Quotients, automorphisms and differential operators

Lárusson

2022

Complex Anal Synerg

Self Cite

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle $${{\mathscr {G}}}$$ G , all over a reduced Stein space X with compatible actions of a reductive complex group on E, $${{\mathscr {G}}}$$ G , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a G-manifold being G-Oka.

“…In fact, any holomorphic differential operator on Q lifts. V has the DP [ 34 , Theorem 2.2]. (The condition that V be admissible in the cited theorem is implied by V being large.)…”

Section: The Linearisation Problemmentioning

confidence: 99%

“…V has the DP [ 34 , Theorem 2.2]. (The condition that V be admissible in the cited theorem is implied by V being large.)…”

Section: The Linearisation Problemmentioning

confidence: 99%

Equivariant Oka theory: survey of recent progress

Lárusson

2022

Complex Anal Synerg

Self Cite

Homotopy principles for equivariant isomorphisms

Lárusson

Trans. Amer. Math. Soc.

2017

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 Characterization of Linearizability for Holomorphic ℂ*-Actions

International Mathematics Research Notices

2021

Let $G$ be a reductive complex Lie group acting holomorphically on $X=\mathbb{C}^n$. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on $\mathbb{C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $\Phi \colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $X/\!\!/G$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $\Phi $ as above. Then $\Phi $ induces a biholomorphism ${\varphi }\colon X/\!\!/G\to V/\!\!/G$ that is stratified, that is, the stratum of $X/\!\!/G$ with a given label is sent isomorphically to the stratum of $V/\!\!/G$ with the same label. The counterexamples to the Linearization Problem construct an action of $G$ such that $X/\!\!/G$ is not stratified biholomorphic to any $V/\!\!/G$. Our main theorem shows that, for a reductive group $G$ with $\dim G\leq 1$, the existence of a stratified biholomorphism of $X/\!\!/G$ to some $V/\!\!/G$ is not only necessary but also sufficient for linearization. In fact, we do not have to assume that $X$ is biholomorphic to $\mathbb{C}^n$, only that $X$ is a Stein manifold.