On the structure of complete k\"ahlerian manifolds furnished with closed conformal vector fields

We show that if a connected compact k\"ahlerian surface $M$ with nonpositive gaussian curvature is furnished with a closed conformal vector field $\xi$ whose singular points are isolated, then $M$ is isometric to a flat torus and $\xi$ is parallel. We also consider the case of a connected complete k\"ahlerian manifod $M$ of complex dimension $n>1$ and furnished with a nontrivial closed conformal vector field $\xi$. In this case, it is well known that the singularities of $\xi$ are automatically isolated and the nontrivial leaves of the distribution generated by $\xi$ and $J\xi$ are totally geodesic in $M$. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of $M$ along it is nonpositive, we show that $\xi$ is parallel and $M$ is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete k\"ahlerian manifolds of complex dimension $n-1$. In particular, the the universal covering of $M$ is isometric to a riemannian product having $\mathbb R^2$ as a factor. We also present a generic example showing that one cannot get rid of the hypothesis on the nonpositivity of the holomorphic sectional curvature along at least one such leaf.