scholarly journals A geometrical characterization of a class of holomorphic vector bundles over a complex torus

1976 ◽  
Vol 61 ◽  
pp. 197-202 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Vector Bundle ◽  
Heisenberg Group ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Projective Bundle ◽  
Geometrical Characterization ◽  
Complex Torus ◽  
Holomorphic Vector Bundles ◽  
Holomorphic Representation

This note is to be a supplement of the preceeding paper in the journal by Matsushima, settling a question raised by him. In his paper he associates a holomorphic vector bundle over a complex torus to a holomorphic representation of what he calls Heisenberg group. We shall show that a simple holomorphic vector bundle is determined in this manner if and only if the associated projective bundle admits an integrable holomorphic connection. A theorem by Morikawa ([3], Theorem 1) is the motivation of this problem and is somewhat strengthened by our result.

scholarly journals A Note on Holomorphic Vector Bundles Over Quotient Manifolds With Respect to Nilpotent Groups

1972 ◽  
Vol 48 ◽  
pp. 183-188
Author(s):  
Hisasi Morikawa
Keyword(s):  
Vector Bundle ◽  
Endomorphism Ring ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Nilpotent Groups ◽  
Projective Bundle ◽  
Analytic Manifold ◽  
Transition Functions ◽  
Holomorphic Vector Bundles ◽  
Complex Analytic Manifold

A holomorphic vector bundle E over a complex analytic manifold is said to be simple, if its global endomorphism ring Endc (E) is isomorphic to C. Projectifying the fibers of E, we get the associated projective bundle P(E) of E, If we can choose a system of constant transition functions of P(Exs), the projective bundle P(E) is said to be locally flat.

scholarly journals Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

2006 ◽  
Vol 13 (1) ◽  
pp. 7-10
Author(s):  
Edoardo Ballico
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Open Neighborhood ◽  
Holomorphic Vector Bundle ◽  
Stein Manifold ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles

Abstract Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂(𝑈).

Topologically trivial holomorphic vector bundles on infinite-dimensional projective varieties

2004 ◽  
Vol 134 (1) ◽  
pp. 33-38
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Finite Rank ◽  
Closed Subset ◽  
Vector Bundles ◽  
Complex Space ◽  
Holomorphic Vector Bundle ◽  
Projective Varieties ◽  
Infinite Dimensional ◽  
Holomorphic Vector Bundles ◽  
Locally Convex

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuous homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.

scholarly journals Kobayashi—Hitchin corrispondence for twisted vector bundles

2021 ◽  
Vol 8 (1) ◽  
pp. 1-95
Author(s):  
Arvid Perego
Keyword(s):  
Vector Bundle ◽  
Compact Manifold ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Kahler Manifolds ◽  
Kähler Metric ◽  
Holomorphic Vector Bundles ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Abstract We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.

scholarly journals A theorem of Matsushima

1974 ◽  
Vol 54 ◽  
pp. 123-134 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Complex Torus ◽  
Frobenius Endomorphism

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

scholarly journals Chern Classes of Projective Modules

1963 ◽  
Vol 23 ◽  
pp. 121-152 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Chern Class ◽  
Vector Bundles ◽  
Stein Manifold ◽  
Projective Modules ◽  
Singular Variety ◽  
Differentiable Functions ◽  
Holomorphic Vector Bundles ◽  
Differentiable Vector

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

scholarly journals BALANCED METRICS ON HOMOGENEOUS VECTOR BUNDLES

2011 ◽  
Vol 08 (07) ◽  
pp. 1433-1438 ◽  
Author(s):  
ROBERTO MOSSA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Balanced Metrics ◽  
Kahler Manifold ◽  
Balanced Metric ◽  
Compact Kähler Manifold ◽  
Homogeneous Variety ◽  
Homogeneous Vector

Let E → M be a holomorphic vector bundle over a compact Kähler manifold (M, ω) and let E = E1 ⊕ ⋯ ⊕ Em → M be its decomposition into irreducible factors. Suppose that each Ej admits a ω-balanced metric in Donaldson–Wang terminology. In this paper we prove that E admits a unique ω-balanced metric if and only if [Formula: see text] for all j, k = 1,…, m, where rj denotes the rank of Ej and Nj = dim H0(M, Ej). We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety (M, ω) and we show the existence and rigidity of balanced Kähler embedding from (M, ω) into Grassmannians.

Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles

2006 ◽  
Vol 49 (1) ◽  
pp. 36-40 ◽  
Author(s):  
Georgios D. Daskalopoulos ◽  
Richard A. Wentworth
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Convergent Sequence ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Hölder Class ◽  
Holomorphic Vector Bundles ◽  
Holder Class

AbstractUsing a modification of Webster's proof of the Newlander–Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Grigori Giorgadze ◽  
Gega Gulagashvili
Keyword(s):  
Vector Bundles ◽  
Riemann Sphere ◽  
Complete Characterization ◽  
Fuchsian System ◽  
Hölder Continuous ◽  
Holomorphic Vector Bundles ◽  
Splitting Type ◽  
Second Order Systems ◽  
The Relationship

Abstract We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

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