Thismonographresultsfromtheauthor'slecturesattheETHduringtheSpring Semester of 1997, when he was presenting a Nachdiplom course on Kahler­ Einsteinmetricsincomplexdifferentialgeometry. Therehasbeenfundamentalprogressincomplexdifferentialgeometryinthe last two decades. The uniformization theory ofcanonicalKahler metrics has beenestablishedinhigherdimensions. Manyapplicationshavebeenfound. One manifestationofthis istheuseofCalabi-Yauspacesinthesuperstringtheory. Theaimofthismonographistogiveanessentiallyself-containedintroduc­ tiontothetheoryofcanonicalKahlermetricsoncomplexmanifolds.Itisalso theauthor'shopetopresentthereaderswithsomeadvancedtopicsincomplex differentialgeometrywhicharehardtobefoundelsewhere. Thetopicsinclude Calabi-Futakiinvariants,ExtremalKahlermetrics,theCalabi-Yautheoremon existenceofKahler Ricci-flat metrics, and recent progresson Kahler-Einstein metricswithpositivescalarcurvature. ApplicationsofKahler-Einsteinmetrics totheuniformizationtheoryarealsodiscussed. Readers with a good general knowledge in differential geometry and par­ tial differential equations should be able to understand the materials in this monograph, I would like tothanktheETH for theopportunityto deliver the lectures in a very stimulating environment. In particular, I thank Meike Akveld for her patience and efficiency in taking notes ofthe lectures and producing the beautifulJb.1EXfile. Withoutherefforts,thismonographcouldneverhavebeen as it is now. I would also like to thank Ms. Nini Wong for her endless pa­ tienceinproof-readingandcorrectingnumeroustyposinearlierversionsofthis monograph. PartofmyworkinvolvedinthismonographwassupportedbyNationalSci­ enceFoundationGrantsDMS-9303999andDMS-9802479,atCourantInstitute ofMathematical Sciencesand MassachusettsInstituteofTechnology. My re­ searchwas alsosupportedbyaSimonsChairFundatMassachusettsInstitute ofTechnology. MIT, April 1999. GangTian Chapter1 IntroductiontoKahlermanifolds 1.1 Kahlermetrics LetM beacompact Coo manifold. ARiemannianmetricgonM isasmooth sectionofT*M @T*M definingapositivedefinitesymmetricbilinearformon TxM for each x E M. In localcoordinatesXl,...,X ,onehas anaturallocal n basis -iL,...,jL forTM, then g is represented by asmooth matrix-valued UXI UX n function {gij},where gij=g(a~i'a~j) . Notethat{gij} ispositivedefinite. Thepair(M,g) isusuallycalledaRieman­ nianmanifold. RecallthatanalmostcomplexstructureJ onM isabundleautomorphism ofthetangentbundleTM satisfyingj2 = - id. Definition1.1 The Nijenhuis tensorN(J) :TM xTM-+TM isgiven by N(v,w) = [v,w]+J[Jv,w]+J[v,Jw]- [Jv,Jw] forv,w vectorfields onM. Analmostcomplexstructure J on M iscalledintegrableifthere isa holo­ morphicstructure(thatisasetofchartswithholomorphictransitionfunctions) such that J corresponds to the induced complex multiplication in TM x C. Clearly,anycomplexstructureinducesanintegrablealmostcomplexstructure. The following theorem is due to Newlander and Nirenberg, see for example Appendix8in [14].