Jumps, folds, and singularities of Kodaira moduli spaces

TL;DR

This paper constructs explicit twistor spaces with jumping rational curves, revealing how their normal bundles change and exploring the geometric and physical implications of these singularities in anti-self-dual manifolds.

Contribution

It provides explicit examples of twistor spaces with jumping rational curves and analyzes their geometric and physical properties, including the structure of associated Ricci-flat manifolds.

Findings

Normal bundle changes from O(1)+O(1) to O(k)+O(2-k)

For k=2, recovers Hitchin's folded hyper-Kahler manifold

Existence of normalisable solutions extending through the fold

Abstract

For any integer $k$ we construct an explicit example of a twistor space which contains a one--parameter family of jumping rational curves, where the normal bundle changes from $O(1)+O(1)$ to $O(k)+O(2-k)$. For $k>3$ the resulting anti--self--dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of $O(k)$. In the case $k=2$ we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three--parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.