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 Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

2020 ◽  
Vol 7 (1) ◽  
pp. 194-214
Author(s):  
Daniele Angella ◽  
Tatsuo Suwa ◽  
Nicoletta Tardini ◽  
Adriano Tomassini
Keyword(s):  
Complex Manifold ◽  
Compact Complex Manifold ◽  
Complex Manifolds ◽  
Dolbeault Cohomology ◽  
Hodge Structures ◽  
Balanced Metric ◽  
The Stability ◽  
Cech Cohomology ◽  
Simply Connected ◽  
Compact Complex Manifolds

AbstractWe construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.

scholarly journals Holomorphic Bundles Trivializable by Proper Surjective Holomorphic Map

2020 ◽  
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu
Keyword(s):  
Algebraic Group ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Compact Complex Manifold ◽  
Holomorphic Map ◽  
Holomorphic Vector Bundles ◽  
Holomorphic Bundles

Abstract Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $\varphi ^* E$ is holomorphically trivial for some surjective holomorphic map $\varphi $, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.

scholarly journals Regularity and Blow-Up Analysis for J-Holomorphic Maps

2003 ◽  
Vol 05 (04) ◽  
pp. 671-704
Author(s):  
Changyou Wang
Keyword(s):  
Blow Up ◽  
Harmonic Maps ◽  
Hermitian Manifold ◽  
Singular Set ◽  
Holomorphic Maps ◽  
Almost Hermitian Manifold ◽  
Holomorphic Map ◽  
Energy Quantization ◽  
Blow Up Analysis ◽  
Almost Kähler

If u∈H1(M,N) is a weakly J-holomorphic map from a compact without boundary almost hermitian manifold (M,j,g) into another compact without boundary almost hermitian manifold (N,J,h). Then it is smooth near any point x where Du has vanishing Morrey norm ℳ2,2m-2, with 2m= dim (M). Hence H2m-2measure of the singular set for a stationary J-holomorphic map is zero. Blow-up analysis and the energy quantization theorem are established for stationary J-holomorphic maps. Connections between stationary J-holomorphic maps and stationary harmonic maps are given for either almost Kähler manifolds M and N or symmetric ∇hJ.

Holomorphic curvatures of twistor spaces

2014 ◽  
Vol 11 (03) ◽  
pp. 1450022
Author(s):  
Danish Ali ◽  
Johann Davidov ◽  
Oleg Mushkarov
Keyword(s):  
Complex Surface ◽  
Hermitian Manifold ◽  
Negative Answer ◽  
Einstein Metric ◽  
Holomorphic Sectional Curvature ◽  
Almost Hermitian Manifold ◽  
Twistor Spaces ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature ◽  
Compact Complex Surface

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

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).

scholarly journals Moduli Space of Brody Curves, Energy and Mean Dimension

2008 ◽  
Vol 192 ◽  
pp. 27-58 ◽  
Author(s):  
Masaki Tsukamoto
Keyword(s):  
Complex Plane ◽  
Moduli Space ◽  
Hermitian Manifold ◽  
Value Distribution ◽  
Holomorphic Map ◽  
Infinite Dimensional ◽  
Mean Dimension ◽  
The Mean ◽  
Moduli Theory ◽  
Bounded Derivative

AbstractA Brody curve is a holomorphic map from the complex plane ℂ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its “mean dimension”. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension.

On 4-dimensional generalized complex space forms

1999 ◽  
Vol 66 (3) ◽  
pp. 379-387 ◽  
Author(s):  
Un Kyu Kim
Keyword(s):  
Sectional Curvature ◽  
Complex Space ◽  
Hermitian Manifold ◽  
Holomorphic Sectional Curvature ◽  
Space Forms ◽  
Generalized Complex ◽  
Complex Forms ◽  
Complex Space Forms

AbstractWe characterize four-dimensional generalized complex forms and construct an Einstein and weakly *-Einstein Hermitian manifold with pointwise constant holomorphic sectional curvature which is not globally constant.

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.

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