Univalent holomorphic mappings on a complex manifold with a C 1 exhaustion function

1999 ◽  
Vol 99 (3) ◽  
pp. 359-369
Author(s):  
Hidetaka Hamada
Keyword(s):  
Complex Manifold ◽  
Holomorphic Mappings ◽  
Exhaustion Function

scholarly journals APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS

2013 ◽  
Vol 24 (14) ◽  
pp. 1350108 ◽  
Author(s):  
KRIS STOPAR
Keyword(s):  
Complex Manifold ◽  
Convex Domain ◽  
Holomorphic Mappings ◽  
Holomorphic Maps ◽  
Convex Domains ◽  
Exceptional Set ◽  
Proper Holomorphic Maps ◽  
Pseudoconvex Boundary ◽  
Oka Principle ◽  
Additional Assumptions

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

COMPACTNESS OF CERTAIN FAMILIES OF PSEUDO-HOLOMORPHIC MAPPINGS INTO ${\mathbb C}^n$

2004 ◽  
Vol 15 (01) ◽  
pp. 1-12 ◽  
Author(s):  
HERVÉ GAUSSIER ◽  
KANG-TAE KIM
Keyword(s):  
Complex Manifold ◽  
Convergence Theorem ◽  
Normal Family ◽  
Holomorphic Mappings ◽  
Complex Structures ◽  
Holomorphic Maps ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Scaling Sequence

We present a normal family theorem for injective almost holomorphic maps from a manifold with almost complex structures into [Formula: see text]. Our theorem implies a new consequence even for the holomorphic mappings of a complex manifold into [Formula: see text], which can be seen as a generalization of the convergence theorem for Frankel's scaling sequence whose images are not necessarily convex. Moreover, our method is closer in spirit to the circle of ideas centered around the classical Montel theorem.

scholarly journals Holomorphic mapping into algebraic varieties of general type

1990 ◽  
Vol 120 ◽  
pp. 155-170 ◽  
Author(s):  
Peichu Hu
Keyword(s):  
Complex Manifold ◽  
General Type ◽  
Holomorphic Mapping ◽  
Holomorphic Mappings ◽  
Canonical Bundle ◽  
Algebraic Varieties ◽  
Algebraic Manifold

We will study holomorphic mappingsfrom a connected complex manifold M of dimension m to a projective algebraic manifold N of dimension n. Assume first that N is of general type, i.e.where KN→N is the canonical bundle of N. If KN is positive, then N is of general type.

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

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