scholarly journals Vector bundles on an elliptic curve and Sklyanin algebras

1998 ◽  
pp. 65-84 ◽  
Author(s):  
B. L. Feĭgin ◽  
A. V. Odesskiĭ
Keyword(s):  
Elliptic Curve ◽  
Vector Bundles ◽  
Sklyanin Algebras

Integrable Systems Associated to a Hopf Surface

2003 ◽  
Vol 55 (3) ◽  
pp. 609-635 ◽  
Author(s):  
Ruxandra Moraru
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Symplectic Geometry ◽  
Vector Bundles ◽  
Integrable Hamiltonian System ◽  
Complex Surface ◽  
Point Of View ◽  
Infinite Cyclic Group ◽  
Completely Integrable Hamiltonian System ◽  
Hopf Surface

AbstractA Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

scholarly journals The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

2010 ◽  
Vol 147 (1) ◽  
pp. 188-234 ◽  
Author(s):  
O. Schiffmann ◽  
E. Vasserot
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Moduli Space ◽  
Vector Bundles ◽  
Hecke Algebras ◽  
Macdonald Polynomials ◽  
Hall Algebra ◽  
Double Affine Hecke Algebras ◽  
Affine Hecke Algebras ◽  
Semistable Vector Bundles

AbstractWe exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras $\ddot {\mathbf {H}}_n$ of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

scholarly journals Simple vector bundles on plane degenerations of an elliptic curve

2012 ◽  
Vol 364 (1) ◽  
pp. 137-174 ◽  
Author(s):  
Lesya Bodnarchuk ◽  
Yuriy Drozd ◽  
Gert-Martin Greuel
Keyword(s):  
Elliptic Curve ◽  
Vector Bundles

scholarly journals Singularities of Projective Embedding (Points of order n on an Elliptic Curve)

1972 ◽  
Vol 45 ◽  
pp. 97-107 ◽  
Author(s):  
Akikuni Kato
Keyword(s):  
Projective Space ◽  
Elliptic Curve ◽  
Vector Bundles ◽  
Neutral Element ◽  
Projective Embedding ◽  
Dual Formulation ◽  
Higher Dimensional ◽  
Dimensional Projective Space ◽  
Dimension One ◽  
Principal Parts

In the Plücker formula for a curve embedded in a higher dimensional projective space, one encounters the notion of stationary point (cf, [B], [W]). W. F. Pohl gave new view point about it in terms of vector bundles and he defined “the singularities of embedding” (cf. [P]). At first, we shall give dual formulation of Pohl’s one by means of the sheaf of principal parts of order n, and next we shall prove the following: If an elliptic curve is embedded in (n — l)-dimensional projective space as a curve of degree n, singularities of projective embedding of order n — 1 are exactly the points of order n with suitable choice of a neutral element on the curve which is an abelian variety of dimension one. The proof is given by making use of the relation between and Schwarzenberger’s secant bundle which we shall also give.

scholarly journals Vector bundles over a real elliptic curve

2016 ◽  
Vol 283 (1) ◽  
pp. 43-62 ◽  
Author(s):  
Indranil Biswas ◽  
Florent Schaffhauser
Keyword(s):  
Elliptic Curve ◽  
Vector Bundles

scholarly journals Vector Bundles on an Elliptic Curve

1971 ◽  
Vol 43 ◽  
pp. 41-72 ◽  
Author(s):  
Tadao Oda
Keyword(s):  
Vector Bundle ◽  
Elliptic Curve ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Closed Field ◽  
List Type ◽  
Algebraically Closed Field ◽  
Frobenius Map

Let k be an algebraically closed field of characteristic p≧ 0, and let X be an abelian variety over k.The goal of this paper is to answer the following questions, when dim(X) = 1 and p≠0, posed by R. Hartshorne: (1)Is E(P) indecomposable, when E is an indecomposable vector bundle on X?(2)Is the Frobenius map F*: H1 (X, E) → H1 (X, E(p)) injective?We also partly answer the following question posed by D. Mumford:(3)Classify, or at least say anything about, vector bundles on X when dim (X) > 1.

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