Gromov’s Oka Principle for Equivariant Maps

• Published in: The Journal of geometric analysis Vol. 31; no. 6; pp. 6102 - 6127
• Main Authors: Kutzschebauch, Frank, Lárusson, Finnur, Schwarz, Gerald W.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Ellipticity; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Interpolation; Invariants; Lie groups; Manifolds; Mathematical analysis; Mathematics; Mathematics and Statistics; Theorems; 32E10; 32E30; Secondary 14L24; 32M10; Runge approximation; 14L30; Reductive group; Complex Lie group; Primary 32M05; Elliptic manifold; Stein manifold; Oka manifold; Equivariant map; Cartan extension; 32Q28
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-020-00520-0

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group G acts on a Stein manifold X and another manifold Y in such a way that Y is G -Oka, then every G -equivariant continuous map X → Y can be deformed, through such maps, to a G -equivariant holomorphic map. Approximation on a G -invariant holomorphically convex compact subset of X and jet interpolation along a G -invariant subvariety of X can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.