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).

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 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.

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.

scholarly journals Homogeneous vector bundles and stability

1986 ◽  
Vol 101 ◽  
pp. 37-54 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Matrix Function ◽  
Vector Bundles ◽  
Hermitian Matrix ◽  
Holomorphic Vector Bundle ◽  
Positive Definite ◽  
Frame Field ◽  
Hermitian Vector Bundle ◽  
Homogeneous Vector

In [5, 6, 7] I introduced the concept of Einstein-Hermitian vector bundle. Let E be a holomorphic vector bundle of rank r over a complex manifold M. An Hermitian structure h in E can be expressed, in terms of a local holomorphic frame field s1, …, sr of E, by a positive-definite Hermitian matrix function (hij) defined by

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.

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 Yang–Mills theory and jumping curves

2015 ◽  
Vol 26 (05) ◽  
pp. 1550029
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Chain Complex ◽  
Complex Manifolds ◽  
Smooth Manifolds ◽  
Hermitian Vector Bundle ◽  
Connected Manifold ◽  
Classical Sense ◽  
Yang Mills ◽  
Simply Connected

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

scholarly journals OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

2019 ◽  
Vol 7 ◽  
Author(s):  
A. ASOK ◽  
J. FASEL ◽  
M. J. HOPKINS
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Algebraic Variety ◽  
Vector Bundles ◽  
Necessary Condition ◽  
Chern Classes ◽  
Smooth Complex ◽  
Steenrod Operations ◽  
Complex Algebraic Variety ◽  
Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

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