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The Bochner tensor is the K\"ahler analogue of the conformal Weyl tensor. Inthis article, we derive local (i.e., in a neighbourhood of almost every point)normal forms for a (pseudo-)K\"ahler manifold with vanishing Bochner tensor.The description is pined down to a new class of symmetric spaces which wedescribe in terms of their curvature operators. We also give a localdescription of weakly Bochner-flat metrics defined by the property that theBochner tensor has vanishing divergence. Our results are based on the localnormal forms for c-projectively equivalent metrics. As a by-product, we alsodescribe all K\"ahler-Einstein metrics admitting a c-projectively equivalentone.

Local description of Bochner-flat (pseudo-)K\"ahler metrics