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Chapter 1 Frobenius Splitting Progress in Mathematics Volume 231 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Michel Brion Shrawan Kumar Frobenius Splitting Methods in Geometry and Representation Theory Birkhauser¨ Boston • Basel • Berlin Michel Brion Shrawan Kumar UniversiteG´ renoble1–CNRS University of North Carolina Institut Fourier Department of Mathematics 38402 St.-Martin d’Heres` Cedex Chapel Hill, NC 27599 France U.S.A. AMS Subject Classifications: 13A35, 13D02, 14C05, 14E15, 14F10, 14F17, 14L30, 14M05, 14M15, 14M17, 14M25, 16S37, 17B10, 17B20, 17B45, 20G05, 20G15 Library of Congress Cataloging-in-Publication Data Brion, Michel, 1958- Frobenius splitting methods in geometry and representation theory / Michel Brion, Shrawan Kumar. p. cm. – (Progress in mathematics ; v. 231) Includes bibliographical references and index. ISBN 0-8176-4191-2 (alk. paper) 1. Algebraic varieties. 2. Frobenius algebras. 3. Representations of groups. I. Kumar, S. (Shrawan), 1953- II. Title. III. Progress in mathematics (Boston, Mass.); v. 231. QA564.B74 2004 516’.3’53–dc22 2004047645 ISBN 0-8176-4191-2 Printed on acid-free paper. c 2005 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in con- nection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/HP) 987654321 SPIN 10768278 Birkhauser¨ is a part of Springer Science+Business Media www.birkhauser.com Contents Preface vii 1 Frobenius Splitting: General Theory 1 1.1 Basic definitions, properties, and examples ............... 2 1.2 Consequences of Frobenius splitting .................. 12 1.3 Criteria for splitting ........................... 20 1.4 Splitting relative to a divisor ...................... 35 1.5 Consequences of diagonal splitting ................... 41 1.6 From characteristic p to characteristic 0 ................ 53 2 Frobenius Splitting of Schubert Varieties 59 2.1 Notation ................................. 60 2.2 Frobenius splitting of the BSDH varieties Zw ............. 64 2.3 Some more splittings of G/B and G/B × G/B ............ 72 3 Cohomology and Geometry of Schubert Varieties 83 3.1 Cohomology of Schubert varieties ................... 85 3.2 Normality of Schubert varieties ..................... 91 3.3 Demazure character formula ...................... 95 3.4 Schubert varieties have rational resolutions .............. 100 3.5 Homogeneous coordinate rings of Schubert varieties are Koszul algebras ........................... 103 4 Canonical Splitting and Good Filtration 109 4.1 Canonical splitting ........................... 111 4.2 Good filtrations ............................. 124 4.3 Proof of the PRVK conjecture and its refinement ............ 142 5 Cotangent Bundles of Flag Varieties 153 5.1 Splitting of cotangent bundles of flag varieties ............. 155 5.2 Cohomology vanishing of cotangent bundles of flag varieties ..... 169 5.3 Geometry of the nilpotent and subregular cones ............ 178 vi Contents 6 Equivariant Embeddings of Reductive Groups 183 6.1 The wonderful compactification .................... 184 6.2 Reductive embeddings ......................... 197 7 Hilbert Schemes of Points on Surfaces 207 7.1 Symmetric products ........................... 208 7.2 Hilbert schemes of points ........................ 216 7.3 The Hilbert–Chow morphism ...................... 220 7.4 Hilbert schemes of points on surfaces ................. 224 7.5 Splitting of Hilbert schemes of points on surfaces ........... 227 Bibliography 231 Index 247 Preface In the 1980s, Mehta and Ramanathan made important breakthroughs in the study of Schubert varieties by introducing the notion of a Frobenius split variety and compatibly split subvarieties for algebraic varieties in positive characteristics. This was refined by Ramanan and Ramanathan via their notion of Frobenius splitting relative to an effective divisor. Even though most of the projective varieties are not Frobenius split, those which are have remarkable geometric and cohomological properties, e.g., all the higher cohomol- ogy groups of ample line bundles are zero. Interestingly, many varieties where a linear algebraic group acts with a dense orbit turn out to be Frobenius split. This includes the flag varieties, which are split compatibly with their Schubert subvarieties, relative to a certain ample divisor; Bott–Samelson–Demazure–Hansen varieties; the product of two flag varieties for the same group G, which are split compatibly with their G-stable closed subvarieties; cotangent bundles of flag varieties; and equivariant embeddings of any connected reductive group, e.g., toric varieties. The Frobenius splitting of the above mentioned varieties yields important geometric results: Schubert varieties have rational singularities, and they are projectively normal and projectively Cohen–Macaulay in the projective embedding given by any ample line bundle (in particular, they are normal and Cohen–Macaulay); the corresponding homogeneous coordinate rings are Koszul algebras; the intersection of any number of Schubert varieties is reduced; the full and subregular nilpotent cones have rational Gorenstein singularities; the equivariant embeddings of reductive groups have rational singularities. Moreover, their proofs are short and elegant. Further remarkable applications of Frobenius splitting concern the representa- tion theory of semisimple groups: the Demazure character formula; a proof of the Parthasarathy–Rango Rao–Varadarajan–Kostant conjecture on the existence of certain components in the tensor product of two dual Weyl modules; the existence of good fil- trations for such tensor products and also for the coordinate rings of semisimple groups in positive characteristics, etc. The technique of Frobenius splitting has proved to be so powerful in tackling nu- merous and varied problems in algebraic transformation groups that it has become an indispensable tool in the field. While much of the research has appeared in journals, nothing comprehensive exists in book form. This book systematically develops the viii Preface theory from scratch. Its various consequences and applications to problems in alge- braic group theory have been treated in full detail bringing the reader to the forefronts of the area. We have included a large number of exercises, many of them covering complementary material. Also included are some open problems. This book is suitable for mathematicians and graduate students interested in geo- metric and representation-theoretic aspects of algebraic groups and their flag varieties. In addition, it is suitable for a slightly advanced graduate course on methods of pos- itive characteristics in geometry and representation theory. Throughout the book, we assume some familiarity with algebraic geometry, specifically, with the contents of the first three chapters of Hartshorne’s book [Har–77]. In addition, in Chapters 2 to 6 we assume familiarity with the structure of semisimple algebraic groups as exposed in the books of Borel [Bor–91] or Springer [Spr–98]. We also rely on some basic results of representation theory of algebraic groups, for which we refer to Jantzen’s book [Jan– 03]. We warn the reader that the text provides much more information than is needed for most applications. Thus, one should not hesitate to skip ahead at will, tracing back as needed. The first-named author owes many thanks to S. Druel, S. Guillermou, S. Inamdar, M. Decauwert, and G. Rémond for very useful discussions and comments on pre- liminary versions of this book. The second-named author expresses his gratitude to A. Ramanathan, V. Mehta, N. Lauritzen and J.F. Thomsen for all they taught him about Frobenius splitting. The second-named author also acknowledges the hospitality of the Newton Institute, Cambridge (England) during January–June, 2001, where part of this book was written. This project was partially supported by NSF. We thank J.F. Thom- sen, W. van der Kallen and two referees for pointing out inaccuracies and suggesting various improvements; and L. Trimble for typing many chapters of the book. We thank Ann Kostant for her personal interest and care in this project and Elizabeth Loew of TEXniques for taking care of the final formatting and layout. Michel Brion and Shrawan Kumar September 2004 Notational Convention Those exercises which are used in the proofs in the text appear with a star. Frobenius Splitting Methods in Geometry and Representation Theory Chapter 1 Frobenius Splitting: General Theory Introduction This chapter is devoted to the general study of Frobenius split schemes, a notion intro- duced by Mehta–Ramanathan and refined further by Ramanan–Ramanathan (see 1.C for more precise references). Any scheme over a field of characteristic p>0 possesses a remarkable endomor- phism, the (absolute) Frobenius morphism F which fixes all the points and raises the functions to their p-th power. A scheme X is called Frobenius split (for short, split),
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