this volume: the use of cubical hyperresolutions and the notion of iterated integrals. They are applied to the construction of mixed Hodge structures on the rational homotopy of algebraic varieties, on the cohomology of a link, on the local cohomology of an analytic space in the neighborhood of a compact algebraic subvariety, and on the vanishing cycles. An.Q.-adic description of the weight filtration on the cohomology of a link is also given, as well as a result showing that mixed Hodge complexes are preserved by truncations. A third subject discussed here is that of the L2- realization of Intersection cohomology and its implication: the existence of pure Hodge structures on these groups. The cases of singular varieties with constant coefficients and of degenerating coefficients on a smooth base are both treated extensively. Finally, connections with arithmetical questions and an update on certain aspects of the Hodge conjecture are included as well. The limitations inherent in the format of the workshop made it impossible to include other important aspects of modern Hodge theory. Among these we should mention the applications to the study of singularities, algebraic cycles, the theory of infinitesimal variations of Hodge structure, and Torelli-type 2 Theorie de Hodge: I, Mathematiciens, Nice 1970; 5-57 ; (1974), Actes du Congres International des II, III, PubL, Math. IHES, (1972),

5 v problems. Some of these questions are extensively discussed in "Topics in Transcendental Algebraic Geometry", P. Griffiths, ed., Annals of Math. Studies, Princeton U. Press (1984) We also refer to this volume as an introduction to the basic concepts and methods. We are very grateful to all the participants who made this a succesful conference and, very specially, the contributors to this volume. Thanks are also due to Dr. Harold Stolberg of the National Science Foundation for his support. October E. Cattani F. Guillen A. Kaplan F. Puerta

6 Table of Contents J. A. Carlson and C. Simpson: Shimura varieties of weight two Hodge structures. E. Cattani, A. Kaplan and W. Schmid: Variations of polarized Hodge structure: asymptotics and monodromy 16 E. Cattani, A. Kaplan and W. Schmid: Some remarks on L 2 and Intersection cohomologi es. 3 2 A. Durfee: The cohomology of links. 42 F. Guillen and F. Puerta: Hodge-Deligne. Hyperresolutions cubigues et applications ala theorie de 49 R. Hain: Iterated integrals and mixed Hodge structures on homotopy groups. 75 R. Hain: Higher Albanese manifolds. 84 R. Hain and S. Zucker: A guide to unipotent variations of mixed Hodge structure. 92 R. Hain and S. Zucker: Truncations of mixed Hodge complexes. 107 M. Kashiwara: Poincare lemma for a variation of polarized Hodge structure. 115 F. Loeser: Evaluation d'integrales et theorie de Hodge. 125 V. Navarro Aznar: cents. Sur les structures de Hodge mixtes associees aux cycles evanes- 143 V. Pati: L 2-cohOmology of algebraic varieties in the Fubini metric. J. H. M. Steenbrink: Some remarks about the Hodge conjecture