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1、ATIYAHANDTODDCLASSESARISINGFROMINTEGRABLEDISTRIBUTIONSZHUOCHEN,MAOSONGXIANG,ANDPINGXUABSTRACT.Inthispaper,westudytheAtiyahclassandToddclassoftheDGmanifold(F[1];dF)corre-spondingtoanintegrabledistributionFTKM=TMRK,whereK=RorC.Weshowthatthesetwoclassesarecanonicallyidenticaltothoseof
2、theLiepair(TKM;F).Asaconsequence,theAtiyahclass0;1ofacomplexmanifoldXisisomorphictotheAtiyahclassofthecorrespondingDGmanifold(T[1];@).XMoreover,ifXisacompactKahlermanifold,thentheToddclassof¨XisalsoisomorphictotheToddclassof0;1thecorrespondingDGmanifold(T[1];@).XCONTENTS1.Introduc
3、tion12.CohomologyofDGmanifoldsoutofintegrabledistributions42.1.Thecontractiondata42.2.ProofofTheoremA102.3.Application103.AtiyahandToddclasses133.1.AtiyahandToddclassesofDGmanifolds133.2.AtiyahandToddclassesoftheLiepair(TKM;F)133.3.Splittingsandconnections143.4.ProofofTheoremB15Refe
4、rences18arXiv:1711.11253v2[math.DG]27Jan20181.INTRODUCTIONAtiyahclasseswereintroducedbyAtiyah[1]tocharacterizetheobstructiontotheexistenceofholomorphicconnectionsonaholomorphicvectorbundleoveracomplexmanifold.In[5],KapranovshowedthattheAtiyahclassX2H1(X;T_End(TX))ofaKahlermanifold¨X
5、inducesanL1algebrastructureonX0; 1X(TX)(see[7,8]forthecomplexmanifoldcase),whichplaysanimportantroleintheformalismofRozansky-Wittentheory.Inhisseminarwork[6],KontsevichindicatedadeeplinkbetweentheToddgenusofcomplexmanifoldsandtheDufloelementofLiealgebras.See[9]fortheformalitytheorem
6、forsmoothDGmanifolds,whichimpliestheKontsevich-DuflotheoremforLiealgebrasandKontsevich’stheoremforResearchpartiallysupportedbyNSFgrantsDMS-1406668andDMS-1707545.12ZHUOCHEN,MAOSONGXIANG,ANDPINGXUcomplexmanifolds[6]underaunifiedframework.InspiredbytheworkofKapranovandKontsevich,Chen,Sti
7、enonandXuintroducedtheAtiyahclassofLiealgebroidpairsin[´3],whichencodesboththeAtiyahclassofcomplexmanifoldsandtheMolinoclass[14]offoliationsasspecialcases.Towardsadifferentdirection,Mehta,StienonandXu[´13]introducedAtiyahclassesandToddclassesofDGmanifolds.TheDufloelementofaLiealgebra
8、gisidentifiedwiththe
