Feynman motives of banana graphs(23)

We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the c

16ALUFFIANDMARCOLLI

whereT=[Gm]istheclassofthemultiplicativegroup,see§

3.ThentheclassofXintheGrothendieckgroupofvarietiesequals [X]=aiTi,Proposition2.2.LetXbeasubsetofprojectivespaceobtainedbyunions,intersections,di erencesoflinearlyembeddedsubspaces.Withnotationasabove,assume c(X)=aiH i.

Thus,adoptingavariableT=H 1intheCSMenvironment,andT=TintheGrothendieckgroupenvironment,theclassescorrespondingtosubsetsasspeci edinthestatementwouldmatchprecisely.

Proof.TheformulaholdsforalinearlyembeddedX=Pr,since

c(P)=((1+H)rr+1 Hr+1)·[P] =((1+H)rr+1 Hr+1)·H r=(1+H 1)r+1 1

.T

SinceembeddedCSMclassesandclassesintheGrothendieckgroupbothsatisfyinclusion-exclusion,thisrelationextendtoallsetsobtainedbyordinaryset-theoreticoperationsperformedonlinearlyembeddedsubspaces,andthestatementfollows.

Proposition2.2applies,forexample,tothecaseofhyperplanearrangementsinPN:forahyperplanearrangement,theinformationcarriedbytheclassintheGrothendieckgroupofvarietiesispreciselythesameastheinformationcarriedbytheembeddedCSMclass.Theseclassesre ectinasubtlewaythecombinatoricsofthearrangement.

Inamoregeneralsetting,itisstillpossibletoenhancetheinformationcarriedbytheCSMclassinsuchawayastoestablishatightconnectionbetweenthetwoenvironments.Forexample,CSMclassescanbetreatedwithinaframeworkwithstrongsimilaritieswithmotivicintegration,[3].

Inanycase,oneshouldexpectthat,inmanyexamples,theworkneededtocomputeaCSMclassshouldalsoleadtoacomputationofaclassintheGrothendieckgroup.Thecomputationsin§3and§4inthispaperwillcon rmthisexpectationforthehypersurfacescorrespondingtobananagraphs.

3.Bananagraphsandtheirmotives

Inthissectionwegiveanexplicitformulafortheclasses[XΓn]ofthebananagraph

hypersurfacesXΓnintheGrothendieckring.Theprocedureweadopttocarryoutthe

computationisthefollowing.WeusetheCremonatransformationof(1.17).ConsiderthealgebraicsimplexΣnplacedinthePn 1ontheright-hand-sideofthediagram(1.17).ThecomplementofthisΣninthegraphhypersurfaceXΓnisisomorphictothecomplementof

thesameunionΣninthecorrespondinghyperplaneLinthePn 1ontheleft-hand-sideof(1.17),byLemma1.7above.Sothisprovidestheeasypartofthecomputation,andonethenhastogiveexplicitlytheclassesoftheintersectionsofthetwohypersurfaceswiththeunionofthecoordinatehyperplanes.The nalformulafortheclass[XΓn]hasa

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