Analytic torsion of all vector bundles over an elliptic curve

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.

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The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

Compositio Mathematica ◽

10.1112/s0010437x10004872 ◽

2010 ◽

Vol 147 (1) ◽

pp. 188-234 ◽

Cited By ~ 32

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.

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Simple vector bundles on plane degenerations of an elliptic curve

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-2011-05354-7 ◽

2012 ◽

Vol 364 (1) ◽

pp. 137-174 ◽

Cited By ~ 2

Author(s):

Lesya Bodnarchuk ◽

Yuriy Drozd ◽

Gert-Martin Greuel

Keyword(s):

Elliptic Curve ◽

Vector Bundles

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ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748015000171 ◽

2015 ◽

Vol 16 (2) ◽

pp. 223-349 ◽

Cited By ~ 6

Author(s):

Jean-Michel Bismut ◽

Xiaonan Ma ◽

Weiping Zhang

Keyword(s):

Vector Bundle ◽

Line Bundle ◽

Differential Form ◽

Toeplitz Operators ◽

Vector Bundles ◽

Direct Image ◽

Analytic Torsion ◽

Leading Term ◽

Positive Line Bundle ◽

Positive Line

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

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

Nagoya Mathematical Journal ◽

10.1017/s0027763000014677 ◽

1972 ◽

Vol 45 ◽

pp. 97-107 ◽

Cited By ~ 3

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.

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Hypercomplex manifolds and hyperholomorphic bundles

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004102006114 ◽

2002 ◽

Vol 133 (3) ◽

pp. 443-457 ◽

Cited By ~ 8

Author(s):

BIRTE FEIX

Keyword(s):

Vector Bundles ◽

Analogous Result ◽

Complex Structure ◽

Zero Section ◽

Analytic Torsion ◽

Totally Geodesic ◽

Real Analytic ◽

The Given ◽

Torsion Free

Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.

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