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 \(X\rightarrow Y\) 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.

我们在发展现代Gromov风格的Oka理论的等变形式方面迈出了第一步。我们定义了复杂流形的标准Oka属性，椭圆率和同伦Runge属性的等变版本，证明它们满足所有预期的基本属性，并提供示例。我们的主要定理是等变Oka原理，即如果有限群G作用在Stein流形X和另一个流形Y上，使得Y为G -Oka，则每个G-等距连续图\（X \ rightarrow Y \ ）可变形，通过这样的地图，到ģ -equivariant全纯地图。G上的近似值-invariant的全纯凸紧子集X和喷气插值沿着ģ的-invariant subvariety X可以被内置到定理。我们推测该定理适用于任意还原复李群的作用，并证明了这一结果的部分结果。