Homotopy principles for equivariant isomorphisms

Kutzschebauch, Frank; Larusson, Finnur; Schwarz, Gerald (2017). Homotopy principles for equivariant isomorphisms. Transactions of the American Mathematical Society, 369(10), pp. 7251-7300. American Mathematical Society 10.1090/tran/6797

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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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