Jumps, folds, and hypercomplex structures

Abstract

We investigate the geometry of the Kodaira moduli space $M$ of sections of $\pi :Z\to {\mathbb{P}}^1$, the normal bundle of which is allowed to jump from $\mathcal{O}(1{)}^n$ to $\mathcal{O}(1{)}^{n-2m}\oplus \mathcal{O}(2{)}^m\oplus {\mathcal{O}}^m$. In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of $M$ extends to a logarithmic connection on $M$.

arXiv commentv.2 substantially rewritten