Mapping cylinders and the Oka principle

Abstract

Abstract. We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are equivalent to being Stein.

Extending holomorphic mappings from subvarieties in Stein manifolds

“…The remaining implications are consequences of definitions and of Proposition 1.2 in [17] concerning the holomorphic extendability of holomorphic maps from a subvariety X 0 ⊂ X to an open neighborhood of X 0 in X. The implication (c)⇒(b) was first proved by Lárusson [29].…”

“…However, using an idea of Lárusson (Theorem 1 in [29]) we show that this addition, as well as the jet interpolation on complex subvarieties, follows automatically (and rather easily) from the Oka property with interpolation. We shall say that a complex manifold Y satisfying the conclusion of proposition 1.2 for any (X, X 0 , K, f 0 ) enjoys the Oka property with jet interpolation and approximation.…”

“…-Lárusson proved that for a Stein manifold, the basic Oka property with interpolation implies ellipticity, i.e., existence of a dominating spray (Theorem 2 in [29]). This also follows from proposition 1.2 and the known result that ellipticity of a Stein manifold is implied by second order jet interpolation ( [21]; proposition 1.2 in [17]).…”

“…I thank F. Lárusson for useful discussions; in particular, the proof of proposition 1.2 is motivated by Theorem 1 in his recent preprint [29]. I also thank B. Drinovec-Drnovšek from whom I have learned the geometric scheme (pushing out by a convex combination of two plurisubharmonic functions) used in the proof of proposition 4.1 and in [10], p. 179, and J. Prezelj-Perman for pointing out an inaccuracy in an earlier version of the paper.…”

Book Review: From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds

“…While affine manifolds are open analogues of smooth algebraic varieties, Stein manifolds form a much larger class within the analytic category. They are sometimes thought of as being natural domains for holomorphic maps with values in arbitrary complex manifolds [27].…”