Extending holomorphic maps from Stein manifolds into affine toric varieties

Lärkäng, Richard; Lárusson, Finnur

doi:10.1090/proc/13108

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…Little is known about the Oka theory of singular targets. The first paper on this topic is [17]. The results there show that analytic Oka theory changes quite dramatically when we move from smooth targets to singular targets.…”

Section: Introduction and Resultsmentioning

confidence: 94%

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

Lárusson¹,

Truong²

2019

Math. Scand.

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We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.Among the properties of a complex manifold Y investigated in Oka theory are the following, where X denotes an arbitrary reduced Stein space.• The approximation property (AP): Every continuous map X → Y that is holomorphic on a holomorphically convex compact subset K of X can be uniformly approximated on K by holomorphic maps X → Y .

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Section: Introduction and Resultsmentioning

confidence: 94%

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

Lárusson¹,

Truong²

2019

Math. Scand.

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“…See [31,Section 2.2] for an example. This is problematic, as Oka properties do not generalise easily to singular spaces [24].…”

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confidence: 99%

“…Our goal in this subsection is to show that the orbit space R n /M is a complex manifold such that the projection map π : R n → R n /M is a principal M -bundle. For this we shall use the following special case of a theorem of Holmann[20, Satz 21,24].…”

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confidence: 99%

Holomorphic Flexibility Properties of Spaces of Elliptic Functions

Bowman

2017

Bull. Aust. Math. Soc.

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Let X be an elliptic curve and P the Riemann sphere. Since X is compact, it is a deep theorem of Douady that the set O(X, P) consisting of holomorphic maps X → P admits a complex structure. If R n denotes the set of maps of degree n, then Namba has shown for n ≥ 2 that R n is a 2n-dimensional complex manifold. We study holomorphic flexibility properties of the spaces R 2 and R 3 .Firstly, we show that R 2 is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a six-sheeted branched covering space of R 3 that is an Oka manifold. It follows that R 3 is C-connected and dominable. We show that R 3 is Oka if and only if P 2 \C is Oka, where C is a cubic curve that is the image of a certain embedding of X into P 2 .We investigate the strong dominability of R 3 and show that if X is not biholomorphic to C/Γ 0 , where Γ 0 is the hexagonal lattice, then R 3 is strongly dominable.As a Lie group, X acts freely on R 3 by precomposition by translations. We show that R 3 is holomorphically convex and that the quotient space R 3 /X is a Stein manifold.We construct an alternative six-sheeted Oka branched covering space of R 3 and prove that it is isomorphic to our first construction in a natural way. This alternative construction gives us an easier way of interpreting the fibres of the branched covering map.The full thesis is available at https://arxiv.org/abs/1609.07184.

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Flexibility Properties of Complex Manifolds and Holomorphic Maps

Forstnerič

2017

Stein Manifolds and Holomorphic Mappings

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Extending holomorphic maps from Stein manifolds into affine toric varieties

Cited by 4 publications

References 13 publications

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

Holomorphic Flexibility Properties of Spaces of Elliptic Functions

Flexibility Properties of Complex Manifolds and Holomorphic Maps

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