  In this paper, we are concerned with studying the existence of invariant
complex manifolds of two-dimensional holomorphic systems. From the geometric
singular perturbation theory we know that if a slow-fast system has associated
a normally hyperbolic compact critical manifold, then there exists a smooth
locally invariant manifold. However, this smooth manifold does not necessarily
have a complex structure. Here, we provide conditions to guarantee the
existence of one-dimensional invariant complex manifolds. Consequently, this
allows us to establish that the centers, foci, and nodes of the reduced problem
are persistent by singular perturbation. The tools used by us are the usual
techniques of Fenichel and Briot-Bouquet Theories.
