Questions on Algebraic Varieties pp 139-260 | Cite as
Moduli of π -Vector Bundles over an Algebraic Curve
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.
Keywords
Vector Bundle Riemann Surface Line Bundle Fundamental Group Unitary RepresentationPreview
Unable to display preview. Download preview PDF.
References
- 1.M. F. Atiyah, Vector bundles over an elliptic curve. Proc. London Math. Soc., Third Series, 7(1957), 412–452.MathSciNetCrossRefGoogle Scholar
- 2.H. Cartan, Quotient d'un espace analytique par un groupe d'automorphismes, Algebraic geometry and topology, A symposium in honour of S. Lefschetz, Princeton University Press, 1957 (Princeton Math. Series, n).Google Scholar
- 3.H. Cartan and S. Eilenberg, Homclogical algebra, Princeton University Press, 1956.Google Scholar
- 4.A. Grothendisck, Sur la mémoire de Weil “Généralisation des fonctions abeliennes, Seminaire Bourbaki, expose 141, (1956–57).Google Scholar
- 5.——, Sur quelques points d'algèbre homologique, Tohoku Math. J., Series 2, 9 (1957), 119–121.MathSciNetGoogle Scholar
- 6.——, Les schémas de Hilbert, Séminaire Bourbaki, expose 221, t. 13, 1960–61.Google Scholar
- 7.——, et J. Dieudonné, Elements de Géometrie algébrique, Publ. Math., Inst. Hautes Etudes Scientifiques.Google Scholar
- 8.D. Mumford, Projective invariants of projective structures and applications. Proc. Intern. Cong. Math., Stockholm, 1962, 526–530.Google Scholar
- 9.——, Geometric invariant theory, Springer-Verlag, 1965.Google Scholar
- 10.——; Abelian varieties (to appear), Oxford University Press, Studies in Mathematics, Tata Institute of Fundamental Research.Google Scholar
- 11.M.S. Narasimhan and C.S. Seshadri, Holomorphic vector bundles on a compact Riemann surface, Math. Ann., 155 (1964), 69–80.MathSciNetzbMATHCrossRefGoogle Scholar
- 12.——, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540–567.MathSciNetCrossRefGoogle Scholar
- 13.A. Selberg, On discontinuous groups in higher dimensional symmetric spaces, Contributions to function theory, Tata Institute of Fundamental Research, 1960.Google Scholar
- J.-P. Serre, Faisceaux algebriques cohérents, Ann. of Math., 61 (1955), 197–278.MathSciNetCrossRefGoogle Scholar
- 15.——, Géometrie analytique et géometric algébrique, Ann. Inst. Fourier, 1955–56, 1–42.Google Scholar
- 16.C.S. Seshadri, Generalized multiplicative meromorphic functions on a complex analytic manifold, J. Indian Math. Soc. 21 (1957), 149–178.MathSciNetGoogle Scholar
- 17.——, Space of unitary vector bundles on a compact Riemann surface, Ann of Math. 82 (1965), 303–336.MathSciNetGoogle Scholar
- 18.——, Mumford's conjecture for GL(20) and applications, Algebraic geometry, Bombay Colloquium (1968), Oxford University Press.Google Scholar
- 19.A. Weill, Généralisation des fonctions abeliennes, J. Math. pures appl. 17 (1938), 47–87.Google Scholar
- 20.——, Remark on the cohomology of groups, Ann. of Math., 80 (1964), 149–157.MathSciNetCrossRefGoogle Scholar