  Complex Finsler vector bundles have been studied mainly by T. Aikou, who
defined complex Finsler structures on holomorphic vector bundles. In this
paper, we consider the more general case of a holomorphic Lie algebroid E and
we introduce Finsler structures, partial and Chern-Finsler connections on it.
First, we recall some basic notions on holomorphic Lie algebroids. Then, using
an idea from E. Martinez, we introduce the concept of complexified prolongation
of such an algebroid. Also, we study nonlinear and linear connections on the
tangent bundle of E and on the prolongation of E and we investigate the
relation between their coefficients. The analogue of the classical
Chern-Finsler connection is defined and studied in the paper for the case of
the holomorphic Lie algebroid.
