Mathematics – Algebraic Geometry
Scientific paper
2006-10-28
Mathematics
Algebraic Geometry
15 pages
Scientific paper
We consider smooth threefolds $Y$ defined over $\Bbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$. Let $X$ be a smooth projective threefold containing $Y$ and $D$ be the boundary divisor with support $X-Y$. We are interested in the following question: What geometry information of $X$ can be obtained from the regular function information on $Y$? Suppose that the boundary $X-Y$ is a smooth projective surface. In this paper, we analyse two different cases, i.e., there are no nonconstant regular functions on $Y$ or there are lots of regular functions on $Y$. More precisely, if $H^0(Y, {\mathcal{O}}_Y)=\Bbb{C}$, we prove that ${1/2}(c_1^2+c_2)\cdot D=\chi({\mathcal{O}}_D)\geq 0$. In particular, if the line bundle ${\mathcal{O}}_D(D)$ is not torsion, then $q=h^1(X, {\mathcal{O}}_X)=0$, ${1/2}(c_1^2+c_2)\cdot D=\chi({\mathcal{O}}_D)=0$, $\chi({\mathcal{O}}_X) >0$ and $K_X$ is not nef. If there is a positive constant $c$ such that $h^0(X, {\mathcal{O}}_X(nD))\geq c n^3$ for all sufficiently large $n$ (we say that $D$ is big or the $D$-dimension of $X$ is 3) and $D$ has no exceptional curves, then $|nD|$ is base point free for $n\gg 0$. Therefore $Y$ is affine if $D$ is big.
No associations
LandOfFree
On threefolds without nonconstant regular functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On threefolds without nonconstant regular functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On threefolds without nonconstant regular functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-652018