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 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 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.

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.

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.

AMPLE VECTOR BUNDLES WITH SECTIONS VANISHING ON PROJECTIVE SPACES OR QUADRICS

1995 ◽  
Vol 06 (04) ◽  
pp. 587-600 ◽  
Author(s):  
ANTONIO LANTERI ◽  
HIDETOSHI MAEDA
Keyword(s):  
Vector Bundle ◽  
Projective Space ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Projective Spaces ◽  
Compact Complex Manifold ◽  
Ample Vector Bundle ◽  
Zero Locus

Let ɛ be an ample vector bundle of rank r≥2 on a compact complex manifold X of dimension n≥r+1 having a section whose zero locus is a submanifold Z of the expected dimension n–r. Pairs (X, ɛ) as above are classified under the assumption that Z is either a projective space or a quadric.

SOLUTION TO $\overline{\partial}$-EQUATIONS WITH EXACT SUPPORT ON PSEUDO-CONVEX MANIFOLDS

2007 ◽  
Vol 04 (03) ◽  
pp. 339-348 ◽  
Author(s):  
OSAMA ABDELKADER ◽  
SAYED SABER
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Smooth Boundary ◽  
Holomorphic Vector Bundle ◽  
Property B ◽  
Pseudo Convex ◽  
Complex Valued ◽  
Satisfying Property

We obtain a solution to the [Formula: see text]-equation with exact support in a domain Ω with C1-smooth boundary satisfying property B in a complex manifold. This is done for complex-valued forms of type (r,s), s ≥ 1 and for forms of type (r,s), q ≤ s ≤ n - q, with values in a holomorphic vector bundle when the domain Ω is strongly q-convex.

scholarly journals A characterization of finite vector bundles on Gauduchon astheno-Kahler manifolds

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Indranil Biswas ◽  
Vamsi Pritham Pingali
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Projective Variety ◽  
Polynomial Equation ◽  
Compact Complex Manifold ◽  
Kahler Manifolds ◽  
Galois Covering ◽  
Finite Vector

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact complex manifold admitting a Gauduchon astheno-Kahler metric.

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.

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