Explicit abelian instantons on \(S^1\)-invariant Kähler Einstein 6-manifolds
This paper deals with explicit abelian instantons on \(S^1\)-invariant Kähler-Einstein 6-manifolds. It is concerned with the setup in the case of a non-compact Kähler-Einstein manifold. The author considers the dimensional reduction of the deformed hermitian Yang-Mills condition on \(S^1\)-invariant Kähler-Einstein 6-manifolds. This makes it possible 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 \(\mathrm{U}(1)\) the author applies this construction to the canonical bundles of \(\mathbb{CP}^2\) and \(\mathbb{CP}^1\times\mathbb{CP}^1\) endowed with the Calabi ansatz metric to find abelian instantons. It is shown 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 this investigation, the author finds a coordinate expression for the holomorphic volume form on \(\mathcal{O}_{\mathbb{CP}^2}(-3)\) and uses it to construct a new special Lagrangian foliation. He constructs explicit examples of deformed hermitian Yang-Mills connections on \(\mathcal{O}_{\mathbb{CP}^2}(-3)\) and finds 1-parameter families of explicit deformed hermitian Yang-Mills connections on certain non-compact \(S^1\)-invariant Kähler Einstein 6-manifolds. \N\NThe consists of the following sections:\N\N1. Introduction.\N\N2. Preliminaries.\N\N3. \(S^1\)-invariant Kähler structures.\N\N4. \(S^1\)-invariant hermitian Yang-Mills connections.\N\N5. Abelian instantons on the canonical bundle of \(\mathbb{CP}^2\).\N\N6. Special Lagrangians in \(\mathcal{O}_{\mathbb{CP}^2}(-3)\).\N\N7. Abelian instantons on the canonical bundle of \(S^2\times S^2\).\N\N8. Abelian instantons on other Kähler Einstein 3-folds.\N\N9. Examples of deformed hermitian Yang-Mills connections.
scientific article; zbMATH DE number 7904738
scientific article