Submaximally symmetric c-projective structures

Abstract

C-projective structures are analogues of projective structures in the complex setting. The maximal dimension of the Lie algebra of c-projective symmetries of a complex connection on an almost complex manifold of C-dimension $n>1$ is classically known to be $2n^2+4n$. We prove that the submaximal dimension is equal to $2n^2-2n+4+2\delta_{3,n}$. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the c-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-K\"ahler metrics is $2n^2-2n+4$, and specializing to the K\"ahler case, we obtain $2n^2-2n+3$. This resolves the symmetry gap problem for metrizable c-projective structures.