In a previous paper, the first author made a general study of the geometry of JM jets bundle, which has a holomorphic structure. In the present paper we define the complex second order Lagrange space (M,L) and we prove the existence of a special complex nonlinear connection, provided by a complex spray deduced from the variational problem. With respect to adapted frames of this (c.n.c.) we emphasize the existence of a N -linear connection, named the Chern-Lagrange connection on (M,L), which is of (1, 0)-type and will play a fundamental role in the study of the complex second order Lagrange spaces. Mathematics Subject Classification 2000: 53B40, 53C60.