A. Van de Ven has pointed out in connection with his recent results(1) that there are not many known examples of compact connected complex 4-manifolds with positive indices. The purpose of this note is to exhibit a series of compact connected complex 4-manfolds M . . . . n, m = 2, 3, 4, ..., with positive indices. Each complex 4-manifold Mn.m is an irregular algebraic surface having a structure...