Algebraic approximations of holomorphic maps from Stein domains to projective manifolds

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