Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds

Abstract

We consider a dimensional reduction of the (deformed) Hermitian Yang-Mills condition on S 1 -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, we apply this construction to the canonical bundle of CP 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 CP 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 O CP 2 (−3). Contents 1. Introduction 2. Preliminaries 3. S 1 -invariant Kähler structures 3.1. Kähler reduction of a Calabi-Yau 3-fold 3.2. The Ricci form and the Einstein condition 3.3. Constant solutions 3.4. Non-constant solutions 4. S 1 -invariant Hermitian Yang-Mills connections 4.1. Kähler reduction of the (deformed) Hermitian Yang-Mills condition 4.2. The abelian case 5. Abelian instantons on the canonical bundle of CP 2 5.1. The Calabi-Yau metric on C 3 5.2. The Calabi-Yau metric on O CP 2 (−3) 5.3. Instantons: Abelian examples 5.4. Special Lagrangians in O CP 2 (−3). 6. Abelian instantons on the canonical bundle of S 2 × S 2 6.1. The Calabi-Yau metric on K 1,−1 6.2. Instantons: Abelian examples 7. Abelian instantons on other Kähler Einstein 3-folds 8. Examples of deformed Hermitian Yang-Mills connections References