Manifold-Valued Holomorphic Approximation

2011 ◽  
Vol 54 (2) ◽  
pp. 370-380 ◽  
Author(s):  
Edgar Lee Stout
Keyword(s):  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Complex Spaces ◽  
Holomorphic Approximation ◽  
Continuous Maps

AbstractThis note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps.

scholarly journals On Holomorphic Maps into a Real Lie Group of Holomorphic Transformations

1970 ◽  
Vol 40 ◽  
pp. 139-146
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Complex Manifold ◽  
Lie Group ◽  
Transformation Group ◽  
Usual Sense ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Complex Spaces ◽  
Lie Transformation ◽  
Lie Subgroup ◽  
Holomorphic Automorphisms

1. Introduction. Let M, N be complex manifolds and G be a group of holomorphic automorphisms of N. In [3] (c.f. p. 74) W. Kaup introduced the notion of holomorphic maps into a family of holomorphic maps between complex spaces. By definition, a map g: M→G is holomorphic if and only if the induced map g̃(x, y): = g(x) (y) (x∈M, y∈N) of M×N into N is holomorphic in the usual sense. The purpose of this note is to give a description of a holomorphic map of a connected complex manifold M into G. We show first the existence of the maximum connected Lie subgroup G0 of G which is a complex Lie transformation group of N.

scholarly journals On holomorphic maps with only fold singularities

2001 ◽  
Vol 164 ◽  
pp. 147-184
Author(s):  
Yoshifumi Ando
Keyword(s):  
Homotopy Class ◽  
Homotopy Equivalent ◽  
Line Bundles ◽  
Jet Space ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Regular Map ◽  
Holomorphic Map ◽  
Fold Map ◽  
Tangent Bundles

Let f : N ≡ P be a holomorphic map between n-dimensional complex manifolds which has only fold singularities. Such a map is called a holomorphic fold map. In the complex 2-jet space J2(n,n;C), let Ω10 denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that Ω10 is homotopy equivalent to SU(n + 1). By using this result we prove that if the tangent bundles TN and TP are equipped with SU(n)-structures in addition, then a holomorphic fold map f canonically determines the homotopy class of an SU(n + 1)-bundle map of TN ⊕ θN to TP⊕ θP, where θN and θP are the trivial line bundles.

REMARK ON THE ARTICLE "SINGULAR LOCI IN HOLOMORPHIC FAMILIES"

1998 ◽  
Vol 09 (07) ◽  
pp. 921-921
Author(s):  
ADAM HARRIS
Keyword(s):  
Essential Role ◽  
Open Neighbourhood ◽  
Complex Manifolds ◽  
Complex Spaces ◽  
Basic Assumptions ◽  
Singular Loci

To the basic assumptions that the map [Formula: see text] is a proper, flat morphism between complex manifolds should be added the hypothesis that the fibres of π are all generically reduced complex spaces. The existence of a non-empty open neighbourhood in [Formula: see text] plays an essential role in the proof of the main theorem, and does not follow automatically from the hypotheses as they were originally stated.

scholarly journals RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450006 ◽  
Author(s):  
GAUTAM BHARALI ◽  
INDRANIL BISWAS
Keyword(s):  
Riemann Surface ◽  
Special Form ◽  
General Class ◽  
Specific Information ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Fiber Space ◽  
Rigidity Result ◽  
Holomorphic Map ◽  
Genus 2

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.

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