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.Comment: The manuscript was updated and revised. This version corrects some errors in Section 4 that do not influence the main results. The exposition is also slightly polishe

C-projective symmetries of submanifolds in quaternionic geometry

“…Note that the Feix-Kaledin construction deals only with manifolds with Weyl curvature of type II. In [9], an explicit model of non-flat c-projective structure with type II Weyl curvature which admits the (sub-) maximal dimension of infinitesimal c-projective symmetries is given for each dimension greater than 2n = 2. The symmetry dimension is 2n 2 − 2n + 4, and this turns out to be equal to the submaximal dimension for general c-projective structures, except for the case 2n = 4.…”

“…In this section we will generalize this result by showing that one may similarly construct a submaximally symmetric quaternionic structure from the submaximally symmetric C-projective structure with curvature of type (1, 1), for some line bundle and connection. We begin by studying the properties of the submaximal type 2 c-projective structure from [9]. Recall that such a structure (for each n > 1) is unique and in local coordinates z i , z i on C n it is given by (the c-projective class of) a complex connection with the only non-vanishing Christoffel symbols: Γ 2 11 = z 1 and Γ 2 11 .…”

“…Proof. The submaximal c-projective model is shown to be unique in [9], and explicit symmetry vector fields are given in section 3 of that paper: the generators are the real and imaginary parts of…”

“…When the drop between maximal and submaximal symmetry dimension is significant, it is called a gap phenomenon, and for parabolic Cartan geometries in general this was studied by B. Kruglikov and D. The in [10]. The dimension of the submaximal symmetry algebra for 4n-dimensional almost quaternionic structures has been computed by the second author, B. Kruglikov and L. Zalabova in [11], and for 2n-dimensional almost c-projective structures by B. Kruglikov, V. Matveev and D. The in [9] (both for n > 1). In these papers the authors also consider the submaximal dimensions of sub-cases of the corresponding structures depending on the type of the Weyl curvature.…”

First BGG operators via homogeneous examples

“…In the real setting, one has to proceed case by case, see e.g. [31,29,30]. There are also known classification of locally homogeneous parabolic geometries of several types in special dimensions.…”