Constraints on automorphism groups of higher dimensional manifolds

In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from Pk(C) by a finite sequence of blow-ups along smooth centers of dimension at most r with k>2r+2 has finite image in GL(H∗(X,Z)). In particular, every holomorphic automorphism f:X→X has zero topological entropy.