# Jumps, folds, and hypercomplex structures

## TL;DR

This paper explores the geometry of a moduli space of sections with jumping normal bundles, identifying conditions for extending the Obata connection to a logarithmic connection, revealing new geometric structures.

## Contribution

It characterizes conditions under which the Obata connection extends to a logarithmic connection on the moduli space with jumping normal bundles.

## Key findings

- Identification of conditions for connection extension
- Analysis of hypercomplex structures in moduli space
- Extension of Obata connection to logarithmic connection

## 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 $M$.