scholarly journals A Note on Holomorphic Vector Bundles Over Complex Tori

1971 ◽  
Vol 41 ◽  
pp. 101-106 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Quotient Group ◽  
Vector Bundles ◽  
Additive Group ◽  
Complex Vector ◽  
Complex Torus ◽  
Holomorphic Vector Bundles ◽  
Complex Tori

Let (ω: Z2r→Cr be an isomorphism of the free additive group of rank 2r into the complex vector n-space such that the quotient group T = Crω/(Z2r) is compact, i.e., Tω is a complex torus.

scholarly journals Heisenberg groups and holomorphic vector bundles over a complex torus

1976 ◽  
Vol 61 ◽  
pp. 161-195 ◽  
Author(s):  
Yozo Matsushima
Keyword(s):  
Vector Space ◽  
Vector Bundles ◽  
Hermitian Form ◽  
Complex Vector Space ◽  
Heisenberg Groups ◽  
Complex Vector ◽  
Complex Torus ◽  
Holomorphic Vector Bundles

Let V be a complex vector space of dimension n, L a lattice of V and E = V/L a complex torus. Let H be a Hermitian form on V. We introduce a multiplication in L × C* by

scholarly journals Unipotent Schottky bundles on Riemann surfaces and complex tori

2014 ◽  
Vol 25 (06) ◽  
pp. 1450056 ◽  
Author(s):  
Carlos Florentino ◽  
Thomas Ludsteck
Keyword(s):  
Riemann Surfaces ◽  
Vector Bundles ◽  
Principal Bundle ◽  
Free Abelian Group ◽  
Degree Zero ◽  
Schottky Groups ◽  
Complex Torus ◽  
Holomorphic Vector Bundles ◽  
Complex Tori ◽  
Unipotent Representations

We study a natural map from representations of a free (respectively, free abelian) group of rank g in GL r(ℂ), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (respectively, complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.

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 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.

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

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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