Affine Algebraic Varieties

TL;DR

This paper establishes new criteria for determining when a complex algebraic variety is affine, based on the absence of complete curves, vanishing higher cohomology, and properties of boundary divisors, with illustrative examples.

Contribution

It provides a comprehensive set of conditions characterizing affineness of algebraic varieties over complex numbers, including new examples and distinctions from previous criteria.

Findings

A variety is affine if it contains no complete curves, has vanishing higher cohomology, and a boundary divisor supporting a big divisor.

Examples show that satisfying only two of the three conditions is insufficient for affineness.

The paper clarifies the role of boundary divisors and cohomology in affineness, independent of the noetherian property of the coordinate ring.

Abstract

In this paper, we give new criteria for affineness of a variety defined over $\Bbb{C}$. Our main result is that an irreducible algebraic variety $Y$ (may be singular) of dimension $d$ ($d\geq 1$) defined over $\Bbb{C}$ is an affine variety if and only if $Y$ contains no complete curves, $H^i(Y, {\mathcal{O}}_Y)=0$ for all $i>0$ and the boundary $X-Y$ is support of a big divisor, where $X$ is a projective variety containing $Y$. We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor $D$ and the affineness of $Y$. If $Y$ is an affine variety, then the ring $\Gamma (Y, {\mathcal{O}}_Y)$ is noetherian. However, to prove that $Y$ is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring $\Gamma (Y, {\mathcal{O}}_Y)$ directly but use the techniques of sheaf and cohomology.