Hamiltonian 2-Forms in Kähler Geometry, I General Theory

We introduce the notion of a hamiltonian 2-form on a Kähler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kähler geometry. In particular, on any Kähler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kähler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms also arise on conformally Einstein Kähler manifolds and provide an Ansatz for extremal Kähler metrics unifying and extending many previous constructions.