Holonomy of the Obata connection on 2-step hypercomplex nilmanifolds

TL;DR

This paper investigates the holonomy of the Obata connection on 2-step hypercomplex nilmanifolds, establishing conditions for flatness and confirming the $ ext{H}$-solvable conjecture in this context.

Contribution

It explicitly computes the curvature tensor, characterizes the holonomy algebra as abelian, and introduces new examples of nilmanifolds with higher nilpotency steps.

Findings

Holonomy algebra is always abelian for 2-step hypercomplex nilmanifolds.

The $ ext{H}$-solvable conjecture holds in this case.

Constructs new examples of $k$-step nilpotent hypercomplex nilmanifolds.

Abstract

We study the holonomy of the Obata connection on 2-step hypercomplex nilmanifolds. By explicitly computing the curvature tensor, we determine the conditions under which the Obata connection is flat, showing that this depends on the nilpotency step of each complex structure. In particular, we show that for 2-step hypercomplex nilmanifolds the holonomy algebra of the Obata connection is always an abelian subalgebra of $\mathfrak{sl}(n, \mathbb{H})$ and we prove that the $\mathbb{H}$-solvable conjecture holds in this case. Furthermore, we provide new examples of $k$-step nilpotent hypercomplex nilmanifolds, with arbitrary $k$, which are not Obata flat.