On automorphisms of blowups of projective manifolds

Original · EN

In this paper we mainly study the following question: For what projective manifold X of dimension ≥ 3 that any f∈ Aut(X) has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where X→ X₀ is a finite blowup along smooth centers, here X₀ is a projective manifold of interest. Here we allow X₀ 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 X₀ (may be upto dim(X₀)-2). Explicit constructions are given in Section SectionBlowupsAndNonVanishingConditions, where we also show that the assumptions in the results in that section are necessary (see Example 6 in Section SectionBlowupsAndNonVanishingConditions). As a consequence, we obtain new examples of manifolds X, whose any automorphism is either of zero topological entropy or is cohomologically hyperbolic.