Etudiant : ALOUANI MOHAMED RIDA

Filière : Master Mathématiques Pures (MMP)

Encadrant : Pr. KADAOUI ABBASSI MOHAMED TAHAR

Annèe : 2025

Résumé : IIt isnaturaltothinkthatgeometry,ormoregenerallymathematics,istooabstract,farfromreality,or evenawasteoftime,especiallywhentalkingaboutcomplicatedconceptslikeEinsteinmanifolds,Kähler manifolds, orCalabiYaumanifolds.Buttherealityisquitetheopposite. Einstein manifolds,Kählermanifolds,andCalabiYaumanifoldsarefundamentalmathematicalcon- cepts thatplayacrucialroleinseveralmodernphysicaltheories,eventhoughtheymayseemabstract and disconnectedfromeverydayreality.Itisnotjusttheory:theGPSinyourphone,forexample,would not workwithoutthisdeepunderstandingofspace-time. GPS satellitesorbittheEarthatveryhighspeedsandarefarfromthesurface,sotimepasses dierentlyforthemcomparedtousontheground.Thisdierence,calledtimedilation,ispreciselywhat Einstein's equationsrelatedtothecurvatureofspace-timepredict. If wedidnotcorrectthisdierenceusingcalculationsbasedonEinsteinmanifoldsandrelativity,the positionscalculatedbyGPSwouldquicklybecomeinaccurate,witherrorsreachingseveralkilometers within afewhours. Einstein manifoldsnaturallyariseinEinstein'sgeneralrelativity,wheretheyareusedtodescribethe curvatureofspace-time.Einstein'sequations, Rμν − 1 2 gμνR + gμνΛ = 8πG c4 Tμν showhowthegeometryoftheuniverseisrelatedtothedistributionofmassandenergy.Foranempty universe,theseequationssimplifyto: Rμν = ( 1 2 R − Λ)gμν whichconstitutestheconditionforanEinsteinmetric. Kähler manifolds,ontheotherhand,aremathematicalobjectsmainlyusedinquantummechanics and supersymmetrytheory.Theyareessentialformodelingcomplexquantumsystems.The L2-complete space ofdierentialformsonthemanifoldisconsideredasthequantumstatespace,andtheLaplacian acts asthesystem'sHamiltonian.Thisstatespaceisnaturallygradedby Z/2Z, separatingevenandodd degree forms.Onecanintroduceasuperalgebraofoperatorsactingonthisspacethatcommuteswith the Hamiltonian.Theseconceptsareessentialforresearchinsupersymmetry,whichseekstounifythe fundamentalforcesofnature. CalabiYaumanifoldsplayacentralroleinstringtheory,whichisanattempttoformulateaunied theory ofphysics,combiningbothgeneralrelativity(gravitation)andquantummechanics.Accordingto this theory,thefundamentalconstituentsoftheuniversearenotpointparticlesbutvibratingstrings, whose dierentvibrationalmodescorrespondtodierentparticles.Oneoftheessentialpostulatesof string theoryisthattheuniversemusthavetendimensionstobemathematicallyconsistent.However, in oureverydayexperience,weperceiveonlyfourdimensions(threeofspaceandoneoftime),which presentsanapparentcontradiction.Toresolvethis,physicistsproposethatthesixadditionaldimensions are compactied, meaningtheyarecurleduponthemselvesatanextremelysmallscaleontheorderof 10−34 metersand arethereforeundetectable.Theuniverseisthenmodeledasaproductspace: M×X, where M is thefour-dimensionalMinkowskispace-time,and X is aCalabiYaumanifoldwithsixreal dimensions (orthreecomplexdimensions).Thesemanifolds,whichsatisfyspecialgeometricconditions (notably havingvanishingRiccicurvature),allowforsupersymmetryanddirectlyinuencethephysical 2 propertiesofparticlesaftercompactication.Thus,althoughouruniverseappearsfour-dimensional, it couldinfactbeten-dimensional,withsixhiddendimensionsfoldedintotheintricategeometryof CalabiYauspaces. These threetypesofmanifolds,althoughtheoreticalandabstract,playacentralroleinmodern physics,particularlyinunderstandingspace-time,gravity,supersymmetry,andthestructureoftheuni- verseatextremelysmallscales.Studyingthesemanifoldsisfarfromawasteoftime;itisessentialto advanceourscienticknowledgeandfuturetechnologies. The mainobjectiveofthisworkistointroduceandexploreKählermanifolds,whicharefundamental objectsinRiemannianandcomplexgeometry,whileemphasizingtheircrucialroleintherelationship betweendierentialgeometryandalgebraicgeometry.Tobegin,wewillintroducethebasicconcepts: complex manifolds,Hermitianmanifolds,symplecticmanifolds,andKählermanifolds,focusingontheir geometric propertiesandthedeeplinksbetweenthemintheframeworkofRiemanniangeometry. A Kählermanifoldisacomplexmanifold M equippedwithaRiemannianmetric h that isHermitian with respecttothecomplexstructure J, meaningthatthemetricsatisesthefollowingcondition: h(JX,JY ) = h(X,Y ), for allvectorelds X and Y . The2-form ω, denedby ω(X,Y ) = h(JX,Y ) is calledtheKählerform.AnessentialpropertyofKählermanifoldsinRiemanniangeometryisthatthis form mustbeclosed,i.e., dω = 0. Thisconditionimpliesthatthecomplexstructure J is parallelwith respecttotheLevi-Civitaconnectionassociatedwiththemetric h. Thiscriteriondenesthestructure of aKählermanifoldandisfundamentalinthestudyofcomplexRiemannianmetrics. Kähler manifolds,althoughlocallymodeledoncomplexEuclideanspace,possessremarkablegeometric properties.ThecanonicalexampleofaKählermanifoldiscomplexprojectivespaceequippedwiththe FubiniStudymetric,whichisaKählerRiemannianmetric.Akeyfeatureofthesemanifoldsisthatany smoothcomplexsubmanifoldofaKählermanifoldnaturallyinheritsaRiemannianandKählerstructure. In particular,smoothcomplexprojectivemanifoldswiththeFubiniStudymetricprovidefundamental examples illustratingthestronginterplaybetweenRiemanniangeometryandcomplexalgebraicgeometry. WealsoexploreseveralaspectsofcompactKählermanifolds,especiallytheCalabiYauconjecture, whichisafundamentalproblemincomplexgeometry.ThisconjecturepredictstheexistenceofRicci- at Kählermetricsoncertainmanifoldsandplaysacentralroleinunderstandingcomplexgeometric structures. WediscussWeitzenböcktechniques,powerfultoolsusedinthestudyofcompactKähler manifolds thatallowanalyzingthegeometricandanalyticpropertiesofdierentialformsandassociated Laplace operators.Finally,wefocusonCalabiYaumanifolds,whicharecompactKählermanifolds possessingaRicci-atKählerform,meaningaKählerformwithzeroRiccicurvature. Tounderstandtheconstructionofthesemanifolds,itisessentialtoconsiderthenotionsofdivisorsand holomorphic linebundles.Adivisoronacomplexmanifold M is aformalsumofanalyticsubvarietiesof codimension1,denedlocallybymeromorphicfunctions.Toeachdivisor D, oneassociatesaholomorphic line bundle [D], whichisacomplexrank-1vectorbundleover M. Concretely,alinebundleisaber bundle over M whose berisacomplexline,andwhoselocaltransitionfunctionsarenon-vanishing holomorphic functions.Thislinkbetweendivisorsandlinebundlesiscentralinalgebraicgeometry, notably inthestudyofthecanonicalbundle KM, whichencodesfundamentalinformationaboutthe complex structureof M. A fundamentalresult,theKodairatheorem,statesthatanycompactprojectiveKählerCalabiYau manifold canbeembeddedintocomplexprojectivespace,meaningthesemanifoldsareinfactalgebraic varieties.Thisprojectivityallowsapplyingpowerfultoolsfromalgebraicgeometrytotheirstudy.For example, theFermatquintic,denedincomplexprojectivespace CP4 bytheequation {[z0, z1, z2, z3, z4] ∈ CP4 : z5 0 + z5 1 + z5 2 + z5 3 + z5 4 = 0}, A smoothhypersurface.Thankstotheadjunctionformula,itscanonicalbundle KX is trivial,meaning the associatedcanonicaldivisoriszero.ThispropertymakestheFermatquinticaclassicexampleof On kählerGeometry3AlouaniMohamedRida a 3-dimensionalCalabiYaumanifold.Thus,understandingdivisorsandlinebundlesnotonlyallows dening thesevarietiesbutalsostudyingtheirdeepgeometricandtopologicalproperties. On