Holonomy of the Obata connection on 2-step 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.

20 pages. To appear in Bull. Lond. Math. Soc. The definition of $\mathbb{H}$-solvability was extended to arbitrary hypercomplex Lie algebras (not necessarily nilpotent). We improved the proof of Theorem 6.1 and introduced Subsection 6.1. We added Lemma 7.1 to simplify the proof of Proposition 7.2