Flat parabolic vector bundles on elliptic curves

Abstract

We describe the moduli space of logarithmic rank 2 connections on elliptic curves with 2 poles.

A Map Between Moduli Spaces of Connections

“…Let E be a rank 2 vector bundle over the elliptic curve C of degree d. By tensoring E with a line bundle L, we can change the degree to any desired value as long as it has the same parity as d. Therefore, the study of moduli spaces of rank 2 connections falls into two cases: odd degree and even degree. Usually the determinant of the bundle is fixed to be either O C in the even case (as in the present paper), or O C (w ∞ ), where w ∞ ∈ C is the identity element for the group structure of C. The moduli space of connections on C with two poles and fixed determinant O C (w ∞ ) has already been described in detail in [6], together with its symplectic structure and apparent map. As pointed out in [7], it is possible to pass from the moduli space in the even degree case to that in the odd degree case.…”

“…Moduli spaces of connections over the Riemann sphere have been extensively studied, in particular as these correspond to spaces of initial conditions for Garnier systems. The elliptic case with one and two poles have been studied in [16] and [6], respectively.…”

Holomorphic connections on holomorphic bundles on Riemann surfaces

“…Holomorphic connections, introduced by Atiyah [2], arise in innumerable contexts in mathematics (see [5,9,11,14,16] and references therein). Their importance can hardly be overemphasized.…”

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We investigate aspects of holomorphic connections on holomorphic principal bundles over a Riemann surface.

Mathematics Subject ClassificationHolomorphic connections, introduced by Atiyah [2], arise in innumerable contexts in mathematics (see [5,9,11,14,16] and references therein). Their importance can hardly be overemphasized.

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The twistor geometry of parabolic structures in rank two

“…Define, at each point y ∈ D, an operation H y , its inverse H −1 y and a partially defined operation P y on M Hod (Y, log D, x). The first H y is the well-known Hecke operation, or elementary transformation, as has been considered in [21] and more recently [13,20,26]. Given (λ, E, ∇, F, β), we set…”