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On the cohomology of moduli spaces of rank two vector bundles over curves


Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, D. (1995). On the cohomology of moduli spaces of rank two vector bundles over curves. In The moduli space of curves. Proceedings of the conference held on Texel Island, Netherlands during the last week of April 1994 (pp. 533-563). Basel: Birkhäuser.

Cite as: http://hdl.handle.net/21.11116/0000-0004-38F7-7
Let C be a compact Riemann surface of genus g. For a fixed line bundle L of degree d over C, denote by \\cal Ng,n,L the moduli space of semistable vector bundles over C of rank n and determinant bundle L. As the topology of \\cal Ng,n,L is uniquely determined by the genus g of C and the parity of d = \\deg (L), one is lead to consider primarily the moduli spaces \\cal Ng,n^+ (for d even) and \\cal N^-g,n (for d odd). -- The present paper deals with the moduli spaces of semistable rank-2 vector bundles over C, i.e., basically with the topological properties of the moduli spaces \\cal N^-g : = \\cal N^-g,2 and \\cal N^+g : = \\cal N^+g,2.\\par The author's main result is a complete description of the additive structure, the multiplicative structure and the intersection theory of the cohomology ring of \\cal N^-g over \\bbfQ. While the additive structure of H^*(\\cal N^-g, \\bbfQ) has been known from the earlier work of Atiyah-Bott (1982) and Harder-Narasimhan (1975), the multiplicative structure was first conjectured by D. Mumford, and affirmatively verified by \\it F. Kirwan [cf. J. Am. Math. Soc. 5, No. 4, 853-906 (1992; Zbl 0804.14010)].\\par The author's proof of Mumford's conjecture on the generators and relations in H^* (\\cal N^-g, \\bbfQ), which is presented here, was already found by him in 1991, as an unpublished result. In spite of F. Kirwán's (published) proof of Mumford's conjecture, the author's approach to the enumerative geometry of the cohomology ring H^* (\\cal N^-g, \\bbfQ) is, nevertheless, of particular interest. This is because it not only provides some additional information about the intersection pairing in the cohomology ring, which may be useful in particular contexts, but also allows some interesting applications that, quite amazingly, provide new proofs of some other great theorems established during the past five years.\\par The first application is another proof of the Verlinde formulas for the dimensions of the spaces of generalized theta functions on \\cal N^-g and \\cal N^+g. The author's proof is, compared to the earlier ones given by Bertram-Szenes (1993), Beauville-Laszlo (1993), Narasimhan-Ramadas (1993), Thaddeus (1994), and Faltings (1994), obtained by a much more direct and computational method which, by its own, is based upon the very explicit enumerative geometry of \\cal N^\\pmg calculated before.\\par Further important applications concern three former conjectures of \\it P. E. Newstead [Trans. Am. Math. Soc. 169, 337-345 (1972; Zbl 0256.14008)] on the Chern classes of the tangent bundle on \\cal N^-g. Although those conjectures of Newstead have been affirmatively answered, in the meantime, namely by D. Gieseker (1984), Thaddeus (1994), and F. Kirwan (1992; loc. cit.), the author's new proofs are (again) very direct and explicitly derived from the structure theory of H^* (\\cal N^-g, \\bbfQ).