Atiyah classes and the essential obstructions in deforming a singular $G_2-$instanton

Abstract

When the rank of the bundle is $\ge 2$, 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 $\ge 3$.

23 pages. Comments are welcome

References in corpus (2)

• $G_2$-instantons over twisted connected sums: an example
• The spectrum of an operator associated with $G_2-$instantons with $1-$dimensional singularities and Hermitian Yang-Mills connections with isolated singularities