Complex Geometry and Dynamics: The Abel Symposium 2013

316 Pages·2015·3.31 MB·English

by  John Erik Fornæss

Abel Symposia 10
John Erik Fornæss
Marius Irgens
Erlend Fornæss Wold Editors
Complex
Geometry
and Dynamics
The Abel Symposium 2013
ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
Moreinformationaboutthisseriesathttp://www.springer.com/series/7462
John Erik Fornæss (cid:129) Marius Irgens (cid:129)
Erlend Fornæss Wold
Editors
Complex Geometry
and Dynamics
The Abel Symposium 2013
123
Editors
JohnErikFornæss MariusIrgens
DepartmentofMathematicalSciences DepartmentofMathematicalSciences
NTNUGløshaugen NTNUGløshaugen
Trondheim,Norway Trondheim,Norway
ErlendFornæssWold
DepartmentofMathematics
UniversityofOslo
Oslo,Norway
ISSN2193-2808 ISSN2197-8549 (electronic)
AbelSymposia
ISBN978-3-319-20336-2 ISBN978-3-319-20337-9 (eBook)
DOI10.1007/978-3-319-20337-9
LibraryofCongressControlNumber:2015955146
Mathematics Subject Classification (2010): 32H04, 37F99, 32W05, 32D15, 53C42, 32H02, 53A10,
32B15, 14H15, 32G15, 37J35, 58A15, 32V40, 32Q45,
32V35, 14J70, 14F10, 12H05, 53A20, 32V25, 32S25,
32T15, 32A07, 32H50, 37F45, 11J97, 14J10, 32U05,
32V10,32V30
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Foreword
TheNorwegiangovernmentestablishedtheAbelPrizeinmathematicsin2002,and
the first prize was awarded in 2003. In addition to honoring the great Norwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstanding
scientificworkinthefieldofmathematics,theprizeshallcontributetowardraising
the status of mathematics in society and stimulate the interest for science among
schoolchildrenandstudents.Inkeepingwiththisobjective,theNielsHenrikAbel
Board has decided to finance annual Abel Symposia. The topic of the Symposia
maybeselectedbroadlyintheareaofpureandappliedmathematics.TheSymposia
should be at the highest international level and serve to build bridges between
thenationalandinternationalresearchcommunities.TheNorwegianMathematical
Societyisresponsiblefortheevents.Ithasalsobeendecidedthatthecontributions
fromthese Symposiashouldbe presentedina seriesofproceedings,andSpringer
Verlag has enthusiastically agreed to publish the series. The Niels Henrik Abel
Boardisconfidentthattheserieswillbeavaluablecontributiontothemathematical
literature.
HelgeHolden
ChairoftheNielsHenrikAbelBoard
Trondheim,Norway
v

Preface
Thetheme ofthe AbelSymposium2013was Complex Geometry,andit was held
at the Norwegian University of Science and Technology, Trondheim, during July
2–5.Theeventattracted43participantsandfeaturedpresentationsby22speakers.
ThescientificagendaprimarilyfocusedongeometricproblemsinSeveralComplex
Variables and Complex Dynamics, including holomorphic laminations/foliations,
the@-equation,CR-geometry,pluripotentialtheory,andfunctiontheory.Theaimof
theAbelSymposiumwastopresentthestateoftheartonthesetopicsandtodiscuss
futureresearchdirections.Thespeakersandtitleswere:
1. Bedford,E.Automorphismsofblowupsofprojectivespace
2. Berndtsson,B.TheopennessproblemandcomplexBrunn-Minkowskiinequali-
ties
3. Błocki,Z.Hörmander’s@-estimate,somegeneralizationsandnewapplications
4. Demailly,J.-P.Onthecohomologyofpseudoeffectivelinebundles
5. Dihn,T.-C.Positiveclosed.p;p/-currentsandapplicationsincomplexdynam-
ics
6. Ebenfelt,P.PartialrigidityofdegenerateCR-embeddingsintospheres
7. Fornstnericˇ,F.ComplexanalysisandtheCalabi-Yauproblem
8. Grushevsky,S. Meromorphic differentialswith realperiodsandthe geometry
ofthemodulispaceofRiemannsurfaces
9. Huang, X. Analyticity of the local hull of holomorphy for a codimension two
real-submanifoldinCn
10. Kohn,J.WeaklypseudoconvexCRmanifolds
11. McMullen,C.Entropyanddynamicsoncomplexsurfaces
12. Merker,J.Siu-YeungholomorphicsectionsofSymmT(cid:2)
X
13. Mok, N. On the Zariski closure of an infinite number of totally geodesic
subvarietesof˝=(cid:2)
14. Nemirovski,S.Topologyandseveralcomplexvariables
15. Ohsawa,T.LeviflatsinHopfsurfaces
16. Sibony,N.DynamicsoffoliationsbyRiemannsurfaces
17. Stensønes,B.Realanalyticdomainsandplurisubharmonicfunctions
18. Ueda,T.Semi-parabolicfixedpointsandtheirbifurcationsincomplexdimen-
sion2
19. Yau,S.-T.Periodintegrals,countingcurves,andmirrorsymmetry
vii
viii Preface
20. Yau, S. Nonconstant CR morphisms between compact strongly pseudoconvex
CRmanifoldsandetalecoveringbetweenresolutionsofisolatedsingularities
21. Yeung,S.-K.Complexhyperbolicityonthemoduliofsomehigher-dimensional
manifolds
22. Zhou,X.SomeresultsonL2-extensionproblemwithoptimalestimate
ThescientificcommitteeconsistedofJohnErikFornæss(NTNU),MariusIrgens
(NTNU), Yum-Tong Siu (Harvard), Erlend F. Wold (Oslo), and Shing-Tung Yau
(Harvard).
During the symposium, a dinner was held in honor of Yum-Tong Siu’s 70th
birthday.Wewouldliketodedicatetheseproceedingstohim.
Theparticipantsatthesymposiumwere:
EricBedford NgaimingMok
BoBerndtsson StefanNemirovski
ZbigniewBłocki TakeoOhsawa
FushengDeng TronOmland
Jean-PierreDemailly MariusOverholt
KlasDiederich NilsØvrelid
Tien-CuongDinh AlexanderRashkovskii
PeterEbenfelt NessimSibony
JohnErikFornæss Yum-TongSiu
FrancForstnericˇ BeritStensønes
DustyGrundmeier ShenghaoSun
SamuelGrushevsky TetsuoUeda
KariHag ErlendF.Wold
PerHag SonyauXie
Siri-MalénHøynes GuowuYao
XiaojunHuang Shing-TungYau
MariusIrgens StephenS.T.Yau
JosephKohn Sai-KeeYeung
ErikLøw Jian-HuaZheng
BenediktMagnusson Xiang-YuZhou
CurtisMcMullen MinxianZhu
JoëlMerker
WewouldliketothanktheNorwegianMathematicalSocietyandtheNilsHenrik
Abel Memorial Fund for giving us the opportunity to host the Abel Symposium.
We would also like to thank the administration at NTNU for their great help
with the organizingand Ruth Allewelt at Springerfor her helpwith preparingthe
proceedings.
Trondheim,Norway JohnErikFornæss
Trondheim,Norway MariusIrgens
Oslo,Norway ErlendFornæssWold
May15,2015
Contents
PseudoautomorphismswithInvariantCurves................................ 1
EricBedford,JefferyDiller,andKyoungheeKim
1 FromCuspidalCurves... ..................................................... 6
2 ...ToBasicCremonaMaps... ............................................... 8
3 ...ToPseudoautomorphisms... .............................................. 12
4 ...ToFormulas................................................................ 14
5 TheConnectionwithCoxeterGroups........................................ 20
6 PseudoautomorphismsonMultiprojectiveSpaces........................... 22
References......................................................................... 26
TheOpennessConjectureandComplexBrunn-Minkowski
Inequalities........................................................................ 29
BoBerndtsson
1 Introduction.................................................................... 29
2 TheBrunn-MinkowskiTheorem ............................................. 30
3 AComplexVariantoftheBrunn-MinkowskiTheorem..................... 31
4 TheOpennessProblem........................................................ 35
5 AConjecturalPictureforStrongOpenness.................................. 41
References......................................................................... 43
Estimatesfor@N andOptimalConstants ....................................... 45
ZbigniewBłocki
1 Introduction.................................................................... 45
2 Estimatesfor@N................................................................. 46
3 OptimalConstants............................................................. 48
References......................................................................... 50
OntheCohomologyofPseudoeffectiveLineBundles........................ 51
Jean-PierreDemailly
1 IntroductionandStatementoftheMainResults............................. 51
2 ApproximationofpshFunctionsandofClosed(1,1)-Currents............. 57
ix