On the topological index of irregular surfaces
  We study the topological index of some irregular surfaces that we call generalized Lagrangian. We show that under certain hypotheses on the base locus of the Lagrangian system the topological index is non-negative. For the minimal surfaces of general type with q=4 and p_g=5 we prove the same statement without any hypotheses. Some similar results for higher dimensional varieties are given. 
M. A. Barja, J. C. Naranjo, G. P. Pirola