Convexity properties of generalized moment maps

Abstract

In this paper, we consider generalized moment maps for Hamiltonian actions on H-twisted generalized complex manifolds introduced by Lin and Tolman [15]. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact H-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward [13] we extend our results to the case of Hamiltonian actions of general compact Lie groups on H-twisted generalized complex orbifolds.

Topology of generalized complex quotients

“…Proposition 1.1 paraphrases a result from Nitta's very interesting recent work [NY07]. Nitta's result was known by the authors to hold under additional hypotheses but his general result came out as a welcome surprise.…”

“…[GGK02]). In [NY07] Nitta actually proved that the components of moment map for Hamiltonian torus actions on compact generalized complex manifolds are abstract nondegenerate moment maps (see also §5). Thus it follows that Kirwan injectivity and surjectivity for the usual equivariant cohomology must hold for GC-Hamiltonian actions.…”

The GIT aspect of generalized Kähler reduction. I

“…In the Kähler case, β 2 is invertible and therefore that (i) 0 is a regular value of µ is equivalent to that (ii) G acts locally freely on µ −1 (0). This equivalence still holds here but needs more technical arguments, which can be found in [16].…”

“…The following algebraic calculation actually has already appeared in [3] [16]. We include it here only because it provides some motivations for our later considerations.…”

Singular Reduction of Generalized Complex Manifolds

“…In Lemma 5.5 of [1], the authors proved the above result in the case that G is a torus; however, their proof holds just as well in the non-abelian case. It relies on viewing the components of µ as the real parts of a pseudo-holomorphic function and applying a version of the Maximum Principle, a course first taken in [17]. A thorough description of this version of the Maximum Principle can be found in Section 4.4 of [8].…”