Algebraic approximations of holomorphic maps from Stein domains to projective manifolds
| dc.creator | Demailly, Jean-Pierre | |
| dc.creator | Lempert, Laszlo | |
| dc.creator | Shiffman, Bernard | |
| dc.date | 1992-12-01 | |
| dc.date | 1993-11-08 | |
| dc.date.accessioned | 2026-07-07T08:57:40Z | |
| dc.date.available | 2026-07-07T08:57:40Z | |
| dc.description | It is shown that every holomorphic map $f$ from a Runge domain $Ω$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_ν$ defined over an exhausting sequence of relatively compact open sets $Ω_ν$ in $Ω$. 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 $A\capΩ$ is algebraic, then $f_ν$ can be taken to coincide with $f$ on $A\capΩ_ν$. The main application of these results, when $Ω$ 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 $f_ν$ can be taken to have their images contained in affine Zariski open subsets of $X$. By using complex analytic methods (pluricomplex potential theory and Hörmander'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 | |
| dc.description | 32 pages, plain-TeX. Note: This paper is a revision of our manuscript of December 11, 1992. The present version contains many technical changes in the first three sections. More general results are obtained with a simpler proof | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9212001 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9212001 | |
| dc.identifier | Duke Math. J. 76 (1994), 333-363 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/147054 | |
| dc.subject | Algebraic Geometry | |
| dc.title | Algebraic approximations of holomorphic maps from Stein domains to projective manifolds | |
| dc.type | text |