Kutzschebauch, F.Lárusson, F.Schwarz, G.2017-09-012017-09-012017Transactions of the American Mathematical Society, 2017; 369(10):7251-73000002-99471088-6850http://hdl.handle.net/2440/107496Article electronically published on May 5, 2017Let 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.en© 2017 American Mathematical SocietyOka principle; geometric invariant theory; Stein manifold; complex Lie group; reductive group; categorical quotient; Luna stratificationJournal article003007481310.1090/tran/67970004072038000162-s2.0-85027004276368268