On automorphisms of blowups of projective manifolds

Abstract

In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\ge 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\to X_0$ 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 upto $dim(X_0)-2$). Explicit constructions are given in Section \ref{SectionBlowupsAndNonVanishingConditions}, where we also show that the assumptions in the results in that section are necessary (see Example 6 in Section \ref{SectionBlowupsAndNonVanishingConditions}). As a consequence, we obtain new examples of manifolds $X$, whose any automorphism is either of zero topological entropy or is cohomologically hyperbolic.

27 pages. Slightly modified the statements and/or proofs of some results, added several new examples including one showing that the assumptions in the results in Section 2 can not be removed

References in corpus (2)

• Automorphisms of rational manifolds of positive entropy with Siegel disks
• On the automorphism group of rational manifolds

Cited by in corpus (1)

• Curves with decomposable normal vector bundles and automorphism groups