PROLONGEMENT D’UN FIBRE HOLOMORPHE HERMITIEN A COURBURE Lp SUR UNE COURBE OUVERTE

1992 ◽  
Vol 03 (04) ◽  
pp. 441-453 ◽  
Author(s):  
OLIVIER BIQUARD
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Weighted Sobolev Space ◽  
Holomorphic Vector Bundle ◽  
Chern Connection ◽  
Hermitian Metrics ◽  
Parabolic Bundle ◽  
Finite Set ◽  
Marked Points

Let X be a Riemann surface and S a finite set of marked points on X. If [Formula: see text] is a hermitian holomorphic vector bundle with Lp-curvature for some p>1, we study the asymptotic behaviour of the Chern connection around the marked points; by solving directly a [Formula: see text]-problem in a weighted Sobolev space, we extend the holomorphic structure of [Formula: see text] over S to get a parabolic bundle. We deduce a proof of the classification of these hermitian metrics.

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)).

PARABOLIC VECTOR BUNDLES AND HERMITIAN-YANG-MILLS CONNECTIONS OVER A RIEMANN SURFACE

1993 ◽  
Vol 04 (03) ◽  
pp. 467-501 ◽  
Author(s):  
JONATHAN A. PORITZ
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Sobolev Space ◽  
Moduli Space ◽  
Vector Bundles ◽  
Weighted Sobolev Space ◽  
Conjugacy Classes ◽  
Stable Bundle ◽  
Yang Mills ◽  
Parabolic Bundles

We study a certain moduli space of irreducible Hermitian-Yang-Mills connections on a unitary vector bundle over a punctured Riemann surface. The connections used have non-trivial holonomy around the punctures lying in fixed conjugacy classes of U (n) and differ from each other by elements of a weighted Sobolev space; these connections give rise to parabolic bundles in the sense of Mehta and Seshadri. We show in fact that the moduli space of stable parabolic bundles can be identified with our moduli space of HYM connections, by proving that every stable bundle admits a unique unitary gauge orbit of Hermitian-Yang-Mills connections.

scholarly journals Non-flat extension of flat vector bundles

2015 ◽  
Vol 26 (14) ◽  
pp. 1550114
Author(s):  
Indranil Biswas ◽  
Viktoria Heu
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Vector Bundles ◽  
Compact Riemann Surface ◽  
Holomorphic Vector Bundle ◽  
Flat Extension

We construct a pair [Formula: see text], where [Formula: see text] is a holomorphic vector bundle over a compact Riemann surface and [Formula: see text] a holomorphic subbundle, such that both [Formula: see text] and [Formula: see text] admit holomorphic connections, but [Formula: see text] does not.

Analysis of statistical coefficients and autoregressive parameters over intrinsic mode functions (IMFs) for epileptic seizure detection

2020 ◽  
Vol 65 (6) ◽  
pp. 693-704
Author(s):  
Rafik Djemili
Keyword(s):  
Involuntary Movement ◽  
Epileptic Seizures ◽  
Intrinsic Mode Functions ◽  
Eeg Signals ◽  
Time Segment ◽  
Feature Extraction Method ◽  
Mode Decomposition ◽  
Finite Set ◽  
Student’S T

AbstractEpilepsy is a persistent neurological disorder impacting over 50 million people around the world. It is characterized by repeated seizures defined as brief episodes of involuntary movement that might entail the human body. Electroencephalography (EEG) signals are usually used for the detection of epileptic seizures. This paper introduces a new feature extraction method for the classification of seizure and seizure-free EEG time segments. The proposed method relies on the empirical mode decomposition (EMD), statistics and autoregressive (AR) parameters. The EMD method decomposes an EEG time segment into a finite set of intrinsic mode functions (IMFs) from which statistical coefficients and autoregressive parameters are computed. Nevertheless, the calculated features could be of high dimension as the number of IMFs increases, the Student’s t-test and the Mann–Whitney U test were thus employed for features ranking in order to withdraw lower significant features. The obtained features have been used for the classification of seizure and seizure-free EEG signals by the application of a feed-forward multilayer perceptron neural network (MLPNN) classifier. Experimental results carried out on the EEG database provided by the University of Bonn, Germany, demonstrated the effectiveness of the proposed method which performance assessed by the classification accuracy (CA) is compared to other existing performances reported in the literature.

scholarly journals Two-Categorical Bundles and their Classifying Spaces

2012 ◽  
Vol 10 (2) ◽  
pp. 299-369 ◽  
Author(s):  
Nils A. Baas ◽  
Marcel Bökstedt ◽  
Tore August Kro
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Vector Spaces ◽  
Bundle Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Principal Bundles

AbstractFor a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

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 The even Clifford structure of the fourth Severi variety

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Maurizio Parton ◽  
Paolo Piccinni
Keyword(s):  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Severi Variety ◽  
Simply Connected

AbstractTheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl

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).

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