Atiyah classes and the essential obstructions in deforming a singular $G_{2}-$instanton

TL;DR

This paper identifies an essential obstruction involving Atiyah classes that prevents the gluing construction of certain $G_{2}$-instantons with 1-dimensional singularities, especially for bundles of rank three or higher.

Contribution

It reveals a new obstruction related to Atiyah classes that impacts the deformation and gluing of $G_{2}$-instantons with singularities.

Findings

Gluing fails if the tangent connection isn't the twisted Fubini-Study connection.

Obstruction appears for bundles of rank ≥ 3.

Atiyah classes play a crucial role in the deformation theory.

Abstract

When the rank of the bundle is $\geq 2$, in a certain sense, we found an essential obstruction for the gluing construction of $G_{2}-$instantons with $1-$dimensional singularities. It involves the Atiyah classes generated by contracting a vector in $\mathbb{C}^{3}$ with the curvature. Intuitively speaking, the gluing does not work if the tangent connection at a component of the $1-$dimensional singular locus is not the twisted Fubini-Study connection on a twisted tangent bundle of $\mathbb{P}^{2}$. Particularly, it fails if the rank of the bundle is $\geq 3$.