Feynman motives of banana graphs(19)

Theclassc(V)residesnaturallyintheChowgroupA V.Forthepurposeofthispaper,thereaderwillmissnothingbyreplacingA Vwithordinaryhomology.

TheChernclassofavarietyVisaclassofevidentgeometricsigni cance:forex-ample,thedegreeofitszero-dimensionalcomponentagreeswiththetopologicalEulercharacteristicofV.ThisfollowsessentiallyfromthePoincar´e-Hopftheorem: c(TV)∩[V] =χ(V).

ItisnaturaltoaskwhetherthereareanalogsoftheChernclassde nedforpossiblysingularvarieties,forwhichatangentbundleisnotnecessarilyavailable.

Somewhatsurprisingly,one ndsthatthereareseveralpossiblede nitions,each‘natu-ral’fordi erentreasons,andallagreeingwitheachotherinthenonsingularcase.IfXisacompleteintersectioninanonsingularvarietyV,itisreasonabletoconsidertheFultonclassc(TV)cvir(X):=

behavesastheclassofa‘virtualtangent

bundle’toX.Itsde nitioncaninfactbeextended(andinmorethanoneway)toarbitraryvarieties,see§4.2.6in[18].

Theclasscvir(X)isinasenseuna ectedbythesingularitiesofX:forahypersurfaceXinanonsingularvarietyV,itisdeterminedbytheclassofXasadivisorinV.

Amuchmorere nedinvariantistheChern-Schwartz-MacPherson(CSM)classofX,whichdependsmorecruciallyonthesingularitiesofX,andwhichwewilluseasameasureofthesingularitiesbycomparisonwithcvir(X).

Thenameoftheclassretainssomeofitshistory.Inthemid-60s,M.-H.Schwartz([29],[30])introducedaclassextendingtosingularvarietiesPoincar´e-Hopf-typeresults,bystudyingtangentframesemanatingradiallyfromthesingularities.IndependentlyofSchwartz’work,GrothendieckandDeligneconjecturedatheoryofcharacteristicclasses ttingatightfunctorialprescription,andintheearly70sR.MacPhersonconstructedaclasssatisfyingthisrequirement([25]).ItwaslaterprovedbyJ.-P.BrasseletandM.-H.Schwartz([14])thattheclassesagree.

InthispaperwedenotetheChern-Schwartz-MacPhersonclassofasingularvarietyXsimplybyc(X)(thenotationcSM(X)isfrequentlyusedintheliterature).c(NXV)

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