scholarly journals STABILITY CONDITIONS ON GENERIC COMPLEX TORI

2012 ◽  
Vol 23 (05) ◽  
pp. 1250035 ◽  
Author(s):  
S. MEINHARDT
Keyword(s):  
Complex Manifold ◽  
Stability Conditions ◽  
Connected Component ◽  
Complex Torus ◽  
Complex Tori ◽  
Simply Connected

In this paper we describe a simply connected component of the complex manifold Stab(X) of stability conditions on a generic complex torus X. A generic complex torus is a complex torus X with Hp,p(X) ∩ H2p(X, ℤ) = 0 for all 0 < p < dim X.

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).

HOLOMORPHIC PRINCIPAL BUNDLES WITH AN ELLIPTIC CURVE AS THE STRUCTURE GROUP

2008 ◽  
Vol 05 (06) ◽  
pp. 851-862 ◽  
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Elliptic Curve ◽  
Real Number ◽  
Complex Manifold ◽  
Positive Real Number ◽  
Differential Forms ◽  
Structure Group ◽  
Positive Real ◽  
Principal Bundles ◽  
Complex Torus ◽  
Dimension One

Let Λ ⊂ ℂ be the ℤ-module generated by 1 and [Formula: see text], where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X. Let θ1 and θ2 be two C∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form [Formula: see text] vanishes, and the elements in H2(X, ℂ) represented by θ1 and θ2 are contained in the image of H2(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle EZ over X, we construct a pair of closed differential forms on X of the above type.

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 Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

2020 ◽  
Vol 7 (1) ◽  
pp. 194-214
Author(s):  
Daniele Angella ◽  
Tatsuo Suwa ◽  
Nicoletta Tardini ◽  
Adriano Tomassini
Keyword(s):  
Complex Manifold ◽  
Compact Complex Manifold ◽  
Complex Manifolds ◽  
Dolbeault Cohomology ◽  
Hodge Structures ◽  
Balanced Metric ◽  
The Stability ◽  
Cech Cohomology ◽  
Simply Connected ◽  
Compact Complex Manifolds

AbstractWe construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.

scholarly journals Einstein hypersurfaces of Kählerian C-spaces

1980 ◽  
Vol 78 ◽  
pp. 153-175
Author(s):  
Yusuke Sakane ◽  
Masaru Takeuchi
Keyword(s):  
Complex Manifold ◽  
Kähler Metric ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Simply Connected ◽  
Kahler Metric

A compact simply connected homogeneous complex manifold is called a C-space. A C-space is said to be kählerian if it carries a Kähler metric. It is known (Matsushima [7]) that a kählerian C-space has always an Einstein Kähler metric which is essentially unique.

scholarly journals Every algebraic Kummer surface is the K3-cover of an Enriques surface

1990 ◽  
Vol 118 ◽  
pp. 99-110 ◽  
Author(s):  
Jong Hae Keum
Keyword(s):  
Crucial Role ◽  
K3 Surfaces ◽  
Abelian Surface ◽  
Enriques Surface ◽  
Kummer Surface ◽  
3 Dimensional ◽  
Complex Torus ◽  
Kummer Surfaces ◽  
Dimension 2 ◽  
Simply Connected

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

scholarly journals On the Definition of Irreducible Holomorphic Symplectic Manifolds and Their Singular Analogs

2021 ◽  
Author(s):  
Martin Schwald
Keyword(s):  
Symplectic Manifolds ◽  
Complex Tori ◽  
Finite Quotients ◽  
Definition Of ◽  
Simply Connected

Abstract In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss holomorphic symplectic, finite quotients of complex tori with ${\operatorname{h}}^0(X,\,\Omega ^{[2]}_X)=1$ and their Lagrangian fibrations. Neither $X$ nor the base can be smooth unless $X$ is a $2$-torus.

scholarly journals Complex hyperkähler structures defined by Donaldson–Thomas invariants

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Tom Bridgeland ◽  
Ian A. B. Strachan
Keyword(s):  
Complex Manifold ◽  
Geometric Structure ◽  
Tangent Bundle ◽  
Total Space ◽  
Stability Conditions ◽  
Link Type ◽  
Hyperkähler Metrics

AbstractThe notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a $$\hbox {CY}_3$$ CY 3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

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.

Stability Conditions Related to a Previously Unrecognized Form of Thrombin

1972 ◽  
Vol 27 (02) ◽  
pp. 361-362 ◽  
Author(s):  
Walter H. Seegers ◽  
Lowell E. McCoy
Keyword(s):  
Stability Conditions
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