The generalized geometry, equivariant over(∂, ̄) ∂-lemma, and torus actions

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an H-twisted generalized complex manifold M. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold M satisfies the over(∂, ̄) ∂-lemma, we prove that they are both canonically isomorphic to (S g^*)^G ⊗ H_H (M), where (S g^*)^G is the space of invariant polynomials over the Lie algebra g of G, and H_H (M) is the H-twisted cohomology of M. Furthermore, we establish an equivariant version of the over(∂, ̄) ∂-lemma, namely the over(∂, ̄)_G ∂-lemma, which is a direct generalization of the d_G δ-lemma [Y. Lin, R. Sjamaar, Equivariant symplectic Hodge theory and d_G δ-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 C Pⁿ and C Pⁿ blown up at a fixed point.