  Let $C$ be a smooth elliptic curve embedded in a smooth complex surface $X$
such that $C$ is a leaf of a suitable holomorphic foliation of $X$. We
investigate complex analytic properties of a neighborhood of $C$ under some
assumptions on complex dynamical properties of the holonomy function. As an
application, we give an example of $(C, X)$ in which the line bundle $[C]$ is
formally flat along $C$ however it does not admit a $C^\infty$ Hermitian metric
with semi-positive curvature.
