On automorphisms of blowups of ${\mathbb{P}}^3$

Abstract

Let $\pi :X\to {\mathbb{P}}^3$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\to {\mathbb{P}}^3$, whose automorphism group $Aut(X)$ has only finitely many connected components. We also present a heuristic argument showing that for a "generic" compact Kähler manifold $X$ of dimension $\ge 3$, the automorphism group $Aut(X)$ has only finitely many connected components.

arXiv comment21 pages. Examples on blowups of $P^2\times P1$ and $P^1\times P^1\times P^1$ included. Combined with recent results of Bayraktar and Cantat, the heuristic argument in the previous version proves a stronger conclusion: For a "generic" compact Kahler manifold $X$ of dimension at least 3, $Aut(X)$ has only finitely many connected components

References in corpus (5)

• Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces
• The Dynamical Degrees of a Mapping
• Constraints on automorphism groups of higher dimensional manifolds
• Automorphisms of rational manifolds of positive entropy with Siegel disks
• On the automorphism group of rational manifolds

Cited by in corpus (7)

• Pseudoautomorphisms with invariant elliptic curves
• Constraints on automorphism groups of higher dimensional manifolds
• Pseudo-automorphisms with no invariant foliation
• On the automorphism group of rational manifolds
• Some properties of the group of birational maps generated by the automorphisms of ${\mathbb{P}}_{\mathbb{C}}^n$ and the standard involution
• Curves with decomposable normal vector bundles and automorphism groups
• Invertible dynamics on blow-ups of P^k