Abelian automorphism groups of threefolds of general type

TL;DR

This thesis investigates abelian automorphism groups of complex threefolds of general type, establishing a linear bound in terms of the canonical divisor's volume and improving previous results for surfaces.

Contribution

It provides a universal linear bound on the size of abelian automorphism groups of threefolds with nef canonical divisor, extending and refining earlier surface case results.

Findings

Bound on automorphism group size proportional to $K^3$

Improved previous bounds for surfaces of general type

Established universal constant for group size limit

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 $\# G \le c K^3$.