The dimension of automorphism groups of algebraic varieties with pseudo-effective log canonical divisors

Abstract

Let $(X,D)$ be a log smooth pair of dimension $n$, where $D$ is a reduced effective divisor such that the log canonical divisor $K_X+D$ is pseudo-effective. Let $G$ be a connected algebraic subgroup of $\mathrm{Aut}(X,D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min \{n-\bar{\kappa }(V),n\}$ with $V:=X\setminus D$. In the dimension two, Shigeru Iitaka claimed in his 1979 Osaka J. Math. paper that $\dim G\le \bar{q}(V)$ for a log smooth surface pair with $\bar{\kappa }(V)=0$ and ${\bar{p}}_g(V)=1$. We (re)prove and generalize this classical result for all surfaces with $\bar{\kappa }=0$ without assuming Iitaka's classification of logarithmic Iitaka surfaces or logarithmic $K3$ surfaces.

minor changes; to appear in Proceedings of the AMS

References in corpus (1)

• Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy