On G-birational rigidity of projective spaces

Abstract

In this paper, we study finite subgroups $G\subset \mathrm{Aut}({\mathbb{P}}^n)$ such that ${\mathbb{P}}^n$ is $G$-birationally rigid. For each $n⩾3$, we prove that $\mathrm{Aut}({\mathbb{P}}^n)$ contains at most finitely many such subgroups up to conjugation. For $n=4,5,7$, we prove that ${\mathbb{P}}^n$ is $G$-birationally superrigid if $G$ is a primitive subgroup isomorphic to ${\mathrm{PSp}}_4({\mathbf{F}}_3)$, ${\mathrm{PSU}}_4({\mathbf{F}}_3)⋊{\mathbf{\mu }}_2$, ${\mathrm{O}}_8^{+}({\mathbf{F}}_2)⋊{\mathbf{\mu }}_2$, respectively.

53 pages. Expanded and revised version. One conjecture is removed, and two new theorems are added (one gave a counter example to the removed conjecture). Proofs of several auxiliary results are revised. All magma scripts are removed, because revised proofs do not use them anymore