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Moduli of vector bundles on curves with parabolic structures. (English) Zbl 0354.14005
Let \(H\) denote the upper half plane and \(\Gamma\) a discrete subgroup of \(\operatorname{Aut} H\). When \(H \bmod\Gamma\) is compact, there is an algebraic interpretation for the unitary representations of \(\Gamma\) in terms of stable and semistable vector bundles over the Riemann surface \(H\bmod\Gamma\). In this paper the author announces an extension of this result to the case where \(H \bmod\Gamma\) has finite measure. For this let \(X\) be a smooth irreducible projective curve defined over an algebraically closed field \(k\), a point of \(X\) and \(V\) a vector bundle over \(X\). The author introduces the concept of a parabolic structure on \(V\) at \(Q\), and states a classification theorem for bundles with parabolic structures completely analogous to those of M. S. Narasimhan and himself for bundles without the added structure [Ann. Math. (2) 82, 540–567 (1965; Zbl 0171.04803); the author, ibid. 85, 303–336 (1967; Zbl 0173.23001) and C. I. M. E., \(3^\circ\) Ciclo Varenna 1969, Quest. algebr. Varieties, 139–260 (1970; Zbl 0209.24503)]. When \(k=\mathbb C\) and \(X-Q=H \bmod\Gamma\), certain of the corresponding moduli spaces can be identified with spaces of unitary representations of \(\Gamma\).
Reviewer: P. E. Newstead

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
14H15 Families, moduli of curves (analytic)
32L05 Holomorphic bundles and generalizations
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
Full Text: DOI
[1] David Mumford, Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 526 – 530.
[2] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965. · Zbl 0147.39304
[3] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540 – 567. · Zbl 0171.04803
[4] C. S. Seshadri, Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 85 (1967), 303 – 336. · Zbl 0173.23001
[5] C. S. Seshadri, Moduli of \pi -vector bundles on an algebraic curve, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 139-260. MR 43 # 6216.
[6] A. Weil, Généralisation des fonctions abéliennes, J. Math. Pures Appl. 17 (1938), 47-87. · JFM 64.0361.02
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