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 P k ( C ) by a finite sequence of blow-ups along smooth centers of dimension at most r with k > 2 r + 2 has finite image in G L ( H ∗ ( X , Z ) ) . In particular, every holomorphic automorphism f : X → X has zero topological entropy.