On automorphisms of blowups of projective manifolds

In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\geq 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\rightarrow X_0$ 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-K\"ahler 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.