Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Affine Algebraic Varieties   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.

31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys