Progress in Mathematics Volume 155

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

Sorin Dragomir Liviu Ornea

Locally Conformal Kähler Geometry

Springer Science+Business Media, LLC

Sorin Dragomir Dipartimento di Matematica Universita degli Studi della Basilicata 85100 Potenza, Italia

Liviu Omea Facultatea de Matematica Universitatea din Bucure§ti Bucure§ti, Romania

Library of Congress Cataloging-in-Publication Data

Dragonrlr,Sorin,1955-Locally confonnal Kähler geometry / Sorin Dragonrlr, Liviu Omea.

p. cm. -- (Progress in mathematics ; v. 155) Includes bibliographica1 references. ISBN 978-1-4612-7387-5 ISBN 978-1-4612-2026-8 (eBook) DOI 10.1007/978-1-4612-2026-8

1. Kählerian manifolds. 2. Geometry, Differential. I. Ornea, Liviu, 1960- . 11. Title. III. Series: Progress in mathematics (Boston, Mass.) ; vol. 155

QA649.D761997 515'.73--dc21

97-27397 CIP

AMS Classification Codes: 53D20, 53C15, 53C40, 53C56.

Printed on acid-free paper

© 1998 Springer Science+Business Media New Y ork Originally published by Birkhäuser Boston in 1998 Softcover reprint ofthe hardcover 1st edition 1998

Copyright is not c1aimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transnrltted, in any fonn or by any means, electronic, mechanical, photocopy­ing, recording, or otherwise, without prior permission of the copyright owner.

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ISBN 978-1-4612-7387-5

Refonnatted from authors' disks by TEXniques, Ine. Boston, MA

987 6 5 432 1

Contents

Introduction

1 L.c.K. Manifolds

2 Principally Important Properties 2.1 Vaisman's conjectures 2.2 Reducible manifolds 2.3 Curvature properties . 2.4 Blow-up . . . . . . . . 2.5 An adapted cohomology

3 Examples 3.1 Hopf manifolds 3.2 The Inoue surfaces . . . . . . . . . . . . 3.3 A generalization of Thurston's manifold 3.4 A four-dimensional solvmanifold 3.5 SU(2) x Sl ............. . 3.6 Noncompact examples ....... . 3.7 Brieskorn & Van de Ven's manifolds

4 Generalized Hopf manifolds

5 Distributions on a g.H. manifold

6 Structure theorems 6.1 Regular Vaisman manifolds 6.2 L.c.K.o manifolds . . . . . . 6.3 A spectral characterization 6.4 k-Vaisman manifolds . . . .

ix

1

7 7

10 11 15 16

21 21 23 25 26 28 28 29

33

41

49

49 56 60 66

vi CONTENTS

7 Harmonic and holomorphic forms 69

69 79

7.1 Harmonic forms ..... . 7.2 Holomorphic vector fields

8 Hermitian surfaces

9 Holomorphic maps 9.1 General properties 9.2 Pseudoharmonic maps 9.3 A Schwarz lemma.

10 L.c.K. submersions 10.1 Submersions from CHf

85

103 103 107 111

121 121

10.2 L.c.K. submersions . . . . . . . . . . . . . . . . . . . . . . 124 10.2.1 An almost Hermitian submersion with total space

s2n-l(c, k) x R, k > -3~ . . . . . . . . . . . . .. 125 10.2.2 An almost Hermitian submersion with total space

R2n-l(c) X R . . . . . . . . . . . . . . . . . . . . . 126 10.2.3 An almost Hermitian submersion with total space

(R x Bn-l)(c,k) x R, k < -3c2 127

10.3 Compact total space . . . . 128 10.4 Total space a g.H. manifold 130

11 L.c. hyper Kahler manifolds 133

12 Submanifolds 147 12.1 Fundamental tensors . . . . . . 147 12.2 Complex and CR submanifolds 153 12.3 Anti-invariant submanifolds . . 158 12.4 Examples . . . . . . . . . . . . 164 12.5 Distributions on submanifolds . 167 12.6 Totally umbilical submanifolds 172

13 Extrinsic spheres 187 13.1 Curvature-invariant submanifolds . 187 13.2 Extrinsic and standard spheres 194 13.3 Complete intersections . 202 13.4 Yano's integral formula .... 212

CONTENTS

14 Real hypersurfaces 14.1 Principal curvatures ..... 14.2 Quasi-Einstein hypersurfaces 14.3 Homogeneous hypersurfaces 14.4 Type numbers ...... . 14.5 L. c. cosymplectic metrics

15 Complex submanifolds 15.1 Quasi-Einstein submanifolds . 15.2 The normal bundle . . . . . . 15.3 L.c.K. and Kahler submanifolds . 15.4 A Frankel type theorem . . 15.5 Planar geodesic immersions

16 Integral formulae 16.1 Hopf fibrations ........ . 16.2 The horizontal lifting technique 16.3 The main result. . . . . . . . .

17 Miscellanea 17.1 Parallel IInd fundamental form. 17.2 Stability ....... . 17.3 f-Structures ..... . 17.4 Parallel f-structure P 17.5 Sectional curvature .. 17.6 L. c. cosymplectic structures 17.7 Chen's class ......... . 17.8 Geodesic symmetries .... . 17.9 Submersed CR submanifolds

A Boothby-Wang fibrations

B Riemannian submersions

Bibliography

vii

219 219 223 225 226 232

239 239 244 251 253 255

257 257 260 267

275 275 277 278 283 285 286 289 291 292

299

303

307

To the memory of Franco Tricerri

Introduction

Let (M2n, J,g) be a Hermitian manifold of complex dimension n, where J denotes its complex structure, and g its Hermitian metric. Cf. P. Libermann, [174], [175], g is a locally conformal Kahler (l.c.K.) metric if g is conformal to some local Kahlerian metric in the neighborhood of each point of M2n.

Precisely, g is l.c.K. if there is an open cover {UihEI of M 2n and a family {JihEI of COO functions Ii : Ui -4 R so that each local metric

is Kahlerian. The main theme of this book is the study of l.c.K. manifolds, i.e. manifolds which carry some l.c.K. metric.

Nowadays complex geometry deals primarily with Kahlerian manifolds, cf. e.g. [291], [21], i.e. manifolds carrying some Kahlerian metric. Nev­ertheless, some readily available complex manifolds, such as complex Hopf manifolds (cf. [132]) admit no global Kahlerian metrics at all. Indeed, let >. E C, 0 < 1>'1 < 1, and n E Z, n ~ 2. Let~.>. be the O-dimensional Lie group generated by the transformation z ~ >.z, z E Cn - {a}. Then (cf. e.g. [162]' vol. II, p. 137) ~.>. acts freely on cn - {a} as a properly discon­tinuous group of complex analytic transformations of cn - {O}, so that the quotient space C~ = (C n - {O})/~.>. becomes a complex manifold. This is the complex Hopf manifold. Since CHr ~ 8 1 X S2n-l (a diffeomorphism) CHr is compact and its first Betti number is bl(CHr) = 1. Then (in view of Theorem 5.6.2 in [106], p. 178) CHr admits no globally defined Kahlerian metrics. In turn (as discovered by W.M. Boothby, [36], for n = 2) any com­plex Hopf manifold CHr admits a globally defined l.c.K. metric go. Indeed, the Hermitian metric

x INTRODUCTION

on Cn - {O} is .6.>.-invariant, and therefore it induces a global Hermitian metric 90 on CHf, the Boothby metric. As observed later by I. Vaisman (cf. [269]) 90 turns out to be l.c.K. Several examples of complex manifolds admitting no global Kahlerian metrics (yet carrying natural global l.c.K. metrics) were subsequently discovered, cf. e.g. the complex Inoue surfaces 8M with the Tricerri metric, cf. [258].

The first eleven chapters of this book report on the main achievements in the theory of l.c.K. manifolds. The last six chapters present the theory of submanifolds in l.c.K. manifolds, as developed by J .L. Cabrerizo & M.F. Andres, [46], S. Ianm~ & K. Matsumoto & 1. Ornea, [133], [134], K. Mat­sumoto, [184], [185], F. Narita, [198], and the authors, cf. [216], [217], and [75], [76], [77], for whom the geometry of the second fundamental form of a submanifold (in particular, of a CR submanifold, in the sense of A. Bejancu, [15]) has been the main interest for quite a few years.

The geometry of l.c.K. manifolds has developed mainly since the 1970s, although, as indicated above, there are early contributions by P. Libermann (going back to 1954). The recent treatment of the subject was initiated by I. Vaisman in 1976. In a long series of papers (cf. [266], [268]-[278]), he established the main properties of l.c.K. manifolds, demonstrated a con­nection with P. Gauduchon's standard metrics, and recognized the Boothby metric as l.c.K., thus being naturally led to the introduction of the general­ized Hopf (g.H.) manifolds. He also explained the relationship between this class of l.c.K. manifolds and the contact metric manifolds. Next, F. Tricerri proved that the blow-up at a point preserves the class, and together with I. Vaisman (cf. [260]) studied the curvature properties of l.c.K. surfaces.

A great amount of research has been produced by E. Bedford & T. Suwa, [20], A. Cordero & M. Fernandez & M. De Leon, [68], T. Kashiwada, [147]­[151], T. Kashiwada & S. Sato, [153] (who showed that the first Betti num­ber of a compact g.H. manifold is odd), B.Y. Chen & P. Piccinni, [64] (who studied foliations naturally occuring on a l.c.K. manifold), S.1. Goldberg & I. Vaisman, [115], C.P. Boyer, [40] (who demonstrated the relationship between anti-self-dual compact complex surfaces and l.c.K. surfaces), H. Pedersen & Y.S. Poon & A. Swann, [226] (who proved that Hermite-Einstein-Weyl man­ifolds are g.H. manifolds), M. Pontecorvo, [231] (who studied conformally flat l.c.K. surfaces), D. Perrone, [227] (who gave a spectral characterization of complex Hopf surfaces), K. Tsukada, [262] (who studied holomorphic vec­tor fields on g.H. manifolds), J.C. Marrero & J. Rocha, [179] (who studied submersions from a l.c.K. manifold), etc.

The most important of the authors' contributions seem to be the (par­tial) classification of totally umbilical submanifolds (in particular of extrinsic

INTRODUCTION xi

spheres) in a g.H. manifold (cf. [134] and [88]), and the classification of all compact minimal CR submanifolds, generically embedded in a complex Hopf manifold with the Boothby metric, which are fibered in tori, and have a flat normal connection, and a second fundamental form of constant length (cf. [11]).

The second named author has had useful discussions with P. Gaudu­chon, s. Ianw~, L. Lemaire, S. Marchiafava, S. Papadima, P. Piccinni, and V. Vuletescu. While this book was written, both authors profited from many useful comments from 1. Vaisman. The authors wish to express their grati­tude to all these people.

The interest of the first named author in l.c.K. geometry has been stim­ulated by discussions with F. Tricerri. Both authors wish to dedicate this book, as a modest reminder, to his memoryl.

Potenza, 22 January 1997

IF. Tricerri died together with his wife and two children in an airplane crash in June 1994.

Locally Conformal Kahler Geometry