Submaximally symmetric c-projective structures

C-projective structures are analogues of projective structures in the almost 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 2n² + 4n. We prove that the submaximal dimension is equal to 2n² - 2n + 4 + 2δ_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-Kähler metrics is 2n² - 2n + 4, and specializing to the Kähler case, we obtain 2n² - 2n + 3. This resolves the symmetry gap problem for metrizable c-projective structures.