on the boundary of the moduli spaces of log hodge structures triviality of the torsornew

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1、ONTHEBOUNDARYOFTHEMODULISPACESOFLOGHODGESTRUCTURES:TRIVIALITYOFTHETORSORTATSUKIHAYAMAAbstract.InthispaperwewillstudythemodulispacesoflogHodgestruc-turesintroducedbyKato-Usui.Thismodulispaceisapartialcompactificationofsomediscretequotientofaperioddomain.Inthecasewheretheperioddo

2、mainisHermitiansymmetric,themodulispaceiswell-knownbecauseitisatoroidalpartialcompactificationintroducedbyMumfordetal.Ontheotherhand,withouttheassumptionofHermitiansymmetricproperty,themodulispaceislesswellunderstood.Thepurposeofthispaperistoshowadistinc-tionbetweenthecaseswher

3、etheperioddomainisHermitiansymmetricandotherwise.Thisfeatureisappearedintrivialityofthetorsor.1.IntroductionLet(H0,F0,h,i0)beapolarizedHodgestructure,andletDbethecorrespond-ingperioddomain.DisahomogeneousspaceforarealLiegroupGandalsoaopenG-orbitintheflagmanifold.ForasubgroupΓof

4、GZ:=Aut(H0,h,i0),ΓactsonDproperlydiscontinuouslyandΓDisanon-compactanalyticspace.Kato-UsuiintroducedpartialcompactificationsofΓDandshowfundamentalpropertiesofthem.Inthispaperwewillstudygeometricstructuresofthesespacesunderthefollowingsituations:(1)DisaHermitiansymmetricspace.

5、(2)Thecase:w=3andhp,q=1(p+q=3,p,q≥0).ForExample,algebraiccurvesorK3surfacescorrespondtothecase(1).Quinticthreefoldsconsideredin[M]correspondtothecase(2).Inthecase(2),Usui([U2])showsglobaltorellitheoremforthequintic-mirrorfamily.1.1.PartialcompactificationsofΓDbyKato-Usui.Apart

6、ialcompactifica-tionofΓDbyKato-Usuiisroughlyaspaceof“nilpotentorbits”uptoΓ.Herea“nilpotentorbit”isdefinedbya“nilpotentcone”ing=Lie(G)andthesubsetofarXiv:0806.4965v1[math.AG]30Jun2008thecompactdualDˇsatisfyingthesomeconditions(seeDefinition3.1).Thereasonwhyweconsideraspaceofnilpo

7、tentorbitsasaextendedperioddomainisrelatedtonilpotentorbittheorem([S]):foraperiodmapϕonapunctureddiskwhosemonodromygroupisgeneratedbyaunipotentelement,alifting˜ϕ(z)isapproxi-matedbyexp(zN)FasImz→∞,whereNistherationalnilpotentelementandFistheelementofDˇ.Infactweadd(R≥0N,exp(CN)

8、F)asapointofboundarytoD.Letσbeanilpotentcone.WedefineDσ:={(τ,Z