The Oka principle for sections of subelliptic submersions

By F. Forstnerič. 2001

53 citations recorded by the source corpus.

Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert proved the Oka principle for maps from Stein manifolds to complex Lie groups and homogeneous spaces, as well as for sections of fiber bundles with homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's result to sections of submersions over a Stein base which admit dominating sprays over small open sets in the base; for proof see [F. Forstneric and J. Prezelj: Oka's principle for holomorphic fiber bundles with sprays, Math. Ann. 317 (2000), 117-154, and the preprint math.CV/0101040]. In this paper we prove the Oka principle for maps from Stein manifolds to any complex manifold Y that admits finitely many sprays which together dominate at every point of Y (such manifold is called subelliptic). The class of subelliptic manifolds contains all the elliptic ones, as well as complements of closed algebraic subvarieties of codimension at least two in a complex projective space or a complex Grassmanian. We also prove the Oka principle for removing intersections of holomorphic maps with closed complex subvarieties A of the target manifold Y, provided that the source manifold is Stein and the manifolds Y and Y\A are subelliptic.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1965eff542cd1c1c08bd46b4c06516d406ff817e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Survey of Oka theory (2010, background): “Gromov’s theorem was generalized to subelliptic submersions in [12].”