Constraints on automorphism groups of higher dimensional manifolds

TL;DR

This paper establishes that certain higher-dimensional rational manifolds have finite automorphism groups with zero topological entropy, based on their construction via blow-ups.

Contribution

It provides new constraints on automorphism groups of rational manifolds obtained through specific blow-up procedures.

Findings

Automorphism groups of these manifolds have finite image in GL(H^*(X,Z)).

Every holomorphic automorphism has zero topological entropy.

Results apply to manifolds obtained from CP^k by blow-ups along low-dimensional centers.

Abstract

In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from CP^k 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 $f:X\to X$ has zero topological entropy.