Hamiltonian 2-forms in Kahler geometry, IV Weakly Bochner-flat Kahler manifolds

Details Hamiltonian 2-forms in Kahler geometry, IV Weakly Bochner-flat Kahler manifolds Hamiltonian 2-forms in Kahler geometry, IV Weakly Bochner-flat Kahler manifolds

2005-11-04

arxiv.org/abs/math/0511119v1

Comm. Anal. Geom. 16 (2008) 91-126.

Mathematics

Differential Geometry

22 pages, 11 figures, sequel to math.DG/0511118, math.DG/0401320 and math.DG/0202280, but partly self-contained

Scientific paper

We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kahler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kahler metric is WBF if and only if its `normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series. It follows that WBF Kahler metrics are extremal. We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kahler 6-manifolds, extending work by the first three authors on weakly selfdual Kahler 4-manifolds. The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kahler manifolds with a hamiltonian 2-form in math.DG/0401320 as well as some of the results in math.DG/0511118.

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