We have classified Bochner-Kähler manifolds of real dimension > 4, which are also Bach flat. In the 4-dimensional case, we have shown that, if the scalar curvature is harmonic, then it is constant. Finally, we show that the gradient of scalar curvature of any Bochner-Kähler manifold is an infinitesimal harmonic transformation, and if it is conformal then the scalar curvature is constant.
Ghosh, Amalendu and Sharma, Ramesh, "Bochner-Kähler and Bach Flat Manifolds" (2019). Mathematics Faculty Publications. 20.
Ghosh, A., & Sharma, R. (2019). Bochner-Kähler and Bach flat manifolds. Archiv der Mathematik, 1-10.