Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface.

*(English)*Zbl 0949.14021From 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.)].

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.)].

Reviewer: P.Cherenack (Rondebosch)

##### 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) |