On the rank of the flat unitary summand of the Hodge bundle

Let f : S -> B be a nonisotrivial fibered surface. We prove that the genus g, the rank u(f) of the unitary summand of the Hodge bundle f*omega(f), and the Clifford index c(f) satisfy the inequality u(f) <= g - c(f). Moreover, we prove that if the general fiber is a plane curve of degree >= 5, then the stronger bound u(f) <= g - c(f) - 1 holds. In particular, this provides a strengthening of bounds proved by M. A. Barja, V. Gonzalez-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth.