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

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 $\mathrm{A}\mathrm{u}\mathrm{t}(X,D)$. 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 $\dim G\le \bar{q}(V)$foralogsmoothsurfacepairwith$\bar{\kappa}(V) = 0$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.