Generalized Geometry, Equivariant ∂∂-lemma, and Torus Actions

In this paper we first consider the Hamiltonian action of a compact connected Lie group on anH-twisted generalized complexmanifold M. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifoldM satisfies the ∂∂-lemma, we prove that they are both canonically isomorphic to (Sg)⊗HH(M), where (Sg ) is the space of invariant polynomials over the Lie algebra g ofG, andHH(M) is the H-twisted cohomology of M. Furthermore, we establish an equivariant version of the ∂∂-lemma, namely ∂G∂-lemma, which is a direct generalization of the dGδ-lemma [LS03] 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 [CL73] in generalized Kähler geometry. We then use it to compute the generalized Hodge numbers for non-trivial examples of generalized Kähler structures onCP and CP blown up at a fixed point.