An Introduction to the Kähler-Ricci Flow

342 Pages·2013·2.438 MB·English

by  Sébastien Boucksom, Philippe Eyssidieux (auth.), Sebastien Boucksom, Philippe Eyssidieux, Vincent Guedj (eds.)

Lecture Notes in Mathematics 2086
Sébastien Boucksom
Philippe Eyssidieux
Vincent Guedj Editors
An Introduction
to the
Kähler-Ricci Flow
Lecture Notes in Mathematics 2086
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304

Se´bastien Boucksom Philippe Eyssidieux
(cid:2)
Vincent Guedj
Editors
An Introduction to the
Ka¨hler–Ricci Flow
123
Editors
Se´bastienBoucksom PhilippeEyssidieux
InstitutdeMathe´matiquesdeJussieu InstitutFourierandInstitutUniversitaire
CNRS-Universite´PierreetMarieCurie deFrance
Paris,France Universite´JosephFourier
Saint-Martind’He`res,France
VincentGuedj
InstitutdeMathe´matiquesdeToulouse
andInstitutUniversitairedeFrance
Universite´PaulSabatier
Toulouse,France
ISBN978-3-319-00818-9 ISBN978-3-319-00819-6(eBook)
DOI10.1007/978-3-319-00819-6
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013946026
MathematicsSubjectClassification(2010):35K96,32Q15,32Q20,32Q25,14E30
(cid:2)c SpringerInternationalPublishingSwitzerland2013
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Preface
In 2010–2011we organized several meetings of our ANR project MACK. These
consisted of series of lectures centered around the Ka¨hler–Ricci flow, which took
place,respectively,in
(cid:129) IMT(Toulouse,France),February2010:minicoursebyH.-D.Cao:
AnintroductiontotheKa¨hler–RicciflowonFanomanifolds;
(cid:129) LATP(Marseille,France),March2010:minicoursebyJ.Song:
TheKa¨hler–Ricciflowoncomplexsurfaces;
(cid:129) CIRM(Luminy,France),February2011:minicourseB.Wein kove:
AnintroductiontotheKa¨hler–Ricciflow;
(cid:129) CIRM(Luminy,France),February2011:minicourseJ.Song:
Ka¨hler–RicciflowandtheMinimalModelProgram;
(cid:129) IMT(Toulouse,France),June2011:minicoursebyD.-H.Phong:
ThenormalizedKa¨hler–RicciflowonFanomanifolds;
(cid:129) IMT(Toulouse,France),June2011:minicoursebyS.BoucksomandV.Guedj:
RegularizingpropertiesoftheKa¨hler–Ricciflow;
(cid:129) FSSM(Marrakech,Morocco),Oct ober2011:minicoursebyC.Imbert:
Introductiontofullynonlinearparabolicequations;
(cid:129) FSSM(Marrakech,Morocco),Oct ober2011:minicoursebyV.Guedj:
ConvergenceoftheKa¨hler–RicciflowonKa¨hler–EinsteinFanomanifolds.
Therewereotherlecturesonmorealgebraicaspects(e.g.,anintroductiontothe
Minimal Model Program and finite generation of the canonical ring by S. Druel
at the CIRM), or on elliptic problems (e.g., Moser–Trudinger inequalities by
B.BerndtssoninMarrakech).Someofthespeakershaveproducedasetoflecture
notes,workinghardtomakethemaccessibletonon-experts.Thisvolumepresents
theminaunifiedway.
v
vi Preface
Itisapleasuretothankalltheparticipantsofthesemeetingsfortheirenthusiasm
andforcreatingaverypleasantatmosphereofwork.Specialthanksofcoursetothe
followinglecturers:
(cid:129) S e´bastienBoucksom(CNRSandIMJ,Paris,France);
(cid:129) HuaiDongCao(LehighUniversity,Bethlehem,USA);
(cid:129) VincentGuedj(IUFandIMT,Toulouse,France)
(cid:129) CyrilImbert(CNRSandUniversit e´Paris-EstCre´teil,France);
(cid:129) DuongHongPhong(ColumbiaUniversity,USA);
(cid:129) JianSong(RutgersUniversity,Piscataway,USA);
(cid:129) BenWeinkove(UniversityofCalifornia,SanDiego,USA).
WeacknowledgefinancialsupportfromtheFrenchANRprojectMACK.Wealso
would like to thank the CIRM and their staff for providing wonderful conditions
ofworkduringthethematicmonth“ComplexandRiemanniangeometry,”as well
as the LATP and the IMT for providing “professeur invite´” positions (resp. for
H.-D.Cao,J.Song,andD.-H.Phong).
Our last meeting in Marrakech was also extremely useful, we thank the orga-
nizers (Said Asserda and Ahmed Zeriahi), the other speakers (B. Berndtsson,
S. Boucksom, A. Broustet, J.-P. Demailly, S. Diverio, N.C. Nguyen, S. Lamy) as
wellasalltheparticipantsfortheirhelpincreatingasuccessfulevent.
Paris,France S.Boucksom
Grenoble,France P.Eyssidieux
Toulouse,France V.Guedj
1July2012
Contents
1 Introduction .................................................................. 1
Se´bastienBoucksom,PhilippeEyssidieux,andVincentGuedj
1.1 Motivation............................................................... 1
1.2 Contents ................................................................. 4
References..................................................................... 5
2 AnIntroductiontoFullyNonlinearParabolicEquations............... 7
CyrilImbertandLuisSilvestre
2.1 Introduction ............................................................. 7
2.2 SchauderEstimatesforLinearParabolicEquations................... 12
2.3 ViscositySolutions:AShortOverview................................ 23
2.4 HarnackInequality...................................................... 42
References..................................................................... 87
3 AnIntroductiontotheKa¨hler–RicciFlow................................ 89
JianSongandBenWeinkove
3.1 Preliminaries ............................................................ 92
3.2 GeneralEstimatesfortheKa¨hler–RicciFlow......................... 106
3.3 MaximalExistenceTimefortheKa¨hler–RicciFlow ................. 121
3.4 ConvergenceoftheFlow ............................................... 128
3.5 TheCaseWhenK IsBigandnef.................................... 140
M
3.6 Ka¨hler–RicciFlowonaProductEllipticSurface ..................... 153
3.7 FiniteTimeSingularities................................................ 163
3.8 TheKa¨hler–RicciFlowandtheAnalyticMMP....................... 173
References..................................................................... 183
4 RegularizingPropertiesoftheKa¨hler–RicciFlow....................... 189
Se´bastienBoucksomandVincentGuedj
4.1 AnAnalyticToolbox.................................................... 190
4.2 SmoothingPropertiesoftheKa¨hler–RicciFlow ...................... 196
4.3 DegenerateParabolicComplexMonge–Ampe`reEquations.......... 207
vii
viii Contents
4.4 A Priori Estimates for Parabolic Complex
Monge–Ampe`reEquations............................................. 210
4.5 ProofoftheMainTheorem............................................. 220
4.6 TheKa¨hler–RicciFlowonaLogTerminalVariety ................... 229
References..................................................................... 236
5 TheKa¨hler–RicciFlowonFanoManifolds............................... 239
Huai-DongCao
5.1 Preliminaries ............................................................ 242
5.2 The(Normalized)Ka¨hler–RicciFlow ................................. 246
5.3 TheLi–Yau–HamiltonInequalitiesforKRF .......................... 258
5.4 Perelman’sEntropyandNoncollapsingTheorems.................... 265
5.5 UniformCurvatureandDiameterEstimatesforNKRF
withNonnegativeBisectionalCurvature............................... 274
5.6 Perelman’sUniformEstimates......................................... 276
5.7 RemarksontheFormationofSingularitiesinKRF................... 288
References..................................................................... 294
6 ConvergenceoftheKa¨hler–RicciFlow
onaKa¨hler–EinsteinFanoManifold...................................... 299
VincentGuedj
6.1 Background.............................................................. 300
6.2 NormalizationofPotentials............................................. 307
6.3 C0-Estimate............................................................. 315
6.4 HigherOrderEstimates................................................. 323
6.5 ConvergenceoftheFlow ............................................... 325
6.6 AnAlternativeApproach............................................... 327
References..................................................................... 332
Chapter 1
Introduction
Se´bastienBoucksom,PhilippeEyssidieux,andVincentGuedj
Abstract This book is the first comprehensive reference on the Ka¨hler–Ricci
flow. It provides an introduction to fully non-linear parabolic equations, to the
Ka¨hler–Ricci flow in general and to Perelman’s estimates in the Fano case, and
alsopresentstheconnectionswiththeMinimalModelprogram.
1.1 Motivation
1.1.1 SomeHistoricalRemarks
AccordingtoWeil[Weil57],thenotionofaKa¨hlermanifold,introducedbyKa¨hler
in 1933 [Ka¨h33], became important through the work of Hodge [Hod41] that
put on a firm footing the theorems of Lefschetz on the topology of complex
projective manifolds [Lef24]. As Hodge remarks, the introduction of a metric
on a complex projective manifold is a somewhat artificial operation. However
artificial, this operationturnedout to be veryfruitful. Kodaira developedHodge’s
ideas into his famous theorem giving a differential geometric characterization of
S.Boucksom((cid:2))
InstitutdeMathe´matiquesdeJussieu,CNRS-Universite´PierreetMarieCurie,4placeJussieu,
75251,Paris,France
e-mail:[e-mail]
P.Eyssidieux
InstitutFourierandInstitutUniversitairedeFrance,Universite´JosephFourier,100ruedesMaths,
38402,Saint-Martind’He`res,France
e-mail:[e-mail]
V.Guedj
InstitutdeMathe´matiquesdeToulouseandInstitutUniversitairedeFrance,Universite´
PaulSabatier,118routedeNarbonne,31062,ToulouseCedex9,France
e-mail:[e-mail]
S.Boucksometal.(eds.),AnIntroductiontotheKa¨hler–RicciFlow, 1
LectureNotesinMathematics2086,DOI10.1007/978-3-319-00819-6 1,
©SpringerInternationalPublishingSwitzerland2013