ON AUTOMORPHISM GROUPS OF ALGEBRAIC VARIETIES - A DYNAMICAL VIEWPOINT

Abstract

Throughout this thesis, we work over the field C of complex numbers. My thesis contains the following three parts.

1. We generalized a surface result, that is, a parabolic automorphism of a compact Kähler surface preserves an elliptic fibration, to hyperkähler manifolds. We gave a criterion for the existence of equivariant fibrations on hyperkähler manifolds from a dynamical viewpoint.

2. We studied virtually solvable groups G of maximal dynamical rank acting on a compact Kähler manifold X. Based on the known Tits alternative type theorem of Zhang (2009), we generalized a finiteness result for the null-entropy subset of a commutative automorphism group due to Dinh–Sibony (2004). We then determined positive-dimensional G-periodic proper subvarieties of the pair (X, G) of maximal dynamical rank. Actually, among other results, we proved that the union of all these positive-dimensional G-periodic proper subvarieties is a Zariski closed subset of X.

3. An upper bound about the dimension of automorphism groups of algebraic varieties with pseudo-effective log canonical divisors was obtained. In dimension two, I generalized a classical result of Iitaka for logarithmic Iitaka surfaces to arbitrary algebraic surfaces of logarithmic Kodaira dimension 0.