论文标题
Matsushima-lichnerowicz类型的定理的定理是符合性类型的广义Kähler歧管
Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type
论文作者
论文摘要
在Kähler的几何形状中，藤（藤本）表明，标态曲率是哈密顿二型差异性的力矩图。在广义的kähler几何形状中，一个人没有合适的levi-civita连接和曲率概念，但是仍然存在矩图的精确框架，标量曲率定义为矩图。然后，一个基本问题是了解具有恒定标态曲率的广义Kähler结构的存在或不存在。在本文中，我们研究了广义复合物歧管的自动形态的代数。我们假设$ h^{1}（m）= 0 $。然后，我们表明，如果概括的复合歧管允许具有恒定标态曲率的符号类型的广义kähler结构，则自动形态的谎言代数是一个还原性的代数。这是Kähler几何形状中Matsushima和Lichnerowicz定理的概括。我们明确计算由立方曲线在$ \ bbb c p^2 $上给出的广义复合结构的自动形态的谎言代数。立方曲线分为9例（见图。$ 1-9 $）。在三种情况下，如图中。 7、8和9，自动形态的谎言代数不是还原性的，并且在三种情况下存在具有恒定标态曲率的符合性类型的广义Kähler结构的障碍。我们还讨论了从普通的kähler歧管$（x，ω）$开始具有恒定标态曲率的变形，并表明，如果$ x $ $ x $的自动化代数是零头的，则具有恒定标态曲率的非平凡的符合符合性类型的kähller结构，并将其作为变形。
In Kähler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized Kähler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized Kähler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that $H^{1}(M)=0$. Then we show that the Lie algebra of the automorphisms is a reductive Lie algebra if a generalized complex manifold admits a generalized Kähler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in Kähler geometry. We explicitly calculate the Lie algebra of the automorphisms of a generalized complex structure given by a cubic curve on $\Bbb C P^2$. Cubic curves are classified into nine cases (see Figure.$1 -- 9$). In the three cases as in Figures. 7, 8 and 9, the Lie algebra of the automorphisms is not reductive and there is an obstruction to the existence of generalized Kähler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary Kähler manifold $(X,ω)$ with constant scalar curvature and show that nontrivial generalized Kähler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of $X$ is trivial.
