Progress in Mathematics Takushiro Ochiai Toshiki Mabuchi Yoshiaki Maeda Junjiro Noguchi Alan Weinstein Editors In Memory of Professor Shoshichi Kobayashi Progress in Mathematics Volume 308 Series Editors Hyman Bass, University of Michigan, Ann Arbor, USA Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Joseph Oesterlé, Université Pierre et Marie Curie, Paris, France Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA More information about this series at http://www.springer.com/series/4848 Takushiro Ochiai • Toshiki Mabuchi • Yoshiaki Maeda Junjiro Noguchi • Alan Weinstein Editors Geometry and Analysis on Manifolds In Memory of Professor Shoshichi Kobayashi Editors Takushiro Ochiai Toshiki Mabuchi Graduate School of Mathematical Sciences Department of Mathematics The University of Tokyo Graduate School of Sciences Tokyo, Japan Osaka University Osaka, Japan Yoshiaki Maeda Tohoku Forum for Creativity Junjiro Noguchi Tohoku University Graduate School of Mathematical Sciences Yokohama, Japan The University of Tokyo Tokyo, Japan Alan Weinstein Department of Mathematics University of California at Berkeley Berkeley, CA, USA ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-319-11522-1 ISBN 978-3-319-11523-8 (eBook) DOI 10.1007/978-3-319-11523-8 Library of Congress Control Number: 2015932660 Mathematics Subject Classification (2010): 53-06, 32Q45, 53C15, 32Q15, 32Q20, 58C10 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Contents Preface .................................................................. vii Part 1: About Shoshichi Kobayashi T. Ochiai In Memory of Professor Shoshichi Kobayashi ........................ 3 H.-H. Wu Events Surrounding the Birth of the Kobayashi Metric .............. 13 H. Kobayashi Academic Genealogy of Shoshichi Kobayashi and Individuals Who Influenced Him .................................... 17 Part 2: Algebraic Geometry and Complex Analysis J. Merker Algebraic Differential Equations for Entire Holomorphic Curves in Projective Hypersurfaces of General Type: Optimal Lower Degree Bound ....................................... 41 J. Noguchi Kobayashi Hyperbolicity and Lang’s Conjecture ..................... 143 T. Ohsawa A Lemma on Hartogs Function and Application to Levi Flat Hypersurfaces in Hopf Surfaces ............................ 153 H. Tsuji On the Extremal Measure on a Complex Manifold .................. 159 P. Vojta A Lang Exceptional Set for Integral Points .......................... 177 K. Yamanoi Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory ................................................. 209 vi Contents J.-H. Yang Geometry and Arithmetic on the Siegel–Jacobi Space ............... 275 S.-T. Yau On the Pseudonorm Project of Birational Classification of Algebraic Varieties ............................................... 327 Part 3: Differential Geometry A. Futaki The Weighted Laplacians on Real and Complex Metric Measure Spaces ..................................................... 343 K. Hasegawa and Y. Kamishima Locally Conformally Ka¨hler Structures on Homogeneous Spaces ..... 353 N. Hitchin A Note on Vanishing Theorems ..................................... 373 G.R. Jensen Dupin Hypersurfaces in Lie Sphere Geometry ....................... 383 T. Mabuchi The Donaldson–Futaki Invariant for Sequences of Test Configurations ................................................. 395 T. Mabuchi and Y. Nitta Strong K-stability and Asymptotic Chow-stability ................... 405 Y. Maeda and S. Rosenberg Traces and Characteristic Classes in Infinite Dimensions ............ 413 R. Miyaoka Moment Map Description of the Cartan–Mu¨nzner Polynomials of Degree Four ...................................................... 437 U. Hertrich-Jeromin and Y. Suyama Ribaucour Pairs Corresponding to Dual Pairs of Conformally Flat Hypersurfaces ..................................... 449 M.S. Tanaka Geometry of Symmetric R-spaces ................................... 471 Preface Prof. Shoshichi Kobayashi was a recognized international leader in differential geometry and complex geometry. As detailed below, throughout a career lasting over five decades, he contributed crucial new ideas which are still fundamental in these researchareas, particularly within complex geometry. Prof.Kobayashi’searlywork contributed to a wide rangeof topics in differ- ential geometry, for example through his well-known work on the study of trans- formationgroupsandontheFrankelconjecture,whichcharacterizedcomplexpro- jective spaces. In complex geometry, his work on Hermitian-Einstein holomorphic vector bundles led to the so-called Kobayashi–Hitchincorrespondence. Themostwell-knownmathematicalobjectbearinghisnameistheKobayashi pseudometric, which he introduced in 1967. This has become an essential tool for thestudyofmappingsbetweencomplexmanifolds.Heidentifiedacrucialproperty of“good”spaces,whichhe calledhyperbolic, inthis theory,andwhichin recogni- tion of this fundamental insight are now known as Kobayashi hyperbolic spaces. Perhapsmostimportantly,hisbooks,especiallythetwovolume“Foundations of Differential Geometry” with Katsumi Nomizu, have taught differential geome- try and complex geometry to generations of students and other researchers. His writing is known for its succinctness and clarity, and the book commonly known as Kobayashi–Nomizuhas been in publication for over 50 years. Over the course of his long career, Prof. Kobayashi produced a most im- pressive mathematical legacy in the form of 35 Ph.D. students, a long list of con- tributions to differential geometry, and many influential monographs. As part of his interest in the dissemination of mathematics, he contributed immensely to thecommunicationbetweenUSandJapanesemathematicians.Hegreatlyenjoyed mathematical discussions with young researchers, and with great warmth always encouragedthem.Prof.ShoshichiKobayashilivesonintheformofhismagnificent mathematical legacy and our memories of a wonderful man. We are very grateful to Professor Shoshichi Kobayashi for his distinguished contributions to differential geometry and complex geometry, and we are pleased tobeabletopublishavolumededicatedtohimconsistingofinvitedcontributions from his PhD students, co-workers and colleagues working on topics close to his researchareas. All papers are refereed to the standards of an excellent journal. Finally,wethankthe editorsatBirkh¨auserfortheirconstantencouragement for this publication. The editors Takushiro Ochiai (chair) Toshiki Mabuchi Yoshiaki Maeda Junjiro Noguchi Alan Weinstein Shoshichi Kobayashi Part 1 About Shoshichi Kobayashi