Matsushima–Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type

Abstract

In K\"ahler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized K\"ahler 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 K\"ahler 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 K\"ahler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in K\"ahler 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 K\"ahler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary K\"ahler manifold $(X,\omega)$ with constant scalar curvature and show that nontrivial generalized K\"ahler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of $X$ is trivial.