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 $\rm {Aut}(X, D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min \{n-\bar {\kappa }(V), n\}$ with $V\coloneq X\setminus D$. In the dimension two, 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.