It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. It turns out that infinitesimal deformations of generalized Kahler 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 Kahler manifold of type $(0,0)$. Poisson structures constructed from a Kahler action of $T^m$ on a Kahler-Einstein manifold give intriguing deformations of generalized Kahler-Einstein structures. In particular, the anticanical divisor consists of three lines on $C P^2$ in general position yields nontrivial examples of generalized Kahler-Einsein structures

中文翻译：

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标量曲率作为广义 Kähler 几何中的矩图

众所周知，标量曲率是作为 Kahler 几何中的矩图出现的。为了进行这个类比，我们在广义 Kahler 几何中引入了矩映射的概念，它给出了广义 Kahler 流形上的广义标量曲率的定义。从矩图的角度，我们得到了广义的 Ricci 形式，它是反正则线丛的第一陈类的代表。事实证明，具有恒定广义标量曲率的广义 Kahler 结构的无穷小变形在紧流形上是有限维的。在 $(0,0)$ 类型的广义 Kahler 流形上给出了广义 Ricci 形式和广义标量曲率的显式描述。由 $T^m$ 在 Kahler-Einstein 流形上的 Kahler 作用构建的泊松结构给出了广义 Kahler-Einstein 结构的有趣变形。特别是，反经典除数由 $CP^2$ 上的三行组成，一般位置产生了广义 Kahler-Einsein 结构的非平凡例子