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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 $$\Gamma$$.