Table Of ContentFields Institute Communications 67
The Fields Institute for Research in Mathematical Sciences
Radu Laza
Matthias Schütt
Noriko Yui
Editors
Arithmetic and
Geometry of K3
Surfaces and
Calabi–Yau
Threefolds
Fields Institute Communications
VOLUME 67
The Fields Institute for Research in Mathematical Sciences
FieldsInstituteEditorialBoard:
CarlR.Riehm,ManagingEditor
EdwardBierstone,DirectoroftheInstitute
MatheusGrasselli,DeputyDirectoroftheInstitute
JamesG.Arthur,UniversityofToronto
KennethR.Davidson,UniversityofWaterloo
LisaJeffrey,UniversityofToronto
BarbaraLeeKeyfitz,OhioStateUniversity
ThomasS.Salisbury,YorkUniversity
NorikoYui,Queen’sUniversity
TheFieldsInstituteisacentreforresearchinthemathematicalsciences,locatedin
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Europe,andseveralcommercialandindustrialpartners.
Forfurthervolumes:
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(cid:129) (cid:129)
Radu Laza Matthias Schu¨tt Noriko Yui
Editors
Arithmetic and Geometry of
K3 Surfaces and Calabi-Yau
Threefolds
123
TheFieldsInstituteforResearch
intheMathematicalSciences
Editors
RaduLaza MatthiasSchu¨tt
MathematicsDepartment Institutfu¨rAlgebraischeGeometrie
StonyBrookUniversity LeibnizUniversita¨tHannover
StonyBrook,NY Hannover,Germany
USA
NorikoYui
DepartmentofMathematicsandStatistics
Queen’sUniversity
Kingston,ON
Canada
ISSN1069-5265 ISSN2194-1564(electronic)
ISBN978-1-4614-6402-0 ISBN978-1-4614-6403-7(eBook)
DOI10.1007/978-1-4614-6403-7
SpringerNewYorkHeidelbergDordrechtLondon
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Preface
The workshop on Arithmetic and geometry of K3 surfaces and Calabi–Yau
threefoldswasheldattheFieldsInstituteandUniversityofTorontofromAugust16
to25,2011.TheworkshopwasorganizedbyCharlesF.Doran(Alberta),Shigeyuki
Kondo¯ (Nagoya), Radu Laza (Stony Brook), James D. Lewis (Alberta), Matthias
Schu¨tt(Hannover),andNorikoYui(Kingston/Fields).
Thisproceedingsvolumeforthe 2011Calabi–YauworkshopiseditedbyRadu
Laza, Matthias Schu¨tt, and Noriko Yui. The editors wish to express their appreci-
ationtoallthecontributorsforpreparingtheirmanuscriptsfortheFieldsCommu-
nications Series, which required extra effort in presenting not only current devel-
opmentsbutalsosomebackgroundmaterialonthediscussedtopics.Allpapersin
thisvolumewerepeer-reviewed.Wearedeeplygratefultoalltherefereesfortheir
effortsinevaluatingthearticles,inparticular,inthelimitedtimeframe.
The workshop was financially supported by various organizations. In addition
to the Fields Institute, the workshop was supported by NSF (grant no. 1100007),
JSPS(Grant-in-Aid(S),No.22224001),andDFG(GRK1463“Analysis,Geometry
and String Theory”). Additionally, several participants have used their individual
grants (e.g., NSF or NSERC) to cover their travel expenses. We are thankful to
all these sponsor organizations:their support made possible the participation of a
largenumberofjuniorparticipantsandofasignificantnumberofresearchersfrom
outsideNorthAmerica.Thisinturnledtoaverydynamicandactiveworkshop.
Some of the articles were copy-edited by Arthur Greenspoon of Mathematical
Reviews.Theeditorsaregratefulforhishelptowardsimprovingboththestylistic
andmathematicalpresentations.
Lastbutnotleast,wethankDebbieIscoeoftheFieldsInstituteforherhelpinre-
formattingarticlesintheSpringerstyleandassemblingthisvolumeforpublication.
StonyBrook,NY RaduLaza
Hannover,Germany MatthiasSchu¨tt
Kingston,ON NorikoYui
v
Introduction
In recent years, research in K3 surfaces and Calabi–Yau varieties has seen
spectacular progress from both arithmetic and geometric points of view, which in
turn continuesto have a huge influence and impact on theoreticalphysics, in par-
ticular,onstringtheory.Theworkshopwasdesignedtobringtogetherexpertsand
junior researchers who are aspiring to become experts for 10 days at the Fields
InstituteinAugust2011toreviewrecentdevelopments,inspiregraduateandpost-
doctoral fellows and young researchers, and also explore future directions of the
subjects.With114(officiallyregistered)participants,therewasawidegeographical
representation,with a very significantpresence of Europeanand Japanese partici-
pants(inadditiontoUSandCanadianparticipants).
The workshopstarted with a 3-dayintroductorysession aimed at graduatestu-
dentsandpostdoctoralfellows,followedbya1weekresearchconferencewithSun-
day off. The introductory lectures were intended to give some background and a
briefoverviewofthevasttopicofCalabi–YauvarietiesandK3surfaces.Atthesub-
sequentresearchconference,therewereintotal35researchtalkspresentedonwide
rangesoftopicsaroundK3,Enriquesandothersurfaces,andCalabi–Yauthreefolds
andhigher-dimensionalvarietiesandmanifolds,someofwhichcanbefoundinthis
volume.
As a consequenceof the significantinterest in the subject, we are organizinga
follow-up extended concentration period on Calabi–Yau varieties, in the form of
a semesterlongthematicprogramCalabi–Yauvarieties:arithmetic,geometryand
physics at the Fields Institute (July to December 2013). This thematic programis
devotedtothearithmeticandgeometryofCalabi–Yauvarietiesandtheconnections
tophysics,especiallystringtheory.
vii
viii Introduction
Scientific Focus oftheWorkshop
The workshopconcentratedon topics (on the geometryand arithmetic of Calabi–
Yauvarieties)thathaveeitherseengreatprogressrecentlyorshownahighpotential
forfutureinventions.Specifically,themajortopicscoveredincluded:
1. Families and degenerations of Calabi–Yau varieties—moduli theoretic and
arithmeticviewpoints.
2. Modularity: Galois representations of Calabi–Yau varieties and their connec-
tions to automorphic forms, in particular to classical, Hilbert, and Siegel
modularforms.
3. Calabi–Yau varieties of CM type and with special automorphisms, especially
K3surfaceswithsymplecticandnon-symplecticautomorphisms.
4. Algebraic cycles and motives: divisors, CM cycles, and motives arising from
K3surfacesandCalabi–Yauthreefolds.
5. Variations of mixed Hodge structures, periods, and Picard–Fuchs differential
equations.
OverviewofThis Volume
In the following paragraphswe give a brief overviewof the volume. There are in
total24articles.Someofthearticlesarewritten-upversionsofthetalkspresentedat
theworkshop,whileothersreportonsubsequentdevelopmentsonthesubjectmatter
oftheworkshop.Roughlythearticlesaredividedintothreecategories,namely:
(cid:129) Introductorylectures
(cid:129) ArithmeticandgeometryofK3,Enriquesandothersurfaces
(cid:129) Arithmetic and geometry of Calabi–Yau threefolds and higher dimensional
varieties
The workshop’sprogramcontained severalother talks on related topics. Addi-
tionaldocumentationfromtheworkshopisavailableonthehomepagemaintained
bytheFieldsInstitute.1
IntroductoryLectures
There are four survey papers by Kondo¯, Lewis, Schu¨tt, and Yui which comprise
a selection of the lectures given by the organizers during the 3-day introductory
periodoftheworkshop.Theselecturesweremostlyaimedatjuniorparticipantsof
the workshop and often geared specifically towards some of the talks to be given
1http://www.fields.utoronto.ca/programs/scientific/11-12/CalabiYau/index.html.
Introduction ix
duringtheresearchconference.Similarlyforthisvolume,thesurveypaperscanbe
usedasastartingpointorasaguidetothesubject.
ThesurveysbyKondo¯ andSchu¨ttreviewthegeometryandarithmeticofK3sur-
faces,includingbasicssuchaslatticetheory.Yui’spaperpresentsthecurrentstatus
on modularity of Calabi–Yau varieties in its different incarnations. The focus lies
onCalabi–Yauvarietiesof dimensionat mostthree.Lewisreviewstranscendental
aspects of algebraic cycles and, specializing to the Calabi–Yau situation, explains
somerecentdevelopmentsinthefield.
ArithmeticandGeometryofK3,Enriques,andOtherSurfaces
Acommonthemeofmanypapersinthissectionareellipticfibrations.Mostnotably,
BertinandLecacheuxclassifyallellipticfibrationsonaspecificK3surface.Froma
similarellipticmodularsurface,AnemaandTopderiveexplicitalgebraiccoverings
ofapointedtorus.Onthemoduliside,BesserandLivne´ relatespecificellipticK3
surfacestoabeliansurfaceswithquaternionicmultiplication;thisproducesexplicit
Shimura curves. Special cycles in moduli spaces of lattice polarized K3 surfaces
aretreatedbyKudla.ThesehigherNoether–Lefschetzlociaretheinputofcertain
generatingserieswhosemodularityisknown.
EllipticfibrationsformalsothekeyingredientforKerr’sapproachestothecom-
putationoftranscendentalinvariantsofindecomposablealgebraicK classes.More-
1
overKerr’sworkbuildsontoricgeometrywhichfeaturesprominentlyinthepaper
ofWhitcheretal.aswell.Herethree-dimensionalreflexivepolytopeswithS sym-
4
metryarerelatedtonaturalone-parameterfamilyofK3surfaceswithsymplecticS
4
action.Picard–FuchsequationsarestudiednotonlyinthispaperbutalsobyGa¨hrs
forcertainone-parameterfamiliesassociatedwithinvertiblepolynomials,usingthe
GKZsystem.
Oguiso’s paper is concerned with a classical problem: it proves that there is a
smoothquarticK3surfaceautomorphismthatisnotderivedfromaCremonatrans-
formation.AlmostasclassicalaproblemforEnriquessurfaces,Dolgachevextends
resultsforcohomologicallyornumericallytrivialautomorphismstoarbitrarychar-
acteristics. Contraryto previousapproaches,the key toolis Lefschetz’fixed point
formula. Enriques surfaces of Hutchinson–Go¨pel type are investigated by Mukai
and Ohashi. Starting from the projective geometry of Jacobian Kummer surfaces,
they give a sextic presentationfor these Enriquessurfaces and then describe their
intrinsicsymmetries.
Onthelessclassicalside,HulekandPloogextendthetheoryofFourier–Mukai
partnerstoincludethepresenceofpolarizations,producingacountingformulafor
thenumberofpartners.Inadifferentdirection,Schoen’spapercomputesinvariants
ofthedegenerationofaproductofellipticcurvesuponsplit-muliplicativereduction.
Heexpatiatesondivisorclassgroup,(co)homology,andPicardgroupoftheclosed
fibers. To close the circle on fibrations, Heijne and Kloosterman study a special
classofsurfaces,theso-calledDelsartesurfaces.Singledoutbytheiraccessibility
toexplicitcomputations,aclassificationofsomespecificfibrationsonthesesurfaces
isgiven.
