Generalized Calabi-Yau manifolds(26)

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

liesinthesamecohomologyclass.

ThenaturalsymmetrygroupoftheproblemisthenthegroupextensionG

2exact→G→Di 0(M).

WewanttodeterminewhenageneralizedCalabi-Yaumanifoldisde nedaccordingtoTheorem6byaMorse-Bottcriticalpoint–non-degeneratetransversetotheorbitsofthegroupG.Weconsiderthetangentspacetothisorbitnext.

Theactionofavector eldonρisjusttheLiederivative

LXρ=dι(X)ρ+ι(X)dρ=d(ι(X)ρ)

sinceρisclosed.Thein nitesimalactionofanexactB- eldB=dξis

dξ∧ρ=d(ξ∧ρ).

ThusthetangentstoanorbitofGatρareforms

ρ˙=d(ι(X)ρ+ξ∧ρ)=d((X+ξ)·ρ)(16)

Becauseoftheinvarianceofthefunctional,ifαisexactandβ=d(ι(X)ρ+ξ∧ρ),thenH(α,β)=0.Supposeconverselythattheexactformβ=dτhasthepropertythatH(α,β)=0forallexactformsα=dψ,thenfrom(15), Jdψ,dτ =± ψ,dJdτ =0

MM

forallψsothat

dJdτ=0.

Thustransversenondegeneracyisequivalenttothefollowingproperty:

De nition5AgeneralizedCalabi-YaumanifoldissaidtosatisfytheddJ-lemmaif

dJdτ=0 dτ=d(ι(X)ρ+ξ∧ρ)

foravector eldXand1-formξ.

Thisconditionmaynotalwaysbesatis ed.Herearetwocaseswhenitis:

Proposition7TheddJ-lemmaholdsif:

a)thegeneralizedCalabi-Yaumanifoldisacomplex3-manifoldwithanonvanishing¯-lemma,orholomorphic3-formandwhichsatis esthe

b)itisasymplectic6-manifoldsatisfyingthestrongLefschetzcondition.

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