Explicit abelian instantons on $S^1$-invariant K\"ahler Einstein $6$-manifolds

TL;DR

This paper studies explicit abelian instantons on $S^1$-invariant K"ahler Einstein 6-manifolds by reducing the Hermitian Yang-Mills equations to 4-dimensional data, leading to new solutions and geometric structures.

Contribution

It provides explicit constructions of abelian $SU(3)$ instantons on certain K"ahler manifolds via dimensional reduction, and introduces new geometric insights such as special Lagrangian foliations.

Findings

Explicit abelian $SU(3)$ instantons on the canonical bundle of $\\mathbb{C}\\mathbb{P}^2$

Families of deformed Hermitian Yang-Mills connections

Coordinate expression for the holomorphic volume form and a special Lagrangian foliation

Abstract

We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on $S^1$-invariant K\"ahler Einstein $6$-manifolds. This allows us to reformulate the (deformed) Hermitian Yang-Mills equations in terms of data on the quotient K\"ahler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}\mathbb{P}^2$. We also find $1$-parameter families of explicit deformed Hermitian Yang-Mills connections. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$.