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Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface. (English) Zbl 0949.14021
From the introduction: Let $$n$$ and $$d$$ be coprime positive integers, and define $${\mathcal M}(n,d)$$ to be the moduli space of (semi)stable holomorphic vector bundles of rank $$n$$, degree $$d$$ and fixed determinant of a compact Riemann surface $$\Sigma$$. The subject of this article is the characterization of the intersection pairings in the cohomology ring $$H^*({\mathcal M}(n,d))$$. A set of generators of this ring was described by M. F. Atiyah and R. Bott in their seminal 1982 paper [Philos. Trans. R. Soc. Lond., A 308, 523-615 (1983; Zbl 0509.14014)] on the Yang-Mills equations on Riemann surfaces.
In 1991, S. K. Donaldson [in: Topological methods in modern mathematics, Proc. Symp. Honor. J. Milnor, Stony Brook 1991, 137-170 (1993; Zbl 0870.57039)] and M. Thaddeus [J. Differ. Geom. 35, No. 1, 131-149 (1992; Zbl 0772.53013)] gave formulas for the intersection pairings between products of these generators in $$H^*({\mathcal M}(2,1))$$ (in terms of Bernoulli numbers). Then using physical methods, E. Witten [J. Geom. Phys. 9, No. 4, 303-368 (1992; Zbl 0768.53042)] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in $$H^*({\mathcal M}(n,d))$$ for general rank $$n$$. These generalized his (rigorously proved) formulas for the symplected volume of $${\mathcal M}(n,d)$$: For instance, the symplectic volume of $${\mathcal M}(2,1)$$ is given by $\text{vol} ({\mathcal M} (2,1))= \biggl(1- \frac{1}{2^{2g-3}} \biggr) \frac {\zeta(2g-2)} {2^{g-2} \pi^{2g-2}}= \frac{2^{g-1}- 2^{2-g}} {(2g- 2)!} |B_{2g-2}|,$ where $$g$$ is the genus of the Riemann surface, $$\zeta$$ is the Riemann zeta function and $$B_{2g-2}$$ is a Bernoulli number. The purpose of this paper is to obtain a mathematically rigorous proof of Witten’s formulas for general rank $$n$$. Our annoucement [L. C. Jeffrey and F. C. Kirwan, Electron. Res. Announc. Am. Math. Soc. 1, No. 2, 57-71 (1995; Zbl 0849.58033)] sketched the arguments we use, concentrating mainly on the case of rank $$n=2$$. The proof involves an application of the nonabelian localization principle [cf. L. C. Jeffrey and F. C. Kirwan, Topology 34, No. 2, 291-327 (1995; Zbl 0833.55009) and E. Witten (loc. cit.)].

##### MSC:
 14H60 Vector bundles on curves and their moduli 14H55 Riemann surfaces; Weierstrass points; gap sequences 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14D20 Algebraic moduli problems, moduli of vector bundles 14H10 Families, moduli of curves (algebraic)
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