A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions

2003 ◽  
Vol 44 (1-2) ◽  
pp. 3-24 ◽  
Author(s):  
Naoto Abe ◽  
Sanae Kurosu
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Affine Immersions

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 Kähler Yamabe minimizers on minimal ruled surfaces

2002 ◽  
Vol 90 (2) ◽  
pp. 180
Author(s):  
Christina W. Tønnesen-Friedman
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Ruled Surface ◽  
Ruled Surfaces

It is shown that if a minimal ruled surface $\mathrm{P}(E) \rightarrow \Sigma$ admits a Kähler Yamabe minimizer, then this metric is generalized Kähler-Einstein and the holomorphic vector bundle $E$ is quasi-stable.

scholarly journals Goresky–Pardon lifts of Chern classes and associated Tate extensions

2017 ◽  
Vol 153 (7) ◽  
pp. 1349-1371 ◽  
Author(s):  
Eduard Looijenga
Keyword(s):  
Vector Bundle ◽  
Hodge Structure ◽  
Holomorphic Vector Bundle ◽  
Mixed Hodge Structure ◽  
Chern Classes ◽  
Analytic Variety ◽  
Algebraic Setting ◽  
Geometric Conditions ◽  
Stable Cohomology ◽  
Complex Analytic Variety

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

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 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.

SOLUTIONS TO ${\bar\partial}$-EQUATION ON STRONGLY q-CONVEX DOMAINS WITH Lp-ESTIMATES

2004 ◽  
Vol 01 (06) ◽  
pp. 739-749 ◽  
Author(s):  
OSAMA ABDELKADER ◽  
SHABAN KHIDR
Keyword(s):  
Vector Bundle ◽  
Convex Domain ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Convex Domains ◽  
Lp Estimates ◽  
Type K ◽  
Kahler Manifold

The purpose of this paper is to construct a solution with Lp-estimates, 1≤p≤∞, to the equation [Formula: see text] on strongly q-convex domain of Kähler manifold. This is done for forms of type (n,s), s≥ max (q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.

scholarly journals On images and inverse images of Weierstrass points

1978 ◽  
Vol 19 (2) ◽  
pp. 125-128
Author(s):  
R. F. Lax
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Compact Riemann Surface ◽  
Holomorphic Vector Bundle ◽  
Compact Complex Manifold ◽  
Trivial Bundle ◽  
Weierstrass Points ◽  
Complex Dimension ◽  
Holomorphic Sections ◽  
Image Position

The classical theory of Weierstrass points on a compact Riemann surface is well-known (see, for example, [3]). Ogawa [6] has defined generalized Weierstrass points. Let Y denote a compact complex manifold of (complex) dimension n. Let E denote a holomorphic vector bundle on Y of rank q. Let Jk(E) (k = 0, 1, …) denote the holomorphic vector bundle of k-jets of E [2, p. 112]. Put rk(E) = rank Jk(E) = q.(n + k)!/n!k!. Suppose that Γ(E), the vector space of global holomorphic sections of E, is of dimension γ(E)>0. Consider the trivial bundle Y × Γ(E) and the mapwhich at a point Q∈Y takes a section of E to its k-jet at Q. Put μ = min(γ(E),rk(E)).

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 Lp-curvature and the Cauchy-Riemann equation near an isolated singular point

2001 ◽  
Vol 164 ◽  
pp. 35-51
Author(s):  
Adam Harris ◽  
Yoshihiro Tonegawa
Keyword(s):  
Vector Bundle ◽  
Singular Point ◽  
Sobolev Inequality ◽  
Plurisubharmonic Function ◽  
Final Section ◽  
Holomorphic Vector Bundle ◽  
Volume Growth ◽  
Holomorphic Extension ◽  
Riemann Equation ◽  
Kähler Form

Let X be a complex n-dimensional reduced analytic space with isolated singular point x0, and with a strongly plurisubharmonic function ρ : X → [0, ∞) such that ρ(x0) = 0. A smooth Kähler form on X \ {x0} is then defined by i∂∂ρ. The associated metric is assumed to have -curvature, to admit the Sobolev inequality and to have suitable volume growth near x0. Let E → X \ {x0} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0, 1)-form with coefficients in E. The main result of this article states that if ξ and the curvature of E are both then the equation has a smooth solution on a punctured neighbourhood of x0. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

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