The moduli space of conically singular instantons over an SU(3)-manifold

Abstract

In this article we study the moduli space of conically singular instantons (or Hermitian Yang--Mills connections) with prescribed tangent connections over a 6-manifold equipped with an $\mathrm{SU}(3)$-structure. That is, we develop a Fredholm deformation theory for such $\mathrm{SU}(3)$-instantons in which we fix the tangent connection but allow the underlying principal bundle (and, in particular, the singular set) to vary. This leads to the existence of a Kuranishi structure for this moduli space. Moreover, we investigate the cokernel of the instanton deformation operator and give under certain assumptions a formula for its dimension. Ultimately, we apply our results to conically singular instantons with structure group $\mathbb{P}\mathrm{U}(n)$ and give a formula for the virtual dimension of their moduli space in terms of sheaf cohomology of certain vector bundles over $\mathbb{P}^2$.