Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June 2001

244 Pages·2003·8.37 MB·English

by  Mark Gross

Universitext
Springer-Ve rlag Berlin Heidelberg GmbH
M.Gross •D.Huybrechts • D.Joyce
Calabi-Yau Manifolds
and Related Geometries
Lectures at a Summer School in Nordfjordeid,
Norway, June 2001
, Springer
MarkGross Daniel Huybrechts
University ofWarwick Universite Paris 7 Denis Diderot
Mathematics Institute Institut de Mathematiques
Coventry CV 4 7A L Topologie et Geometrie Algebriques
United Kingdom 2, place Jussieu
e-mail: [email protected] 75251 Paris Cedex 05
e-mail: [email protected]
DominicJoyce
Lincoln College
OxfordOX13DR
United Kingdom
e-mail: [email protected]
Editors:
Geir Ellingsrud LorenOIson
Universtity of Oslo Universtity ofTroms0
Department ofMathematics Institute of Mathematics and Statistics
P.O. Box 1053 Blindern 9037 Troms0, Norway
0316 Oslo, Norway e-mail: [email protected]
e-mail: [email protected]
Kristian Ranestad Stein A. Str0mme
Universtity of Oslo Universtity ofBergen
Department ofMathematics Department ofMathematics
P.O. Box 1053 Blindern 5008 Bergen, Norway
0316 Oslo, Norway e-mail: [email protected]
e-mail: [email protected]
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ISBN 978-3-540-44059-8 ISBN 978-3-642-19004-9 (eBook)
DOI 10.1007/978-3-642-19004-9
Mathematics Subiect Classification (2000): 14J32,32Q25,53C26, 53C29, 53C38
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Preface
Each summersince1996,algebraicgeometers and algebraistsin Norwayhave
organised a summer school in Nordfjordeid, a small place in the western
part of Norway. In addition to the beauty ofthe place, located between the
mountains, closeto the fjord and not far from the Norway's largest glacier,
a reason for going there is that Sophus Liewas born and spent his fewfirst
years in Nordfjordeid, so it has a flavour ofboth the exotic and pilgrimage.
It is also convenient:the municipality ofEid has created a conferencecentre
named afterSophus Lie,aimedat attractingactivitiesto fillthe summer term
of the local boarding school.
The summer schools are a joint effort ofthe four universities in Norway
- the universities of Tromse, Trondheim, Bergen and Oslo. They are pri
marily meant as trainingforNorwegiangraduatestudents, but haveoverthe
years attracted increasing numbers ofstudents fromotherpartsofthe world,
adding the value ofbeing international to the schools.
The themes ofthe schools havebeen varied, but build around some cen
tral topics in contemporary mathematics. The format of the school has by
nowbecome tradition - three international experts givingindependent, but
certainly connected,seriesoftalks with exercisesessionsin the evening,over
fiveor six days.
In 2001the organising committee consisted ofStein Arild Stromme from
the University of Bergen, Loren Olson from the University of Tromse, and
Kristian Ranestad and Geir Ellingsrud from the University of Oslo. We
wanted to make a summer schoolgiving the students insight in some of the
newinteractionsbetweendifferentialand algebraicgeometry.The three topics
wefinallychose, Riemannian holonomyand calibrated geometry, Calabi-Yau
manifolds and mirror symmetry, and Compact hyperkiihler manifolds, are
parts of the fascinating current development of mathematics, and wethink
they illustrate wellthe modern interplay between differential and algebraic
geometry.
Wewerefortunate enough toget positive answerswhenweasked Dominic
Joyce, Mark Gross and Daniel Huybrechts to give the courses, and we are
thankful for the great job the three lecturers did, both on stage in Nord
fjordeid and by writing up the nicenotes whichhavenowdevelopedinto this
book.
Geir Ellingsrud
Contents
Preface V
Part I. Riemannian Holonomy Groups and Calibrated Geometry
Dominic Joyce
1 Introduction.... .. .... ..... ..... .... ............ ..... ........ 3
2 Introduction to Holonomy Groups. ............................. 4
3 Berger's Classification ofHolonomyGroups 10
4 Kahler Geometry and Holonomy ............................... 14
5 The Calabi Conjecture........................................ 20
6 The Exceptional Holonomy Groups............................. 26
7 Introduction to Calibrated Geometry " 31
8 Calibrated Submanifolds in IRn ... ••.. . ••.. .... . ...•.. .. .. .. .. . 36
9 Constructions ofSL m-folds in em ............................. 41
10 Compact Calibrated Submanifolds 50
11 Singularities ofSpecial Lagrangian m-folds ...................... 57
12 The SYZConjecture,and SL Fibrations 63
Part II. Calabi-Yau Manifolds and Mirror Symmetry
Mark Gross
13 Introduction................................................. 71
14 The Classical Geometry of Calabi-Yau Manifolds 72
15 Kahler Moduli and Gromov-Witten Invariants. .................. 93
16 Variation and Degeneration ofHodge Structures 102
17 A Mirror Conjecture 121
18 Mirror Symmetry in Practice 122
19 The Strominger-Yau-Zaslow Approach to Mirror Symmetry 140
Part III. Compact Hyperkahler Manifolds
Daniel Huybrechts
20 Introduction 163
21 Holomorphic Symplectic Manifolds 164
22 Deformations of Complex Structures 172
23 The Beauville-Bogomolov Form 177
24 Cohomology ofCompact Hyperkahler Manifolds 190
25 Twistor Space and Moduli Space 196
VIII Contents
26 Projectivity ofHyperkahler Manifolds 203
27 Birational Hyperkahler Manifolds 213
28 The (Birational) Kahler Cone 221
References.................................................... 227
Index 237
Part I
Riemannian Holonomy Groups
and Calibrated Geometry
Dominic Joyce
M. Gross et al., Calabi-Yau Manifolds and Related Geometries
© Springer-Verlag Berlin Heidelberg 2003
1 Introduction
The holonomy group Hol(g) of a Riemannian n-manifold (M,g) is a global
invariant which measures the constant tensors on the manifold. It is a Lie
subgroup of SO(n), and for generic metrics Hol(g) = SO(n). If Hol(g) is
a proper subgroup of SO(n) then we say 9 has special holonomy. Metrics
with special holonomy are interestingfora number ofdifferent reasons. They
include Kiihler metrics with holonomy U(m), which are the most natural
class ofmetrics on complex manifolds, Calabi-Yau manifoldswith holonomy
SU(m), and hyperkiihler manifoldswith holonomy Sp(m).
Calibrated submanifoldsare a class of k-dimensional submanifolds N of
a Riemannian manifold (M,g) defined using a closed k-form ip on M called
a calibration. Calibrated submanifolds are automatically minimal submani
folds. Manifolds with special holonomy (M,g) generally comeequipped with
one or more natural calibrations ip, which then define interesting classes of
submanifolds in M. One such class is special Lagrangian submanifolds(SL
m-folds) of Calabi-Yau manifolds.
This part of the book is an expanded version of a course of 8 lectures
given at Njordfjordeid in June 2001. The first half of the course discussed
Riemannian holonomy groups, focussingin particularon Kahler and Calabi
Yau manifolds.The second halfdiscussedcalibratedgeometry and calibrated
submanifolds of manifolds with special holonomy,focussing in particular on
SL m-folds of Calabi-Yau m-folds. The final lecture surveyed research on
the SYZ Conjecture, which explains Mirror Symmetry between Calabi-Yau
3-folds using special Lagrangian fibrations, and so made contact with Mark
Gross' lectures.
Ihaveretained this basicformat, sothat the first half§1-§6isonRieman
nian holonomy,and the second half §7-§12on calibrated geometry, finishing
with the SYZ Conjecture. The principal aim of the first half is to provide a
firmgroundingin Kahler and Calabi-Yaugeometryfrom the differential geo
metric point ofview, to serve as background for the more advanced,algebro
geometric material discussed in Parts II and III below. Therefore I have
treated other subjects such as the exceptional holonomygroups fairly briefly.
In the second half I shall concentrate mainly on SL m-folds in em and
Calabi-Yaum-folds.Thisispartlybecauseofthe focusofthe bookonCalabi
Yaumanifolds and the link with Mirror Symmetry, partly because ofmyown
research interests, and partly because more work has been done on special
Lagrangian geometry than on other interesting classesofcalibrated subman
ifolds,so there issimply more to say.
This is not intended as an even-handed survey of a field, but is biassed
in favour of my own interests, and areas of research I want to promote in
future. Therefore my own publications appear more often than they deserve,
whilst more significant work is omitted, through my oversight or ignorance.
I apologize to other authors who feelleft out.
4 I Riemannian HolonomyGroups and Calibrated Geometry
This isparticularly true in the secondhalf. Muchofsections 9,11and 12
isan account ofmyown research programme into the singularities ofSLm
folds.It therefore has a provisional,unfinishedquality, with someconjectural
material. My excusefor including it in a book isthat I believethat a proper
understanding ofSLm-folds and their singularities willlead to exciting new
discoveries- newinvariants ofCalabi-Yau3-folds,and newchapters in the
Mirror Symmetrystorywhichareobscureat present,but are hinted at in the
Kontsevich Mirror Symmetry proposal and the SYZConjecture. SoI would
liketo get more people interested in this area.
We begin in §2with some background from Differential Geometry, and
define holonomy groups of connections and of Riemannian metrics. Section
3 explains Berger's classification of holonomy groups of Riemannian man
ifolds. Section 4 discusses Kahler geometry and Ricci curvature of Kahler
manifolds and defines Calabi-Yau manifolds, and §5 sketches the proof of
the Calabi Conjecture, and how it is used to construct examples of Calabi
Yau and hyperkahler manifolds via Algebraic Geometry. Section 6 surveys
the exceptional holonomygroups G and Spin(7).
2
Thesecond halfbegins in §7with an introduction to calibrated geometry.
Section 8 coversgeneral properties ofcalibrated submanifolds in IRn and §9
,
em.
construction ofexamples ofSLm-folds in Section 10discussescompact
calibrated submanifolds inspecial holonomymanifolds,and §11the singular
ities ofSL m-folds. Finally, §12brieflyintroduces String Theory and Mirror
Symmetry, explains the SYZ Conjecture, and summarizes some research on
the singularities ofspecial Lagrangian fibrations.
2 Introduction to Holonomy Groups
We begin by giving some background from differential and Riemannian ge
ometry,principally to establish notation, and moveon to discussconnections
on vector bundles, parallel transport, and the definition ofholonomygroups.
Somesuitable reading for this section ismy book [113, §2-§3].
2.1 Tensors and Forms
Let M be a smooth n-dimensional manifold, with tangent bundle TM and
cotangent bundle T*M. Then TM and T*M are vectorbundlesover M. If
E is a vector bundle over M, we use the notation COO(E) for the vector
space ofsmooth sections of E.Elements ofCOO(TM) are called vectorfields,
and elements of COO(T*M) are called I-forms. By taking tensor products of
the vector bundles TM and T*M we obtain the bundles of tensorson M.
A tensorT on M is a smooth section of a bundle ®kTM @ ®lT*M for
some k,l EN.
It is convenient to write tensors using the index notation. Let U be an
open set in M, and (Xl,...,z") coordinateson U.Then ateach point x E U,