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We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kähler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kähler 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 Kähler metrics are extremal. We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kähler 6-manifolds, extending work by the first three authors on weakly selfdual Kähler 4-manifolds. The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kähler manifolds with a hamiltonian 2-for

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