We generalize our elliptic characterization of Oka manifolds to Oka maps. The generalized characterization can be considered as an affirmative answer to the relative version of Gromov’s conjecture. ...As an application, we unify previously known Oka principles for submersions; namely the Gromov type Oka principle for subelliptic submersions and the Forstnerič type Oka principle for holomorphic fiber bundles with CAP fibers. We also establish the localization principle for Oka maps which gives new examples of Oka maps.
Homotopy principles for equivariant isomorphisms Kutzschebauch, Frank; Lárusson, Finnur; Schwarz, Gerald
Transactions of the American Mathematical Society,
10/2017, Letnik:
369, Številka:
10
Journal Article
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Let G be a reductive complex Lie group acting holomorphically on Stein manifolds X and Y. Let p_X\colon X\to Q_X and p_Y\colon Y\to 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 \phi sends the Luna strata of Q_X isomorphically onto the corresponding Luna strata of Q_Y. Fix \phi . We demonstrate two homotopy principles in this situation. The first result says that if there is a G-diffeomorphism \Phi \colon X\to Y, inducing \phi , which is G-biholomorphic on the reduced fibres of the quotient mappings, then \Phi 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 \Phi \colon X\to Y which induces a continuous family of G-equivariant biholomorphisms of the fibres p_X{^{-1}}(q) and p_Y{^{-1}}(\phi (q)) for q\in Q_X and if X satisfies an auxiliary property (which holds for most X), then \Phi 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.
The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind ...of theorems for Riemann surfaces of second kind is an estimate of the number of irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analog can be interpreted as a quantitatve statement on the limitation for Gromov’s Oka principle. For any finite open Riemann surface
X
(maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from
X
to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. The bound depends on a conformal invariant of the Riemann surface. If
X
σ
is the
σ
-neighbourhood of a skeleton of an open Riemann surface with finitely generated fundamental group, then the number of irreducible holomorphic mappings up to homotopy from
X
σ
to the twice punctured complex plane grows exponentially in
1
σ
.
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. Assume that we have a biholomorphism Q := QX ➞ QY and ...an open cover {Ui} of Q and G-biholomorphisms ${\Phi _i}:p_X^{ - 1}\left( {{U_i}} \right) \to p_Y^{ - 1}\left( {{U_i}} \right)$ inducing the identity on Ui. There is a sheaf of groups A on Q such that the isomorphism classes of all possible Y is the cohomology set H¹ (Q, A). The main question we address is to what extent H¹ (Q, A) contains only topological information. For example, if G acts freely on X and Y, then X and Y are principal G-bundles over Q, and Grauert's Oka principle says that the set of isomorphism classes of holomorphic principal G-bundles over Q is canonically the same as the set of isomorphism classes of topological principal G-bundles over Q. We investigate to what extent we have an Oka principle for H¹ (Q, A).
The conformal module of conjugacy classes of braids is an invariant that appeared earlier than the entropy of conjugacy classes of braids, and is inversely proportional to the entropy. Using the ...relation between the two invariants, we give a short conceptional proof of an earlier result on the conformal module. Mainly, we consider situations, when the conformal module of conjugacy classes of braids serves as obstruction for the existence of homotopies (or isotopies) of smooth objects involving braids to the respective holomorphic objects, and present theorems on the restricted validity of Gromov’s Oka principle in these situations.
We give necessary and sufficient conditions for solving the spectral Nevanlinna–Pick lifting problem. This reduces the spectral Nevanlinna–Pick problem to a jet interpolation problem into the ...symmetrized polydisc.
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