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

Abstract

Let ( X , D ) (X,D) be a log smooth pair of dimension n n , where D D is a reduced effective divisor such that the log canonical divisor K X + D K_X + D is pseudo-effective. Let G G be a connected algebraic subgroup of A u t ( X , D ) \rm {Aut}(X, D) . We show that G G is a semi-abelian variety of dimension ≤ min { n − κ ¯ ( V ) , n } \le \min \{n-\bar {\kappa }(V), n\} with V ≔ X ∖ D V\coloneq X\setminus D . In the dimension two, Iitaka claimed in his 1979 Osaka J. Math. paper that dim ⁡ G ≤ q ¯ ( V ) \dim G\le \bar {q}(V) for a log smooth surface pair with κ ¯ ( V ) = 0 \bar {\kappa }(V) = 0 and p ¯ g ( V ) = 1 \bar {p}_g(V) = 1 . We (re-)prove and generalize this classical result for all surfaces with κ ¯ = 0 \bar {\kappa }=0 without assuming Iitaka’s classification of logarithmic Iitaka surfaces or logarithmic K 3 K3 surfaces.