Moduli of cubic surfaces and Hodge theory

Arnaud Beauville

Description

Niveau: Supérieur, Doctorat, Bac+8
Moduli of cubic surfaces and Hodge theory [After Allcock, Carlson, Toledo] Arnaud BEAUVILLE Introduction This is a detailed version of three lectures given at the annual meeting of the Research Group “Complex Algebraic Geometry” at Luminy in October 2005. The aim was to explain, in a way as elementary as possible, the work of Allcock, Carlson, Toledo [ACT] which describes, using Hodge theory, the moduli space of cubic surfaces in P3 as a quotient of the complex ball in C4 . That work uses a number of different techniques which are quite basic in algebraic geometry: Hodge theory of course, monodromy, differential study of the period map, geometric invariant theory, Torelli theorem for the cubic threefold . . . One of our aims is to explain these techniques by illustrating how they work in a concrete and relatively simple situation. As a result, these notes are quite different from the original paper [ACT]. While that paper contains a wealth of interesting and difficult results (on the various moduli spaces which can be considered, the corresponding monodromy group, their description by generators and relations), we have concentrated on the main theorem and the basic methods involved, at the cost of being sometimes sketchy on the technical details of the proof. We hope that these notes may serve as an introduction to this nice subject.

• linear


• surfaces modulo


• v? ???


• moduli space


• group pgl


• hermitian unimodular

Sujets

• Beauville
• Organizational studies
• Linear
• Moduli space

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 19
• Langue: English

Extrait

ModuliofcubicsurfacesandHodgetheory [AfterAllcock,Carlson,Toledo] ArnaudB EAUVILLE Introduction Thisisadetailedversionofthreelecturesgivenattheannualmeetingofthe ResearchGroup“ComplexAlgebraicGeometry”atLuminyinOctober2005.The aimwastoexplain,inawayaselementaryaspossible,theworkofAllcock,Carlson, Toledo[ACT]whichdescribes,usingHodgetheory,themodulispaceofcubicsurfaces inP 3 asaquotientofthecomplexballinC 4 .Thatworkusesanumberofdifferent techniqueswhicharequitebasicinalgebraicgeometry:Hodgetheoryofcourse, monodromy,differentialstudyoftheperiodmap,geometricinvarianttheory,Torelli theoremforthecubicthreefold...Oneofouraimsistoexplainthesetechniquesby illustratinghowtheyworkinaconcreteandrelativelysimplesituation. Asaresult,thesenotesarequitedifferentfromtheoriginalpaper[ACT].While thatpapercontainsawealthofinterestinganddifficultresults(onthevarious modulispaceswhichcanbeconsidered,thecorrespondingmonodromygroup,their descriptionbygeneratorsandrelations),wehaveconcentratedonthemaintheorem andthebasicmethodsinvolved,atthecostofbeingsometimessketchyonthe technicaldetailsoftheproof.Wehopethatthesenotesmayserveasanintroduction tothisnicesubject. Inthenextsectionwewillmotivatetheconstructionbydiscussingamore complicatedbutmoreclassicalcase,namelyquarticsurfacesinP 3 .In § 2wewill explainthemainresult;attheendofthatsectionwewillexplainthestrategyof theproof,andatthesametimetheplanofthesenotes. 1.Motivation:thecaseofquarticsurfaces Asannounced,westartbyrecallingbrieflythedescriptionofthemodulispace ofquarticsurfacesinP 3 .Referencesinclude[BHPV],[X],or[B2]forashort introduction. (1.1)AquarticsurfaceS ⊂ P 3 isaK3surface,whichmeansthatitadmitsa unique(uptoascalar)holomorphic2-formω,whichisnon-zeroateverypoint. TheonlyinterestingcohomologyofSisthelatticeH 2 (S,Z),endowedwiththe unimodularsymmetricbilinearformdefinedbythecup-product.Moreoverthevector spaceH 2 (S,C)=H 2 (S,Z) ⊗ CadmitsaHodgedecomposition H 2 (S,C)=H 2,0 ⊕ H 1,1 ⊕ H 0,2 , 1

whichisdeterminedbythepositionofthelineH 2,0 =CωinH 2 (S,C)(wehave H 0,2 =H 2,0 ,andH 1,1 istheorthogonalofH 2,0 ⊕ H 0,2 ).ThepointisthatH 2 (S,C) dependsonlyonthetopologyofS,whilethepositionofH 2,0 =Cωdependsheavily onthecomplexstructure. Tobemoreprecise,wedenotebyLalatticeisomorphictoH 2 (S,Z)forallS. Wefixavectorh 0 ∈ Lofsquare4(theyareallconjugateunderO(L)).Amarked quarticsurfaceisapair(S,σ)ofaquarticSandanisometryσ:L − ∼ → H 2 (S,Z) suchthatσ(h 0 )=h,theclassinH 2 (S,Z)ofaplanesection.Wedenoteby M f themodulispaceofmarkedquarticsurfaces;itisnotdifficulttoseethatitisa complexmanifold.ThegroupΓofautomorphismsofLwhichfixh 0 actson M f byγ ∙ (S,σ)=(S,σ ◦ γ − 1 );thequotient M := M f /Γistheusualmodulispace ofquarticsurfaces,thatis,theopensubsetofP(H 0 (P 3 , O P 3 (4))parameterizing smoothquarticsurfacesmodulotheactionofthelineargroupPGL(4). (1.2)Theadvantageofworkingwith M f isthatwecannowcomparethe Hodgestructuresofdifferentsurfaces.Given(S,σ),weextendσtoanisomorphism L C − ∼ → H 2 (S,C)andput ℘˜(S,σ)=σ − 1 (H 2,0 )=σ − 1 ([ω]) ∈ P(L C ). Themap℘˜iscalledtheperiodmap,forthefollowingreason:chooseabasis (e 1 ,...,e 22 )ofL ∗ ,sothatL C isidentifiedwithC 22 .Putγ i = t σ − 1 (e i ),viewed asanelementofH 2 (S,Z);then ZZ  ℘˜(S,σ)=ω:...:ω ∈ P 21 . γγ 221 R Thenumbers γ i ωareclassicallycalledthe“periods”ofω. R Sinceωisholomorphicwehaveω ∧ ω=0,and S ω ∧ ω¯>0;moreover,since ωisoftype(2,0)andhoftype(1,1),wehaveω.h=0inH 2 (S,C).Inotherwords, ℘˜(S,σ)liesinthesubvarietyΩofP(L C ),calledtheperioddomain,definedby Ω= { [x] ∈ P(L C ) | x 2 =x.h 0 =0,x.x¯>0 } . TheactionofΓonL C preservesΩ,andthemap℘˜isΓ-equivariant.Thus wehaveacommutativediagram: M f −− ℘˜ −→ Ω    yy M−− ℘ −→ Ω/Γ. 2