An example of non-compact totally complex submanifolds of compact quaternionic Kähler symmetric spaces
Yuuki Sasaki
TL;DR
The paper addresses the problem of constructing non-compact totally complex submanifolds of maximal dimension inside compact quaternionic Kähler symmetric spaces (excluding $\mathbb{H}P^{n}$). It develops a construction based on a Helgason sphere and its polar to produce a rank-2 submanifold $N$ that is a holomorphic line bundle over a Hermitian symmetric space $H(p)$, with totally geodesic fibers, yielding a ruled, cohomogeneity-one, totally complex submanifold satisfying $2\dim N = \dim M$. A key result is that, except for the ambient spaces $M=G_{2}(\mathbb{C}^{n})$ or $M=G_{2}/SO(4)$, no compact submanifold of the same dimension can contain $N$ as an open part; in the $G_{2}(\mathbb{C}^{n})$ case, $N$ sits as an open part of $\mathbb{C}P^{n-2}$. This work expands the catalog of non-compact totally complex submanifolds in compact quaternionic Kähler symmetric spaces and highlights a rich interaction between Lie group actions, twistor geometry, and submanifold theory.
Abstract
Totally complex submanifolds of a quaternionic Kähler manifold are analogous to complex submanifolds of a Kähler manifold. In this paper, we construct an example of a non-compact totally complex submanifold of maximal dimension of a compact quaternionic Kähler symmetric space, except for quaternionic projective spaces. A compact Lie group acts on our example isometrically, and this action is of cohomogeneity one. Our example is a holomorphic line bundle over some Hermitian symmetric space of compact type. Moreover, each fiber is a totally geodesic submanifold of the ambient quaternionic Kähler symmetric space and our example is a ruled submanifold. Our construction relies on the action of a subgroup of the isometry group and a maximal totally geodesic sphere with maximal sectional curvature known as a Helgason sphere. Furthermore, we prove that there exist no compact submanifolds of the same dimension that contain our example as an open part, except where the ambient quaternionic Kähler symmetric space is a complex Grassmannian.