Revista de la
Unión Matemática Argentina
A rigidity result for Kählerian manifolds endowed with closed conformal vector fields
Antonio Caminha
Volume 60, no. 2 (2019), pp. 469–484

Download PDF


We show that if a connected compact Kählerian surface $M$ with nonpositive Gaussian curvature is endowed 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 Kählerian manifod $M$ of complex dimension $n > 1$ and endowed 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 Kählerian manifolds of complex dimension $n-1$. In particular, the universal covering of $M$ is isometric to a Riemannian product having $\mathbb{R}^2$ as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of $\xi$.