Generalized geometry, equivariant $\bar{\partial }\partial$-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 $\bar{\partial }\partial$-lemma, we prove that they are both canonically isomorphic to $(S\g^*)^G\otimes H_H(M)$, 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 $\bar{\partial }\partial$-lemma, namely ${\bar{\partial }}_G\partial$-lemma, which is a direct generalization of the $d_G\delta$-lemma 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 in generalized Kähler geometry. We then use it to compute the generalized Hodge numbers for non-trivial examples of generalized Kähler structures on $\C¶^n$ and $\CP^n$ blown up at a fixed point.

to appear in the Journal of Geometry and Physics, 27 pages, a few typos and small mistakes corrected, added a few more references

References in corpus (8)

• Reduction of Courant algebroids and generalized complex structures
• Symmetries in generalized Kähler geometry
• Topological sigma-models with H-flux and twisted generalized complex manifolds
• Generalized geometry and the Hodge decomposition
• Reduction of Generalized Complex Structures
• Hamiltonian symmetries and reduction in generalized geometry
• Submanifolds of generalized complex manifolds
• Vector fields, torus actions and equivariant cohomology

Cited by in corpus (7)

• The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction
• Topology of generalized complex quotients
• The Equivariant cohomology theory of twisted generalized complex manifolds
• Convexity properties for generalized moment maps I
• Convexity properties of generalized moment maps
• Chern-Weil homomorphism in twisted equivariant cohomology
• Generalized complex hamiltonian torus actions: Examples and constraints