辛型广义Kähler流形自同构的李代数的Matsushima-Lichnerowicz型定理
Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type
摘要：
在Kähler几何中，Fujiki-Donaldson证明了数量曲率是作为哈密顿量同态的矩映射而产生的。在广义Kähler几何中，人们没有合适的Levi-Civita连接和曲率的概念，但是仍然存在一个精确的矩映射框架，并且数量曲率被定义为矩映射。然后一个基本的问题是理解具有常数数量曲率的广义Kähler结构的存在性或不存在性。本文研究广义复流形的自同构的李代数。我们假设$H^{1}(M)=0$。证明了如果一个广义复流形上存在一个具有常数量曲率的辛型广义Kähler结构，则其自同构的李代数是约化李代数.这是Kähler几何中Matsushima和Lichnerowicz定理的推广。本文明确地计算了由$\Bbb C P^2$上的三次曲线给出的广义复结构的自同构的李代数。三次曲线分为九种情况（见图）。$1 -- 9$）。在三种情况下，如图。7，8和9，自同构的李代数不是约化的，并且在这三种情况下存在具有常数数量曲率的辛型广义Kähler结构。我们还讨论了从一个普通的Kähler流形$(X,ω)$与常数数量曲率曲率的变形，并表明，非平凡的广义Kähler结构的辛型常数数量曲率曲率出现的变形，如果$X$的自同构的李代数是平凡的。
英文摘要：
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.
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