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.

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A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact Lie group action on a generalized Calabi-Yau manifold and construct a reduced generalized Calabi-Yau structure on the reduced space. As an application, we show some relationship between generalized moment maps and the Bergman kernels, and prove the Duistermaat-Heckman formula for a torus action on a generalized Calabi-Yau manifold.

In this paper, we shall prove the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure. This generalizes the result of Cho, Futaki and Ono [5] for compact toric Sasaki manifolds. Introduction.The aim of this paper is to show the uniqueness of Sasaki-Einstein metrics up to the action of the identity component of the automorphism group for the transverse holomorphic structure. A Sasaki manifold is a Riemannian manifold (S, g) whose cone metric ḡ = dr 2 + r 2 g on C(S) = S × R + is Kählerian. Sasakian geometry sits naturally in two aspects of Kähler geometry, since for one thing, (S, g) is the base of the cone manifold (C(S), ḡ ) which is Kählerian, and for another thing any Sasaki manifold is contact, and the one dimensional foliation associated to the characteristic Reeb vector field admits a transverse K ähler structure. A Sasaki-Einstein manifold then admits a one dimensional Reeb foliation with a transverse Kähler-Einstein metric, which is studied from viewpoints of geometry and mathematical physics. Boyer, Galicki, Kollár and Thomas obtained Sasaki-Einstein metrics on a family of the links of hypersurfaces of Brieskorn-Pham type, which include exotic spheres [3, 4]. Gauntlett, Martelli, Sparks and Waldram discovered that there exist irregular toric Sasaki-Einstein manifolds which are not obtained as total spaces of line orbibundles on Kähler-Einstein orbifolds [8, 9]. These toric examples are much explored by Futaki, Ono and Wang [7], who showed that, for any compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasakian structure on which a Sasaki-Einstein metric exists. Furthermore, Cho, Futaki and Ono proved in [5] the uniqueness of Sasaki-Einstein metrics on compact toric Sasaki manifolds up to the action of the identity component of the automorphism group for the transverse holomorphic structure by showing that the argument of Guan [10] is valid also for the space of Kähler potentials for the transverse Kähler structure. In the present paper, we shall prove such uniqueness without toric assumption: THEOREM A. Let (S, g) be a compact Sasaki manifold with a Sasakian structure S = {g, ξ, η, Φ}. Assume that the set E of all Sasaki-Einstein metrics compatible with g is non-empty. Then the identity component of the automorphism group for the transverse holomorphic structure acts transitively on E .

For a polarized algebraic manifold (X, L), let T be an algebraic torus in the group Aut(X) of all holomorphic automorphisms of X. Then strong relative K-stability (cf. [6]) will be shown to imply asymptotic relative Chow-stability. In particular, by taking T to be trivial, we see that asymptotic Chow-stability follows from strong K-stability.