Algebraic approximations of holomorphic maps from Stein domains to projective manifolds Jean-Pierre Demailly 1 L´aszl´oLempert 2 Universit´ede Grenoble I Purdue University Institut Fourier, BP 74 Department of Mathematics U.R.A.188duC.N.R.S. WestLafayette,IN47907,U.S.A. 38402 Saint-Martin d’H`eres, France Bernard Shiffman 3 Johns Hopkins University Department of Mathematics Baltimore, MD 21218, U.S.A. Key words: affine algebraic manifold, algebraic approximation, algebraic curve, complete pluripolar set, Eisenman metric, H¨ormander’s L2-estimates for ∂, holo- morphic map, holomorphic retraction, holomorphic vector bundle, hyperbolic space, Kobayashi pseudodistance, Kobayashi-Royden pseudometric, Nash alge- braic map, Nash algebraic retraction, plurisubharmonic function, projective alge- braic manifold, quasi-projective variety, Runge domain, Stein manifold. A.M.S. Classification 1985: 32E30, 32H20, 14C30 Table of contents arXiv:alg-geom/9212001v3 8 Nov 1993 1. Introduction ................................................... ........................ p. 2 2. Holomorphic and Nash algebraic retractions ............................................ p. 8 3. Nash algebraic approximation on Runge domains in affine algebraic varieties...........p. 13 4. Nash algebraic approximations omitting ample divisors ................................ p. 19 5. Exhaustion of Stein manifolds by Runge domains of affine algebraic manifolds ......... p. 27 References ................................................... ........................... p. 30 1 Research partially supported by Institut Universitaire de France 2 Research partially supported by National Science Foundation Grant No. DMS-9303479 3 Research partially supported by National Science Foundation Grant No. DMS-9204037 1 1. Introduction The present work, which was motivated by the study of the Kobayashi pseudodistance on algebraic manifolds, proceeds from the general philosophy that analytic objects can be approximated by algebraic objects under suitable restric- tions.