  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 $d=\partial+\overline{\partial}$-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 $\C^\ast$.
