Abstract
Let X be a smooth algebraic curve, proper over ℓ the field complex numbers (or equivalently a compact Riemann surface) of genus g. Let J be the Jacobian of X; it is a group variety of dimension g and its underlying set of points is the set of divisor classes (or equivalently isomorphic classes of line bundles) of degree zero.
It is a classical result that the underlying topological space of J can be identified with the set of (unitary) characters of the fundamental group π1 (X) into ℓ (i.e. homomorphisms of π1(X) into complex numbers of modulus one) and therefore J = S1 × … × S1, g times, as a topological manifold S1 being the unit circle in the complex plane.
The purpose of these lectures is to show how this result can be extended to the case of unitary representations of arbitrary rank of Fuchsian groups with compact quotient.
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Seshadri, C. (2010). Moduli of π -Vector Bundles over an Algebraic Curve. In: Marchionna, E. (eds) Questions on Algebraic Varieties. C.I.M.E. Summer Schools, vol 51. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11015-3_5
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