Riemann-Poisson and Kähler-Poisson complex manifolds and structures

In this paper we consider structures of complex Poisson brackets on the space of smooth functions in a $n$-dimensional complex manifold generated by the $(1,1)$-form $d=\mathrm{\partial }+\bar{\mathrm{\partial }}$-closed and non-degenerate (with non-holomorphic and non-antiholomorphic coefficients). In this case, we have view the compatibility between complex Poisson and Riemannian structures and an example is giving in ${\text{C}}^{\ast }$.