arXiv math/06108812006-10-28math.AGmath.CV

关于一类三重折叠的$D$ -维

On the $D$-dimension of a certain type of threefolds

Jing Zhang

备注：: 14 pages

摘要：

设$Y$是一个3维代数流形，对于所有的$j\geq 0$，$i>0$和$h^0(Y, {\mathcal{O}}_Y) > 1$都有$H^i(Y, Ω^j_Y)=0$.设$X$是$Y$的光滑完备化，使得边界$X-Y$是$X$上有效除数$D$的支集，且具有简单的法向交叉.我们证明了$X$的$D$ -维数不可能是2，即，在$Y$上，要么任意两个非常数正则函数代数相关，要么存在三个代数独立的非常数正则函数。其次，如果$X$的$D$维数大于1，则$Y$的相关方案同构于Spec $Γ(Y, {\mathcal{O}}_Y)$。进一步证明了任意维数$d\geq 1$的代数流形$Y$是仿射的当且仅当对所有$j\geq 0$，$i>0$，$H^i(Y, Ω^j_Y)=0$是正则可分的，即，对于$Y$上的任何两个不同点$y_1$、$y_2$，存在$Y$上的正则函数$f$，使得$f(y_1)\neq f(y_2)$。

英文摘要：

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective divisor $D$ on $X$ with simple normal crossings. We prove that the $D$-dimension of $X$ cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebraically independent nonconstant regular functions on $Y$. Secondly, if the $D$-dimension of $X$ is greater than 1, then the associated scheme of $Y$ is isomorphic to Spec$Γ(Y, {\mathcal{O}}_Y)$. Furthermore, we prove that an algebraic manifold $Y$ of any dimension $d\geq 1$ is affine if and only if $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$ and it is regularly separable, i.e., for any two distinct points $y_1$, $y_2$ on $Y$, there is a regular function $f$ on $Y$ such that $f(y_1)\neq f(y_2)$.

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