Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 082, 31 pages      arXiv:0708.2186      http://dx.doi.org/10.3842/SIGMA.2007.082

Monodromy of a Class of Logarithmic Connections on an Elliptic Curve

Francois-Xavier Machu
Mathématiques - bât. M2, Université Lille 1, F-59655 Villeneuve d'Ascq Cedex, France

Received March 22, 2007, in final form August 06, 2007; Published online August 16, 2007

Abstract
The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their monodromy, differential Galois group and the underlying rank-2 vector bundle. The latter is described in terms of elementary transforms. The question of its (semi)-stability is addressed.

Key words: elliptic curve; ramified covering; logarithmic connection; bielliptic curve; genus-2 curve; monodromy; Riemann-Hilbert problem; differential Galois group; elementary transformation; stable bundle; vector bundle.

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