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 â Ì â -lemma, we prove that they are both canonically isomorphic to ( S g â ) G â H H ( M ) , where ( S g â ) 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 â Ì â -lemma, namely the â Ì G â -lemma, which is a direct generalization of the d G ÎŽ -lemma [Y. Lin, R. Sjamaar, Equivariant symplectic Hodge theory and d G ÎŽ -lemma, J. Symplectic Geom. 2 (2) (2004) 267â278] for Hamiltonian symplectic manifolds with the Hard Lefschetz property. Second we consider the torus action on a compact generalized KĂ€hler manifold which preserves the generalized KĂ€hler 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 KĂ€hler manifolds, Invent. Math. 21 (1973) 303â309] in generalized KĂ€hler geometry. We then use it to compute the generalized Hodge numbers for non-trivial examples of generalized KĂ€hler structures on C P n and C P n blown up at a fixed point.
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