Explicit abelian instantons on S1-invariant Kähler Einstein 6-manifolds

We consider the dimensional reduction of the (deformed) Hermitian Yang–Mills condition on S1-invariant Kähler Einstein 6-manifolds. This allows us to reformulate the (deformed) Hermitian Yang–Mills equations in terms of data on the quotient Kähler 4-manifold. In particular, when the gauge group is U(1) we apply this construction to the canonical bundles of CP2 and CP1×CP1 endowed with the Calabi ansatz metric to find abelian instantons. We show that these are determined by a suitable subset of the spectrum of the zero section and are explicitly given in terms of certain hypergeometric functions. As a by-product of our investigation we find a coordinate expression for the holomorphic volume form on OCP2(−3) and use it to construct a new special Lagrangian foliation. We also find 1-parameter families of explicit deformed Hermitian Yang–Mills connections on certain non-compact S1-invariant Kähler Einstein 6-manifolds.