Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kahler manifolds of symplectic type

In Kahler geometry, Fujiki-Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized Kahler 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 Kahler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold (M, J). We assume that Hodd(M, C) = 0. Then we showthat the Lie algebra of the automorphisms is a reductive Lie algebra if (M, J) admits a generalized Kahler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in Kahler geometry. We explicitly calculate the Lie algebra of the automorphisms of a generalized complex structure given by a cubic curve on CP2. 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 Kahler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary Kahler manifold (X, omega) with constant scalar curvature and show that nontrivial generalized Kahler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of X is trivial. We show the Hessian formula of generalized extremal Kahler structures and obtain the decomposition theorem of the Lie algebra of the reduced automorphisms of a generalized extremal Kahler manifold.