The monoid of effective divisor classes on a complex torus

1981 ◽  
pp. 208-231 ◽  
Author(s):  
Jeffrey A. Rosoff
Keyword(s):  
Effective Divisor ◽  
Complex Torus

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).

scholarly journals Remarks on quasi-polarized varieties

1989 ◽  
Vol 115 ◽  
pp. 105-123 ◽  
Author(s):  
Takao Fujita
Keyword(s):  
Line Bundle ◽  
Effective Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Projective Scheme

Let V be a variety, which means, an irreducible reduced projective scheme over an algebraically closed field of any characteristic. A line bundle L on V is said to be nef if LC ≧ 0 for any curve C in V. Thus, “nef” is never an abbreviation of “numerically equivalent to an effective divisor”. L is said to be big if k(L) = n = dim V. In case L is nef, it is big if and only if Ln > 0 (cf. [F7; (6.5)]. When L is nef and big, the pair (V, L) will be called a quasi-polarized variety.

scholarly journals A theorem of Matsushima

1974 ◽  
Vol 54 ◽  
pp. 123-134 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Complex Torus ◽  
Frobenius Endomorphism

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

Kähler structures on complex torus

2000 ◽  
Vol 10 (2) ◽  
pp. 257-267 ◽  
Author(s):  
Meng-Kiat Chuah
Keyword(s):  
Complex Torus ◽  
Kähler Structures

p-ADIC PERIODS OF MODULAR ELLIPTIC CURVES AND THE LEVEL-LOWERING THEOREM

2008 ◽  
Vol 04 (01) ◽  
pp. 15-23 ◽  
Author(s):  
S. TAKAHASHI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Cusp Form ◽  
Rational Numbers ◽  
Fermat's Last Theorem ◽  
Complex Torus ◽  
Essential Tool ◽  
Fermat’S Last Theorem ◽  
Level Lowering

An elliptic curve defined over the field of rational numbers can be considered as a complex torus. We can describe its complex periods in terms of integration of the weight-2 cusp form corresponding to the elliptic curve. In this paper, we will study an analogous description of the p-adic periods of the elliptic curve, considering the elliptic curve as a p-adic torus. An essential tool for the proof of such a description is the level-lowering theorem of Ribet, which is one of the main ingredients used in the proof of Fermat's Last Theorem.

scholarly journals Todd genera of complex torus manifolds

2012 ◽  
Vol 12 (3) ◽  
pp. 1777-1788
Author(s):  
Hiroaki Ishida ◽  
Mikiya Masuda
Keyword(s):  
Complex Torus ◽  
Torus Manifolds

scholarly journals RANK ONE CONNECTIONS ON ABELIAN VARIETIES, II

2012 ◽  
Vol 23 (12) ◽  
pp. 1250125
Author(s):  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE ◽  
A. K. RAINA
Keyword(s):  
Line Bundle ◽  
Exact Sequence ◽  
Abelian Varieties ◽  
Explicit Construction ◽  
The Other ◽  
Holomorphic Line Bundle ◽  
Canonical Isomorphism ◽  
Complex Torus ◽  
Rank One

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle [Formula: see text], where α is the addition map on A × A, and p1 is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

On the structure of generalized Albanese varieties

1992 ◽  
Vol 111 (2) ◽  
pp. 267-272
Author(s):  
Hurit nsiper
Keyword(s):  
Algebraic Group ◽  
Class Group ◽  
Effective Divisor ◽  
Closed Field ◽  
Projective Surface ◽  
Mapping Property ◽  
Rational Map ◽  
Algebraically Closed Field ◽  
Albanese Map ◽  
Albanese Variety

Given a smooth projective surface X over an algebraically closed field k and a modulus (an effective divisor) m on X, one defines the idle class group Cm(X) of X with modulus m (see 1, chapter III, section 4). The corresponding generalized Albanese variety Gum and the generalized Albanese map um:X|m|Gum have the following universal mapping property (2): if :XG is a rational map into a commutative algebraic group which induces a homomorphism Cm(X)G(k) (1, chapter III, proposition 1), then factors uniquely through um.

scholarly journals Directed harmonic currents for laminations on certain compact complex surfaces

2014 ◽  
Vol 25 (04) ◽  
pp. 1450032 ◽  
Author(s):  
Carlos Pérez-Garrandés
Keyword(s):  
Riemann Surfaces ◽  
Complex Surfaces ◽  
Harmonic Current ◽  
Complex Torus ◽  
Harmonic Currents ◽  
Compact Complex Surfaces ◽  
Closed Current

Let ℒ be a Lipschitz lamination by Riemann surfaces embedded in M. If M is a complex torus, ℂℙ1 × ℂℙ1 or 𝕋1 × ℂℙ1 and there is no directed closed current then there exists a unique directed harmonic current of mass one. Moreover, if ℒ is embedded in M = ℂℙ1 × ℂℙ1 and has no compact leaves, then there is no directed closed current. If ℒ is not Lipschitz, then slightly weaker results are obtained.

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