On threefolds without nonconstant regular functions

Abstract

We consider smooth threefolds $Y$ defined over $\Bbb{C}$ with $H^i(Y, Ω^j_Y)=0$ 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 ${1/2}(c_1^2+c_2)\cdot D=χ({\mathcal{O}}_D)\geq 0$. In particular, if the line bundle ${\mathcal{O}}_D(D)$ is not torsion, then $q=h^1(X, {\mathcal{O}}_X)=0$, ${1/2}(c_1^2+c_2)\cdot D=χ({\mathcal{O}}_D)=0$, $χ({\mathcal{O}}_X) >0$ 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.