  A compact complex manifold X is said to be rationally cohomologically
rigidified if its automorphism group Aut(X) acts faithfully on the cohomology
ring H*(X,Q). In this note, we prove that, surfaces of general type with
irregularity q>2 are rationally cohomologically rigidified, and so are minimal
surfaces S with q=2 unless K^2=8X. This answers a question of Fabrizio Catanese
in part.
  As examples we give a complete classification of surfaces isogenous to a
product with q=2 that are not rationally cohomologically rigidified. These
surfaces turn out however to be rigidified.
