The author studies the surjectivity of the exponential function for complex algebraic (in particular, complex semisimple) Lie groups. The main result is that for complex algebraic, or more generally, complex splittable Lie groups the surjectivity of the exponential function is equivalent to the connectedness in the Lie group of the centralizers of the nilpotent elements in the Lie algebra. This implies that the only complex semisimple Lie groups with surjective exponential function are isomorphic to finite products of the adjoint groups of SL\((n,{\mathbb C})\).