Moduli for principal bundles over algebraic curves. I. (English) Zbl 0901.14007

This paper together with part II [ibid. 106, No. 4, 421-449 (1996)] forms the thesis of the author, who died in 1993. The results of this thesis have been used by many mathematicians, but they are published for the first time in these two papers.
Let \(G\) be a connected reductive algebraic group over \(\mathbb{C}\), and \(X\) a projective nonsingular irreducible complex curve. The subject of the thesis is the construction of the moduli spaces of semistable principal \(G\)-bundles over \(X\). In particular, for \(G= \text{GL}(n)\), we get the well-known moduli spaces of semistable vector bundles on \(X\), constructed by D. Mumford, M. S. Narasimhan and C. S. Seshadri [cf. M. S. Narasimhan and C. S. Seshadri, Ann. Math., II. Ser. 82, 540-567 (1965; Zbl 0171.04803); C. S. Seshadri, Ann. Math., II. Ser. 85, 303-336 (1967; Zbl 0173.23001)]. The author introduced the notion of stable principal \(G\)-bundle and constructed moduli spaces of such bundles as analytic varieties [A. Ramanathan, Math. Ann. 213, 129-152 (1975; Zbl 0284.32019)]. In this thesis the moduli spaces are constructed as algebraic varieties using the geometric invariant theory of Mumford. The more general notion of semistable principal \(G\)-bundles is used to obtain projective coarse moduli spaces.
In the first paper some general results on principal \(G\)-bundles are given. The notion of semistability for these bundles is introduced here. An equivalence relation on the set of principal \(G\)-bundles of a given topological type is defined and a kind of Jordan-Hölder theorem for semistable principal \(G\)-bundles is proved.
In the second paper the construction of the moduli spaces is done. It is reduced to a quotient space problem, and geometric invariant theory is used. A coarse projective moduli space is obtained for the set of semistable principal \(G\)-bundles of a given topological type, and it is proved that two principal \(G\)-bundles give the same point in this moduli space if and only if they are equivalent.


14D20 Algebraic moduli problems, moduli of vector bundles
14H60 Vector bundles on curves and their moduli
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H10 Families, moduli of curves (algebraic)
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