Algebraic approximations of holomorphic maps from Stein domains to projective manifolds
Algebraic approximations of holomorphic maps from Stein domains to projective manifolds
复制标题
从 Stein 域到射影流形的全纯映射的代数近似
DOI：
10.1215/s0012-7094-94-07612-6
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发表时间：
1992
影响因子：
2.5
通讯作者：
B. Shiffman
中科院分区：
文献类型：
--
作者：
J. Demailly;L. Lempert;B. Shiffman
证明了从仿射代数簇$S的Runge区域$\Omega$到射影代数流形$X$的每个全纯映射$f_\nu$都是定义在$\Omega$中的相对紧开集序列$\Omega_\nu$上的Nash代数映射的一致极限.还给出了一个相对形式：如果$S$中有一个代数子簇$A$(不一定是约化的)，使得$f$对$A\CAP\Omega$的限制是代数的，则$f_\nu$可视为与$A\CAP\Omega_\nu$上的$f$重合.当单位圆为单位圆时，这些结果的主要应用是证明拟射影代数流形$Z$的Kobayashi伪距离和Kobayashi-Royden无穷小度量仅根据$Z$中的闭代数曲线是可计算的.类似地，拟射影代数流形的$p$维Eisenman度量可以用它的$p$维代数子簇的Eisenman体积来计算。文中讨论的另一个问题是，是否可以认为近似的$f_\nu$的像包含在$X$的仿射Zariski开子集中。通过使用复分析方法(多重位势理论和Ormander的$L^2估计)，我们证明了如果$f$是一个嵌入(与$\dim S和$dim X$)，并且如果$X$上有充足的线丛$L$使得
It is shown that every holomorphic map $f$ from a Runge domain $\Omega$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_\nu$ defined over an exhausting sequence of relatively compact open sets $\Omega_\nu$ in $\Omega$. A relative version is also given: If there is an algebraic subvariety $A$ (not necessarily reduced) in $S$ such that the restriction of $f$ to $A\cap\Omega$ is algebraic, then $f_\nu$ can be taken to coincide with $f$ on $A\cap\Omega_\nu$. The main application of these results, when $\Omega$ is the unit disk, is to show that the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric of a quasi-projective algebraic manifold $Z$ are computable solely in terms of the closed algebraic curves in $Z$. Similarly, the $p$-dimensional Eisenman metric of a quasi-projective algebraic manifold can be computed in terms of the Eisenman volumes of its $p$-dimensional algebraic subvarieties. Another question addressed in the paper is whether the approximations $f_\nu$ can be taken to have their images contained in affine Zariski open subsets of $X$. By using complex analytic methods (pluricomplex potential theory and H\"ormander's $L^2$ estimates), we show that this is the case if $f$ is an embedding (with $\dim S<\dim X$) and if there is an ample line bundle $L$ on $X$ such that