Normal bundles to Laufer rational curves in local Calabi-Yau threefolds

TL;DR

This paper proves a conjecture relating the normal bundles of Laufer rational curves in local Calabi-Yau threefolds to the Hessian of a superpotential at critical points, providing explicit geometric and algebraic insights.

Contribution

It establishes a formula for the normal bundle of Laufer rational curves in local Calabi-Yau threefolds based on the Hessian of a superpotential, confirming a conjecture by Ferrari.

Findings

Normal bundle computed via Hessian of superpotential

Validation of Ferrari's conjecture in specific geometric settings

Explicit relation between deformation theory and superpotential critical points

Abstract

We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.