Topologically trivial holomorphic vector bundles on infinite-dimensional projective varieties

2003 ◽  
Vol 44 (1-2) ◽  
pp. 35-40
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Projective Varieties ◽  
Infinite Dimensional ◽  
Holomorphic Vector Bundles

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.

HOMOGENEOUS VECTOR BUNDLES AND COWEN-DOUGLAS OPERATORS

1993 ◽  
Vol 04 (03) ◽  
pp. 503-520 ◽  
Author(s):  
DAVID R. WILKINS
Keyword(s):  
Vector Bundles ◽  
Symmetric Domain ◽  
Linear Operators ◽  
Open Unit ◽  
Bounded Symmetric Domain ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Holomorphic Vector Bundles ◽  
Class B

In this paper we obtain an algebraic classification of all homogeneous Hermitian holomorphic vector bundles of arbitrary rank over a bounded symmetric domain. This classification result is used in order to classify, up to unitary equivalence, all irreducible homogeneous bounded linear operators on a separable infinite-dimensional Hilbert space that belong to the Cowen-Douglas class B2 (∆), where ∆ is the open unit disk.

scholarly journals COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

2012 ◽  
Vol 23 (04) ◽  
pp. 1250037 ◽  
Author(s):  
MICHELE BOLOGNESI ◽  
SONIA BRIVIO
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Low Rank ◽  
Coherent Systems ◽  
Projective Varieties ◽  
Complex Curve ◽  
Smooth Complex ◽  
Moduli Theory ◽  
Modular Varieties ◽  
Small Genus

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

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