Homotopy principles for equivariant isomorphisms

Abstract

Abstract. Let G be a reductive complex Lie group acting holomorphically on Stein manifolds X and Y . Let p X : X → Q X and p Y : Y → Q Y be the quotient mappings. When is there an equivariant biholomorphism of X and Y ? A necessary condition is that the categorical quotients Q X and Q Y are biholomorphic and that the biholomorphism ϕ sends the Luna strata of Q X isomorphically onto the corresponding Luna strata of Q Y . Fix ϕ. We demonstrate two homotopy principles in this situation. The first result says that if there is a G-diffeomorphism Φ : X → Y , inducing ϕ, which is G-biholomorphic on the reduced fibres of the quotient mappings, then Φ is homotopic, through G-diffeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y . The second result roughly says that if we have a G-homeomorphism Φ : X → Y which induces a continuous family of G-equivariant biholomorphisms of the fibres p X −1 (q) and p Y −1 (ϕ(q)) for q ∈ Q X and if X satisfies an auxiliary property (which holds for most X), then Φ is homotopic, through Ghomeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y . Our results improve upon those of [KLS15] and use new ideas and techniques.

A homotopy theorem for Oka theory

“…These two key results are in Cartan's exposition of Grauert's work Proposition 1 and 2 (see [3], p. 109). [14], p. 7293).…”

“…The Oka principle for equivariant isomorphisms: In this setting it is hard to show that the given inclusion Φ ֒→ Ψ is a local weak homotopy equivalence. Theorem 1.3, p. 7253 in [14] reduces the proof to the case where we have X = Y for given Stein G-manifolds X and Y , where G is a complex reductive Lie group. Having this, one needs to extend some Lemmata from [14] in Section 3 and 5 to analogous parametric versions.…”

“…Theorem 1.3, p. 7253 in [14] reduces the proof to the case where we have X = Y for given Stein G-manifolds X and Y , where G is a complex reductive Lie group. Having this, one needs to extend some Lemmata from [14] in Section 3 and 5 to analogous parametric versions. These necessary adaptations are simple once Section 3 and 5 in [14] are understood.…”

“…In concrete terms, Grauert's result is proved using the Oka principle for principal G-bundles [3,10]. All others depend on extensions of Grauert's work, namely on the Oka principle for admissible pairs of sheaves [4] in the case of Forster and Ramspott's and Leiterer's results, on the Oka principle for elliptic submersions [9,12] in the case of Gromov's result, and on the Oka principle for equivariant isomorphisms [14] in the case of the result due to Kutzschebauch, Lárusson and Schwarz. The first three theorems can be proved alternatively with Forstnerič's Oka principles for stratified fiber bundles [7], a generalization of Gromov's work to stratified settings including more general fibers and possibly non-smooth base spaces.…”

“…Its proof builds on ideas of Gromov [11,12] and on the work of Forstnerič and Prezelj [9], who have carried out many steps of the proof of Theorem 1 in the special case of elliptic submersions. References to the sections in [3,4,9,12,14] providing the analytic key ingredients from the assumptions in Theorem 1 are given in the appendix. A convenient notion to formulate any question arising naturally in Oka theory is the notion of a sheaf of topological spaces.…”