Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kaehler manifolds of symplectic type
Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kaehler manifolds of symplectic type
复制标题
辛型广义凯勒流形自同构李代数的Matsushima-Lichnerowicz型定理
DOI：
10.1007/s00208-021-02299-z
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发表时间：
2022
影响因子：
1.4
通讯作者：
Ryushi Goto
中科院分区：
文献类型：
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作者：
Mimura Masato;Tokushige Norihide;Ryushi Goto
在Kähler几何中，Fujiki-Donaldson表明标量曲率作为哈密顿微分同态的矩映射出现。在广义Kähler几何中，人们对Levi-Civita连接和曲率没有很好的概念，但是矩映射仍然存在一个精确的框架，标量曲率被定义为矩映射（Goto, J Sympl Geom 18(1):147 - 190,2020）。那么一个基本问题就是如何理解具有常标量曲率的广义Kähler结构的存在与否。本文研究了一类广义复流形的自同构的李代数，并在此基础上证明了该自同构的李代数是一个约化李代数，如果该类李代数具有常数标量曲率的广义Kähler辛型结构。这是Kähler几何中Matsushima定理和Lichnerowicz定理的推广。我们显式地计算了由三次曲线给出的广义复结构自同构的李代数。三次曲线分为九种情况，如第7节的图1-9所示。在图7-9所示的三种情况下，自同构的李代数不是约化的，并且在这三种情况下具有常标量曲率的辛型广义Kähler结构存在障碍。我们还讨论了从具有常数标量曲率的普通Kähler流形出发的变形，并证明了具有常数标量曲率的辛型非平凡广义Kähler结构，如果x的自同构的李代数是平凡的，就会产生变形。给出了广义极值Kähler结构的Hessian公式，得到了广义极值Kähler流形约化自同构的Lie代数分解定理。
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 good notions of Levi-Civita connection and curvature, however there still exists a precise framework for the moment map and the scalar curvature is defined as the moment map (Goto, J Sympl Geom 18 (1):147–190, 2020). 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 manifoldWe assume thatThen we show that the Lie algebra of the automorphisms is a reductive Lie algebra ifadmits 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. Cubic curves are classified into nine cases as in Figs. 1–9 in Sect. 7. In the three cases as in Figs. 7–9, 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 manifoldwith 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 ofXis trivial. We show the Hessian formula of generalized extremal Kähler structures and obtain the decomposition theorem of the Lie algebra of the reduced automorphisms of a generalized extremal Kähler manifold.