Abelian automorphism groups of threefolds of general type

Abstract

This thesis is devoted to the study of abelian automorphism groups of surfaces and $3$-folds of general type over complex number field $\Bbb C$. We obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of general type whose canonical divisor $K$ is numerically effective, and we improve on Xiao's results on abelian automorphism groups of minimal smooth projective surfaces of general type. More precisely, the main results in this thesis are the following. {\bf Theorem 3.0.} Let $X$ be a smooth 3-fold of general type over the complex number field, $K$ the canonical divisor of $X$. Let $G$ be an abelian group of automorphisms of $X$. Suppose $K$ is nef. Then there exists a universal constant coefficient $c$ such that $\# G \le c K^3$.