The generalized geometry, equivariant [formula omitted]-lemma, and torus actions

Abstract

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 ∂ ̄ ∂ -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 ∂ ̄ ∂ -lemma, namely the ∂ ̄ 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 P n and C P n blown up at a fixed point.