Normal Projective Surfaces and Dynamics of Automorphism Groups of Projective Varieties

Abstract

In Chapters 1 and 2, we studied normal projective surfaces with only quotient singularities over the complex number field. Log del Pezzo surface plays the role as the ?opposite? of surface of general type. The complete classification of log del Pezzo surfaces of Cartier index 3 and rank 2 is given in Theorem 1. Log Enriques surface is a generalization of K3 and Enriques surface. In Theorem 2, we classified all the rational log Enriques surfaces of rank 18 by giving concrete models for the realizable types of these surfaces.

In Chapter 3, we studied the relation between the geometry of a variety and its automorphism group. In particular, we prove some slightly finer Tits alternative theorems for automorphism groups of compact Kaehler manifolds (Theorems 3.1, 3.2, 3.3), give sufficient conditions for the existence of equivariant fibrations of surfaces for the dimension reduction purpose (Theorem 3.4), determine the uniqueness of automorphisms on surface (Theorem 3.5), and confirm, to some extent, the belief that a compact Kaehler manifold has lots of symmetries only when it is a torus or its quotient (Theorem 3.6).