Complex quaternionic manifolds and c-projective structures

TL;DR

This paper explores complex quaternionic manifolds with specific holonomy, showing how certain c-projective classes induce quaternionic structures and characterizing special connections within this framework.

Contribution

It establishes a link between c-projective classes and quaternionic manifolds via the Feix--Kaledin construction, and characterizes distinguished $U^*(2n)$ connections.

Findings

Real analytic connections with type (1,1) curvature induce quaternionic structures.

Characterization of the $U^*(2n)$ connection in the quaternionic setting.

Holonomy of induced connections is contained in $GL(n,bH)U(1)$.

Abstract

We discuss complex quaternionic manifolds, i.e., those that have holonomy $GL(n,\mathbb{H})U(1)$, which naturally arise via quaternionic Feix--Kaledin construction. We show that for a fixed c-projective class, any real analytic connection with type $(1,1)$ curvature induces, via quaternionic Feix--Kaledin construction, an $S^1$-invariant connection with holonomy contained in $GL(n,\mathbb{H})U(1)$. As an application, we characterize in this setting the distinguished $U^*(2n):=SL(n,\mathbb{H})U(1)$ connection studied in Battaglia \cite{Bat} and Hitchin \cite{Hit3}.