Let M be an open Riemann surface and A be the punctured cone in C n \ {0} on a smooth projective variety Y in P n−1 . Recently, Runge approximation theorems with interpolation for holomorphic immersions M → C n , directed by A, have been proved under the assumption that A is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by A from a smooth affine curve M into C n . The Oka property is naturally replaced by the stronger assumption that A is algebraically elliptic, which it is if Y is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.

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The subject of this paper is the homotopy principle, also called the h-principle or the Oka-Grauert principle, concerning sections of certain holomorphic fiber bundles on Stein manifolds. We give a proof of a theorem of Gromov (1989) from sec. 2.9 in [Gro]; see theorems 1. 3 and 1.4 below. This result, which extends the work of H. Grauert from 1957 ([Gr3], [Gr4], [Car]), has been used in the proofs of the embedding theorem for Stein manifolds into Euclidean spaces of minimal dimension [EGr], [Sch].1.1 Definition. Let h: Z → X be a holomorphic mapping of complex manifolds. A section of h is any map f : X → Z such that h• f is the identity on X. We say that sections of h satisfy the h-principle (or the Oka-Grauert principle) if each continuous section f 0 : X → Z can be deformed to a holomorphic section f 1 : X → Z through a continuous one parameter family (a homotopy) of continuous sections f t : X → Z (0 ≤ t ≤ 1), and any two holomorphic sections f 0 , f 1 : X → Z which are homotopic through continuous sections are also homotopic through holomorphic sections. If this holds for a trivial bundle Z = X × F → X, we say that maps X → F satisfy the h-principle.1.2 Definition. (Gromov [Gro]) A (dominating) spray on a complex manifold F is a holomorphic vector bundle p: E → F , together with a holomorphic map s: E → F , such that s is the identity on the zero section F ⊂ E, and for each x ∈ F the derivative Ds(x) maps E x (which is a linear subspace of T x E) surjectively onto T x F .The following result can be found in sec. 2.9 of [Gro].1.3 Theorem. If F is a complex manifold which admits a spray, then the sections of any locally trivial holomorphic fiber bundle with fiber F over any Stein manifold satisfy the h-principle. In particular, mappings from Stein manifolds into F satisfy the h-principle.Stronger results are given in theorem 1.4 and corollary 1.5 below. In the sequel [FP] to this paper we give a proof of Gromov's Main Theorem ([Gro], sect. 4.5) to the effect that the h-principle holds for sections of holomorphic submersions h: Z → X, where X is Stein and each point x ∈ X has a neighborhood U ⊂ X such that Z|U = h −1 (U ) admits a fiber-spray (see def. 3.1 below).For non-specialists we recall that a complex manifold is called Stein (after Karl Stein, 1951 [Ste]) if it has 'plenty' of global holomorphic functions. For the precise definition