Frobenius Splitting Methods in Geometry and Representation Theory
The theory of Frobenius splittings has made a significant impact in the study of the geometry of flag varieties and representation theory. This work, unique in book literature, systematically develops the theory and covers all its major developments. Key
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Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein
Michel Brion Shrawan Kumar
Frobenius Splitting Methods in Geometry and Representation Theory
Birkh¨auser Boston • Basel • Berlin
Michel Brion Universit´e Grenoble 1 – CNRS Institut Fourier 38402 St.-Martin d’H`eres Cedex France
Shrawan Kumar University of North Carolina Department of Mathematics Chapel Hill, NC 27599 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, 1958Frobenius 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.
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Contents Preface 1
Frobenius Splitting: General Theory 1.1 Basic definitions, properties, and examples . 1.2 Consequences of Frobenius splitting . . . . 1.3 Criteria for splitting . . . . . . . . . . . . . 1.4 Splitting relative to a divisor . . . . . . . . 1.5 Consequences of diagonal splitting . . . . . 1.6 From characteristic p to characteristic 0 . .
vii
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1 2 12 20 35 41 53
2
Frobenius Splitting of Schubert Varieties 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Frobenius splitting of the BSDH varieties Zw . . . . . . . . . . . . . 2.3 Some more splittings of G/B and G/B × G/B . . . . . . . . . . . .
59 60 64 72
3
Cohomology and Geometry of Schubert Varieties 3.1 Cohomology of Schubert varieties . . . . . . . . . . 3.2 Normality of Schubert varieties . . . . . . . . . . . . 3.3 Demazure character formula . . . . . . . . . . . . . 3.4 Schubert varieties have rational resolutions . . . . . 3.5 Homogeneous coordinate rings of Schubert varieties are
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