Tuyen Trung Truong
Abstract: In this paper we mainly study the following question: For what projective manifold \(X\) of dimension \(\geq 3\) that any \(f\in \operatorname{Aut}(X)\) has zero topological entropy? In a previous paper we showed heuristically that for a "generic" compact Kähler manifold \(X\) of dimension \(\geq 3\), the automorphism group \(\operatorname{Aut}(X)\) has only finitely many connected components, in particular any automorphism of its has zero topological entropy. This was proved using a non-vanishing condition of nef cohomology classes. In the current paper, we use generalized non-vanishing conditions to study the case where \(X\rightarrow X_0\) is a finite blowup along smooth centers, here \(X_0\) is a projective manifold of interest. Here we allow \(X_0\) to be either one of the following manifolds: it has Picard number 1, or a Fano manifold, or it is a projective hyper-Kähler manifold. We also allow the centers of blowups to have large dimensions relative to that of \(X_0\) (may be up to \(\dim(X_0)-2\)). As a consequence, we obtain new examples of manifolds \(X\), whose any automorphism is either of zero topological entropy or is cohomologically hyperbolic.
ArXiv: 1301.4957