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.

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.

scholarly journals Notes on projective structures on complex manifolds

1989 ◽  
Vol 116 ◽  
pp. 63-88 ◽  
Author(s):  
Björn Gustafsson ◽  
Jaak Peetre
Keyword(s):  
Riemann Surface ◽  
Linear Transformation ◽  
Projective Structure ◽  
Complex Manifolds ◽  
Projective Structures ◽  
Fractional Linear Transformation ◽  
Local Coordinates

Consider a Riemann surface X equipped with a projective structure, that is, a covering of X with coordinate neighborhoods U and corresponding (holomorphic) local coordinates {t} such that in the intersection U ∩ U′ of any two such coordinate neighborhoods U and U′ change of local coordinates is mediated by a fractional linear transformation

scholarly journals Analytic cycles and generically finite holomorphic maps

1995 ◽  
Vol 52 (3) ◽  
pp. 457-460
Author(s):  
Yingchen Li
Keyword(s):  
Holomorphic Mappings ◽  
Biholomorphic Mapping ◽  
Holomorphic Maps ◽  
Holomorphic Map ◽  
Homology Groups ◽  
Normal Complex ◽  
One To One

We study the behaviour of analytic cycles under generically finite holomorphic mappings between compact analytic spaces and prove that if two compact and normal complex analytic spaces have the same analytic homology groups, then any generically one to one holomorphic map between them must be a biholomorphic mapping. This generalises an old theorem of Ax and Borel.

scholarly journals RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

2019 ◽  
pp. 1-16
Author(s):  
Xiaokui Yang
Keyword(s):  
Complex Manifold ◽  
Hermitian Manifold ◽  
Compact Complex Manifold ◽  
Holomorphic Sectional Curvature ◽  
Vanishing Theorems ◽  
Holomorphic Maps ◽  
Holomorphic Map ◽  
Holomorphic Bisectional Curvature ◽  
Tautological Line Bundle ◽  
Compact Complex Manifolds

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$ . In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

scholarly journals Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

2017 ◽  
Vol 28 (09) ◽  
pp. 1740004 ◽  
Author(s):  
Antonio Alarcón ◽  
Finnur Lárusson
Keyword(s):  
Riemann Surface ◽  
Convex Hull ◽  
Minimal Surface ◽  
Cohomology Class ◽  
Holomorphic Maps ◽  
De Rham Cohomology ◽  
Surface Theory ◽  
Open Riemann Surface ◽  
De Rham ◽  
Smooth Locus

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

Theta constants of two kinds on a compact Riemann surface of genus 2

1970 ◽  
Vol 23 (1) ◽  
pp. 381-407 ◽  
Author(s):  
Harry E. Rauch ◽  
Hershel M. Farkas
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Genus 2

A FAMILY OF SURFACES CONSTRUCTED FROM GENUS 2 CURVES

2007 ◽  
Vol 18 (05) ◽  
pp. 585-612 ◽  
Author(s):  
CHAD SCHOEN
Keyword(s):  
Riemann Surface ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
Analytic Variety ◽  
Genus 2 ◽  
Complex Analytic Variety ◽  
Projective Manifolds ◽  
Genus 2 Curves ◽  
Complex Projective

We consider the deformations of the two-dimensional complex analytic variety constructed from a genus 2 Riemann surface by attaching its self-product to its Jacobian in an elementary way. The deformations are shown to be unobstructed, the variety smooths to give complex projective manifolds whose invariants are computed and whose images under Albanese maps (re)verify an instance of the Hodge conjecture for certain abelian fourfolds.

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