Kirwan injectivity and surjectivity are two important results in equivariant symplectic geometry. For a symplectic manifold \((M,\omega)\), a Hamiltonian action by a connected Lie group \(G\) on \((M,\omega)\) is regulated by a moment map \(\mu:M\rightarrow \mathfrak{g}^\ast\) taking values in the dual of the Lie algebra of \(G\). Contracting by \(\xi\in \mathfrak{g}\) produces a real valued function \(\mu^\xi:M\rightarrow \mathbb{R}\) called a component of the moment map. If \(G\) is compact, then for any \(\xi\in \mathfrak{g}\), \(\mu^\xi\) is a Morse-Bott function and may be used to study the equivariant topology of \(M\). In [F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31. Princeton, New Jersey: Princeton University Press (1984; Zbl 0553.14020)], using ideas of M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond., A 308, 523–615 (1983; Zbl 0509.14014)], Kirwan demonstrated that a Hamilton action on a compact symplectic manifold \(M\) is equivariantly formal. In particular, the equivariant cohomology of \(M\) with rational coefficients satisfies a noncanonical isomorphism \(H^\ast_G(M)\cong H^\ast(M)\otimes H^\ast(BG)\), as graded \(H^\ast(BG)\)-modules, where \(BG\) is the classifying space for \(G\). Furthermore, if \(G=T\) is a torus, and \(i:M^T\hookrightarrow M\) denotes inclusion of the fixed point set, the localization map in equivariant cohomology \(i^\ast:H^\ast_T(M)\rightarrow H^\ast_T(M^T)\) is an injection, a result known as Kirwan injectivity. Kirwan also showed that the map \(k:H_G(M)\rightarrow H_G(\mu^{-1}(0))\) induced by inclusion is a surjection. This result is known as Kirwan surjectivity and the map \(k\) is known as the Kirwan map. If \(0\) is a regular value of \(\mu\), then \(H_G(\mu^{-1}(0))\cong H(M//G)\), where \(M//G=\mu^{-1}(0)/G\) is the symplectic quotient, so \(H(M//G)\) is describable as a quotient ring \(H_G(M)/ker (k)\).
In this paper, the authors generalize Kirwan injectivity and surjectivity to Hamiltonian actions on compact generalized complex manifolds, in the sense of Y. Lin and S. Tolman [Commun. Math. Phys. 268, No. 1, 199–222 (2006; Zbl 1120.53049)]. In the paper of Lin and Tolman [op.cit], a definition was introduced of a Hamiltonian action and a generalized moment map in both generalized complex and generalized Kähler geometries. In the present paper, the authors study the twisted equivariant cohomology using Morse theory, in the more general case of Hamiltonian actions on compact complex manifolds. The main results of paper are the following theorems.
Theorem 1.2 (Equivariant Formality). Consider the Hamiltonian action of a compact connected group \(G\) on a compact \(H\)-twisted generalized complex manifold \(M\). Then we have a noncanonical isomorphism \(H_G(M;H+\alpha)\cong H(M;H)\otimes H(BG)\), where \(\alpha\) is the moment 1-form of the Hamiltonian action.
Theorem 1.3 (Kirwan Injectivity). Let \(T\) be a compact torus and let \(M\) be a compact \(H\)-twisted generalized Hamiltonian \(T\)-space with induced equivariant 3-form \(H+\alpha\), and let \(i:M^T\rightarrow M\) denote the inclusion of the fixed point set. Then the induced map \(i^\ast:H_T(M;H+\alpha)\rightarrow H_T(M^T; H+\alpha)\cong H(M^T;H)\otimes H(BT)\) is an injection.
Theorem 1.4 (Kirwan Surjectivity). Let \(M\) be a compact \(H\)-twisted generalized Hamiltonian \(T\)-space with induced equivariant 3-form \(H+\alpha\) and moment map \(\mu\), where \(T\) is a compact torus. For a regular value \(c\in \mathfrak{t}^\ast\) of \(\mu\) we have: \(H_F(M;H+\alpha)\rightarrow H(\mu^{-1}(c)/T;\widetilde{H})\) is a surjection, where \(\widetilde{H}\) is the twisting 3-form inherited through reduction.
These results are established more generally for compact nondegenerate abstract moment maps with compatible equivariantly closed 3-form. “We expect that Kirwan surjectivity remains true for the Hamiltonian action of a compact connected Lie group on a compact twisted generalized complex manifold, and we hope to return to this question in later work.”
At the end the authors discuss one ‘possible application’ of their results: for establishing a possible close relationship between the deformation theory of the generalized complex manifold \(M\) and that of its generalized complex quotient.