  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.

Differential Geometry