The generalized geometry, equivariant ∂̄∂-lemma, and torus actions

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an HH-twisted generalized complex manifold MM. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold MM satisfies the ∂̄∂-lemma, we prove that they are both canonically isomorphic to (Sg∗)G⊗HH(M)(Sg∗)G⊗HH(M), where (Sg∗)G(Sg∗)G is the space of invariant polynomials over the Lie algebra gg of GG, and HH(M)HH(M) is the HH-twisted cohomology of MM. Furthermore, we establish an equivariant version of the ∂̄∂-lemma, namely the ∂̄G∂-lemma, which is a direct generalization of the dGδdGδ-lemma [Y. Lin, R. Sjamaar, Equivariant symplectic Hodge theory and dGδdGδ-lemma, J. Symplectic Geom. 2 (2) (2004) 267–278] for Hamiltonian symplectic manifolds with the Hard Lefschetz property.Second we consider the torus action on a compact generalized Kähler manifold which preserves the generalized Kähler structure and which is equivariantly formal. We prove a generalization of a result of Carrell and Lieberman [J.B. Carrell, D.I. Lieberman, Holomorphic vector fields and compact Kähler manifolds, Invent. Math. 21 (1973) 303–309] in generalized Kähler geometry. We then use it to compute the generalized Hodge numbers for non-trivial examples of generalized Kähler structures on CPnCPn and CPnCPn blown up at a fixed point.