Homotopy principles for equivariant isomorphisms
Date
2017
Authors
Kutzschebauch, F.
Lárusson, F.
Schwarz, G.
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Journal article
Citation
Transactions of the American Mathematical Society, 2017; 369(10):7251-7300
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Frank Kutzschebauch, Finnur Lárusson, and Gerald W. Schwarz
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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.
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Dissertation Note
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Article electronically published on May 5, 2017
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© 2017 American Mathematical Society