Generalized Calabi-Yau manifolds(23)

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphis

6

6.1ThevariationalproblemThevolumefunctional

Wede nedabovethefunctionφonU Λev/odV (Λ6V)1/2.Untwistingbytheone-dimensionalvectorspace(Λ6V)1/2,thereisacorrespondingopenset,whichwestillcallU,inΛev/odV and,sinceφishomogeneousofdegree2,aninvariantfunction

φ:U→Λ6V .

ThebilinearsymplecticformnowtakesvaluesinΛ6V alsoandsothederivativeatρofφisalinearmapfromΛev/odV toΛ6V whichcanbewritten

Dφ(ρ˙)= ρ ,ρ˙ .(13)

SupposeMisacompactoriented6-manifold,andρisaform,eitheroddoreven,butingeneralofmixeddegree,whichliesateachpointofMintheopensubsetUdescribedabove.Following[11]weshallcallsuchaformstable.Wecanthende neavolumefunctional V(ρ)=φ(ρ).

M

Theorem6Aclosedstableformρ∈ ev/od(M)isacriticalpointofV(ρ)initscohomologyclassifandonlyifρ+iρ de nesageneralizedCalabi-YaustructureonM.

Proof:Takethe rstvariationofV(ρ):

δV(ρ˙)=Dφ(ρ˙)= ρ ,ρ˙

MM

from(13).Thevariationiswithina xedcohomologyclasssoρ˙=dα.Thus

δV(ρ˙)= ρ ,dα =σ( ρ)∧dα

MM

from(4).ByStokes’theoremthisis

±dσ( ρ)∧α=±σ(dρ )∧α=± dρ ,α

MMM

sincefromitsde nitionσcommuteswithd.

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