Gromov’s Oka Principle for Equivariant Maps

Abstract

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.

Equivariant Oka theory: survey of recent progress

“…[ 24 , Example 2.7] If , , is a polynomial such that vanishes nowhere on , then the affine algebraic manifold has the algebraic density property and is, therefore, Oka [ 17 ]. The fixed-point set W of the involution of X is smooth, given by the formula , and is a double branched covering of with branch locus .…”

“…[ 24 , Proposition 2.1] Let a complex reductive group G act holomorphically on a complex manifold Y . If G acts trivially on Y , then Y is G -Oka if and only if Y is Oka.…”

“…Here are three ways to construct new equivariantly Oka manifolds from old. The first two are from [ 24 ]. The localisation principle in (c) is due to Kusakabe [ 19 , Theorem A.5].…”

“…The first main theorem of [ 24 ] is Theorem G for the case when is a finite group. Let us give a rough sketch of the proof of part (a).…”

“…Parts (b) and (c) then follow by equivariant adaptations of standard methods. The proof of Theorem G is completed in [ 24 , Sect. 5].…”