Abstract
The cohomology ring of the moduli space M(n,d) of semistable bundles of coprime rank n and degree d over a Riemann surface M of genus g ≥ 2 has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when n>2. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these ‘dual’ Mumford relations form a complete set for the rational cohomology ring of M(3,d) and show (Sect. 5) that the Pontryagin ring vanishes in degree 12g-8 and above.
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References
Atiyah, M. F. and Bott, R.: The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A308 (1982) 523–615.
Baranovsky, V.: Cohomology ring of the moduli space of stable vector bundles with odd determinant, Izv. Russ. Acad. Nauk.58(4) (1994 204–210.
Earl, R. A.: A note on the cohomology of the moduli of rank two stable bundles, (1996 preprint).
Harder, G. and Narasimhan, M. S.: On the cohomology groups of moduli spaces of vector bundles over curves, Math. Ann.212 (1975) 215–248.
Hartshorne, R.: Algebraic Geometry, GTM 52, Springer-Verlag, New York, Heidelberg, Berlin, 1977.
King, A. D. and Newstead, P. E.: On the cohomology of themoduli space of rank 2 vector bundles on a curve, (1995 preprint).
Kirwan, F. C.: On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Math.24 (1986) 221–275.
Kirwan, F. C.: Cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc.5 (1992) 853–906.
Milnor, J. W. and Stasheff, J. D.: Characteristic Classes, Ann. of Math. Stud. Vol. 76, Princeton University Press Princeton NJ, 1974.
Neeman, A.: The topology of quotient varieties, Ann. Math. ( 2 )122 (1985) 419–459.
Newstead, P. E.: Introduction to moduli problems and orbit spaces, Tata Inst. Lect.51 (1978).
Newstead, P. E.: Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc.169 (1972) 337–345.
Shatz, S. S.: The decomposition and specialization of algebraic families of vector bundles, Compositio Math.35 (1977) 163–187.
Siebert, B. and Tian, G.: Recursive relations for the cohomology ring of moduli spaces of stable bundles, (1994 preprint).
Thaddeus, M.: Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom.35 (1992) 131–149.
Weitsman, J.: Geometry of the intersection ring of the moduli space of flat connections and the conjectures of Newstead and Witten, (1993 preprint).
Zagier, D.: On the Cohomology of moduli spaces of rank 2 vector bundles over curves, (1995 preprint).
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EARL, R. The Mumford relations and the moduli of rank three stable bundles. Compositio Mathematica 109, 13–48 (1997). https://doi.org/10.1023/A:1000101030261
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DOI: https://doi.org/10.1023/A:1000101030261