[Paper Review] Abelian automorphism groups of threefolds of general type

Jin-Xing Cai|ArXiv.org|Feb 15, 1995

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper establishes a linear bound on the size of abelian automorphism groups of smooth threefolds of general type over ℂ, provided the canonical divisor K is nef. Using techniques from algebraic geometry and group actions on varieties, the author improves upon Xiao's earlier results by proving that the order of such a group G satisfies #G ≤ cK³ for a universal constant c, thus providing a sharp quantitative constraint on symmetry in these threefolds.

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

Motivation & Objective

• To investigate the structure and size of abelian automorphism groups acting on smooth threefolds of general type over ℂ.
• To extend and refine existing bounds on automorphism groups in algebraic geometry, particularly for threefolds.
• To improve upon Xiao's earlier results concerning abelian automorphism groups of minimal smooth surfaces of general type.
• To establish a universal linear bound in terms of the self-intersection number K³ for such groups under the nefness condition on K.
• To provide a quantitative understanding of symmetries in higher-dimensional algebraic varieties of general type.

Proposed method

• The study employs techniques from algebraic geometry, particularly focusing on the geometry of canonical divisors and group actions on complex threefolds.
• The author analyzes the action of abelian groups G on smooth threefolds X of general type, assuming the canonical divisor K is numerically effective (nef).
• Using the nefness of K, the paper derives constraints on the possible order of G through intersection theory and representation theory of finite abelian groups.
• The proof relies on the structure of the Néron-Severine model and the behavior of G-invariant linear systems on X.
• A key step involves bounding the dimension of the space of G-invariant sections of |mK| for large m, leading to a bound on #G.
• The final bound is derived via a universal constant c such that #G ≤ cK³, independent of the specific threefold.

Experimental results

Research questions

• RQ1 What is the maximal possible size of an abelian automorphism group acting on a smooth threefold of general type over ℂ?
• RQ2 How does the nefness of the canonical divisor K influence the structure and size of abelian automorphism groups on threefolds of general type?
• RQ3 Can Xiao's results on surfaces of general type be extended and improved to the case of threefolds with nef canonical divisors?
• RQ4 Is there a universal linear bound on the order of abelian automorphism groups in terms of K³ for threefolds of general type with K nef?
• RQ5 What geometric and cohomological constraints arise from the action of an abelian group on a threefold of general type?

Key findings

• The paper establishes a linear upper bound on the order of abelian automorphism groups of smooth threefolds of general type, given by #G ≤ cK³ for a universal constant c.
• The bound holds under the assumption that the canonical divisor K is numerically effective (nef), which is a key geometric constraint.
• The result improves upon Xiao's earlier bounds for minimal smooth surfaces of general type, extending the scope to threefolds.
• The proof relies on the interplay between group actions, linear systems, and intersection theory on threefolds.
• The bound is effective and uniform across all such threefolds, making it applicable in classification and moduli problems.
• The result provides a sharp quantitative limit on the symmetry of threefolds of general type with nef canonical bundle.