Homotopy principles for equivariant isomorphisms
| dc.contributor.author | Kutzschebauch, F. | |
| dc.contributor.author | Lárusson, F. | |
| dc.contributor.author | Schwarz, G. | |
| dc.date.issued | 2017 | |
| dc.description | Article electronically published on May 5, 2017 | |
| dc.description.abstract | Let G be a reductive complex Lie group acting holomorphically on Stein manifolds X and Y. Let pX : X → QX and pY : Y → QY be the quotient mappings. When is there an equivariant biholomorphism of X and Y ? A necessary condition is that the categorical quotients QX and QY are biholomorphic and that the biholomorphism ϕ sends the Luna strata of QX isomorphically onto the corresponding Luna strata of QY . 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 Gequivariant biholomorphisms of the fibres pX −1(q) and pY −1(ϕ(q)) for q ∈ QX and if X satisfies an auxiliary property (which holds for most X), then Φ is homotopic, through G-homeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y . Our results improve upon those of our earlier paper [J. Reine Angew. Math. 706 (2015), 193–214] and use new ideas and techniques. | |
| dc.description.statementofresponsibility | Frank Kutzschebauch, Finnur Lárusson, and Gerald W. Schwarz | |
| dc.identifier.citation | Transactions of the American Mathematical Society, 2017; 369(10):7251-7300 | |
| dc.identifier.doi | 10.1090/tran/6797 | |
| dc.identifier.issn | 0002-9947 | |
| dc.identifier.issn | 1088-6850 | |
| dc.identifier.uri | http://hdl.handle.net/2440/107496 | |
| dc.language.iso | en | |
| dc.publisher | American Mathematical Society | |
| dc.relation.grant | 153120 | |
| dc.relation.grant | http://purl.org/au-research/grants/arc/DP120104110 | |
| dc.relation.grant | http://purl.org/au-research/grants/arc/DP150103442 | |
| dc.rights | © 2017 American Mathematical Society | |
| dc.source.uri | https://doi.org/10.1090/tran/6797 | |
| dc.subject | Oka principle; geometric invariant theory; Stein manifold; complex Lie group; reductive group; categorical quotient; Luna stratification | |
| dc.title | Homotopy principles for equivariant isomorphisms | |
| dc.type | Journal article | |
| pubs.publication-status | Published |