Hamiltonian 2-forms in Kahler geometry, I General Theory

Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul

Hamiltonian 2-forms in Kahler geometry, I General Theory

Authors

Apostolov, Vestislav

Calderbank, David M. J.

Gauduchon, Paul

Description

We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kaehler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms draw attention to a general construction of Kaehler metrics which interpolates between toric geometry and Calabi-like constructions of metrics on (projective) line bundles. They also arise on conformally-Einstein Kaehler manifolds. We explore these connections and ramifications.
LaTeX, 41 pages; updated to improve exposition and coherence with part II (math.DG/0401320)