资源描述：
《lectures on calabi-yau and special lagrangian geometry - dominic joyce》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、LecturesonCalabi{YauandspecialLagrangiangeometryDominicJoyceLincolnCollege,Oxford,OX13DR1IntroductionCalabi{Yaum-foldsarecompactRicci-
atK•ahlermanifolds(M;J;g)ofcomplexdimensionm,withtrivialcanonicalbundleKM.Takentogether,thecomplexstructureJ,K•ahlermetricg,
2、andaholomorphicsectionofKMmakeuparich,fairlyrigidgeometricalstructurewithveryinterestingproperties
3、forinstance,Calabi{Yaum-foldsoccurinsmooth, nite-dimensionalmodulispaceswithknowndimension.UsingAlgebraicGeometryandYau'ssolutionoftheCalabiConjecture,onecancon
4、structCalabi{Yaum-foldsinhugenumbers.StringTheorists(aspeciesoftheoreticalphysicist)areveryinterestedinCalabi{Yau3-folds,andhavemadesomeextraordinaryconjecturesaboutthem,inthesubjectknownasMirrorSymmetry.SpecialLagrangianm-folds(SLm-folds)areadistinguishedcla
5、ssofrealm-dimensionalminimalsubmanifoldsNinCalabi{Yaum-foldsM.Theyarefairlyrigidandwell-behaved,sothatcompactSLm-foldsoccurinsmoothmodulispacesofdimensionb1(N),forinstance.TheexistenceofcompactspecialLagrangianm-foldsingeneralCalabi{Yaum-foldsisnotwellunderst
6、ood,buttherearereasonstobelievetheyareveryabundant.TheyareimportantinStringTheory,andareexpectedtoplayaarXiv:math.DG/0108088v29Nov2001r^oleintheeventualexplanationofMirrorSymmetry.Inthispaperweaimtodothreethings.Firstly,weintroduceCalabi{Yaumanifolds,goingvia
7、RiemannianholonomygroupsandK•ahlergeometry.Secondly,weintroducespecialLagrangiansubmanifolds,goingviacalibratedgeometry.Finally,wesurveyanareaofcurrentresearchonthesingularitiesofSLm-folds,anditsapplicationtoMirrorSymmetryandtheSYZConjecture.Exercisesaregiven
8、attheendofeachsection.IhopethepaperwillbeusefultograduatestudentsinGeometry,StringTheorists,andotherswhowishtolearnthesubject.Apartfromthelasttwosections,thepaperisintendedasastraightforwardexpositionofstandardmate-rial,totakethereaderfromastartingpointofagoo
9、dbackgroundinDi erentialGeometryasfarastheboundariesofcurrentresearchinanexcitingarea.OurapproachtoCalabi{Yaum-foldsisresolutelyDi erentialGeometric.1Thatis,weregardthemassmoothrealmanifo
