Scalar curvature as moment map in generalized Kähler geometry

Abstract

It is known that the scalar curvature arises as the moment map in Kähler geometry. In pursuit of the analogy, we develop the moment map framework in generalized Kähler geometry of symplectic type. Then we establish the definition of the scalar curvature on a generalized Kähler manifold of symplectic type from the moment map view point. We also obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. We show that infinitesimal deformations of generalized Kähler structures with constant generalized scalar curvature are finite dimensional on a compact manifold. Explicit descriptions of the generalized Ricci form and the generalized scalar curvature are given on a generalized Kähler manifold of type (0, 0). Poisson structures constructed from a Kähler action of T m on a Kähler-Einstein manifold give rise to intriguing deformations of generalized Kähler-Einstein structures. In particular, the anticanonical divisor of three lines on CP 2 in general position yields nontrivial examples of generalized Kähler-Einstein structures. IntroductionLet (X, ω) be a compact symplectic manifold with a symplectic structure ω. An almost complex structure J is compatible with ω if a pair (J, ω) gives an almost Kähler structure on M . We denote byC ω the set of almost complex structures which are compatible with ω. ThenC ω is an infinite dimensional Kähler manifold on which Hamiltonian diffeomorphisms of (M, ω) actC ω preserving the Kähler structure. Each J ∈ C ω gives a Riemannian metric g(J) and we denote by s(J) the scalar curvature of g(J) which is regarded as a function on C ω . Then the following theorem was established in Kähler geometry by Fujiki and Donaldson.The scalar curvature is the moment map onC ω for the action of Hamiltonian diffeomorphisms.The moment map framework in Kähler geometry suggests that the existence of constant scalar curvature Kähler metrics is inevitably linked with the certain stability in algebraic geometry which leads to wellknown Donaldson-Tian-Yau conjecture in Kähler geometry.Generalized Kähler geometry is a successful generalization of ordinary Kähler geometry which is equivalent to bihermitian geometry satisfying the certain torsion conditions. Many interesting examples of generalized Kähler manifolds were already constructed by holomorphic Poisson structures [Go1], [Go2], [Go3], [Go4], [Gu1], [Hi1], [Hi2], [Lin1]. * 1 Thus ψ is given by ψ = e b+ √ −1ω , where ω is a real symplectic form. A pair (J , J ψ ) is called a generalizedKähler structure of symplectic type. We can obtain further generalization of moment map framework for any d-closed nondegenerate, pure spinor ψ. J . Since h 0,2 · φ α = 0, we haveThus we have e · φ α , N · h · φ α s = 0. Then it follows N · h · φ α , e · φ α s = 0.Lemma 7.10. Let N be as in before. Then we haveProof. Let {e i } be a local basis of E J . Since dφ α = η α · φ α + N · φ α , then we have N · φ α , e i · e j · e k · φ α s = dφ α , e i · e j · e k · φ α s = − e i · e j · dφ α , e k · φ α s Since the co...

Differential calculus for generalized geometry and geometric Lax flows

“…The above system has been studied in mathematics and physics literatures, for instance, Garcia-Fernandez-Streets [9], and references therein. We note that an equation similar to (1.9) appeared in [29] citing communication with Gualtieri (reference [11] ibid.) and the equivalence with the generalized Ricci flow appeared in [9] (Remark 4.8 ibid.…”

Toric generalized Kähler structures. III

“…Goto's approach to scalar curvature and its refinement. In [10], Goto provided a notion of scalar curvature for GK manifolds of symplectic type in terms of generalized pure spinors defining the underlying geometry. The goal of this section is thus two-fold: on one side we briefly recall Goto's definition and on the other side we refine it in such a way that, with the help of bi-spinors, we can compute the scalar curvature in terms of the underlying biHermitian data.…”

“…In [10], Goto actually used a different splitting which was neither the metric splitting nor the symplectic splitting. The generalized pure spinor of J 2 was chosen to be Ψ = e B− √ −1ω where B is a closed real 2-form.…”

“…In [5], L. Boulanger considered toric GK structures of symplectic type and defined a notion of scalar curvature for these structures using an analogous version of the philosophy in Kähler geometry that scalar curvature should be the moment map in an infinite-dimensional GIT theory to find cscK metrics. In [10], R. Goto investigated general GK structures of sympletic type and succeeded in finding an analogue of Ricci form in terms of local data from generalized pure spinors defining the geometry. Goto was able to extract a scalar curvature out of the Ricci form and prove in general that his version of scalar curvature is the moment map of the abovementioned infinite-dimensional GIT theory.…”

The Riemannian and symplectic geometry of the space of generalized Kähler structures

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On a compact complex manifold (M, J) endowed with a holomorphic Poisson tensor π J and a deRham class α ∈ H 2 (M, R), we study the space of generalized Kähler (GK) structures defined by a symplectic form F ∈ α and whose holomorphic Poisson tensor is π J . We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger [10] and Goto [36] to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki-Mabuchi extremal vector field [25] and Calabi-Lichnerowicz-Matsushima result [12,55,58] for the Lie algebra of the group of automorphisms of (M, J, π J ).

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