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Stable bundles revisited. (English) Zbl 0757.32013

Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 1-18 (1991).
[For the entire collection see Zbl 0743.00053.]
This article was entered twice in ZMATH database (both reviews are given below).
This is a survey article discussing several topics related to the moduli variety of vector bundles on algebraic curves. The first topic is a computation of the Poincaré polynomial of this variety which can be done along two completely different lines. M. F. Atiyah and the author [Philos. Trans. R. Soc. Lond., A 308, 523–615 (1983; Zbl 0509.14014)] worked over \(\mathbb{C}\) and they have used the equivariant Morse theory for this purpose. G. Harder and M. S. Narasimhan [Math. Ann. 212, 215–248 (1975; Zbl 0324.14006)] and later Usha V. Desale and S. Ramanan [Math. Ann. 216, 233–244 (1975; Zbl 0293.14005)] have used the reduction of the moduli variety into finite characteristic and then they have applied the Weil conjectures. This method is rather popular now. Recently it was used by L. Goettsche for the computation of the Betti numbers of the Hilbert schemes and by K. Yoshioka for the computation of the Betti numbers of the moduli of vector bundles over some algebraic surfaces. – The author of the survey shows that there exists a striking and mysterious analogy between these two methods.
The second part of the paper is devoted to a discussion of the Verlinde formula for the dimension of the space of sections of some line bundles associated with theta-functions.
A.N. Parshin (Moskva)
As the author remarks in the beginning “the topological classification of complex vector bundles over a Riemann surface is of course very simple ... a classification in the complex analytic category leads to ... subtle phenomena which links with number theory, gauge theory and conformal field theory ...”. He starts with line bundles. The classical results are reviewed such as Jacobian varieties, \(\theta\)-divisors, Poincaré series and the dimension formula for \(H^ 0\) of \(k\theta\).
To extend these results to vector bundles he reviews the notion of stability by Mumford. The isomorphism classes of stable bundles \(E\) were known to admit a structure of a smooth algebraic variety \(J^ n(M)\), \(n\) denoting the rank. \(J^ n(M)\) falls into components \(J^ n_ k(M)\), where \(k=c_ 1(E)\). When \(n\) and \(k\) are relatively prime, \(J^ n_ k\) is compact. Otherwise one usually compactifies \(J^ n_ k\) by using semistable bundles, the compactification also denoted by \(J^ n_ k\). He then explains how the notion of stability helps to explain the structure of all bundles. This part of analysis also plays an essential role in the approaches to the problem of computing \(H^*(J^ n_ k)\) as taken by Harder, Narasimhan, Atiyah and himself. The Yang-Mills version of this topic relates the complex structures on \(E\) to the connections on a principal \(U(n)\) bundle \(P\), which was contained in the joint work of Atiyah and himself.
The question of extending the classical \(\theta\) divisor to the higher rank bundles has been studied by mathematicians and physicists. Not much concerned with some technical difficulties, there is a formula due to E. Verlinde computing \(\dim H^ 0(J^ n_{n(g-1)}(M);k\theta)\). The author describes this formula in the context of representation as developed by A. Szenes and himself. Finally he explains how the papers of N. J. Hitchin [Commun. Math. Phys. 131, No. 2, 347–380 (1990; Zbl 0718.53021)] and S. Axelrod, S. Della Pietra, and E. Witten [J. Differ. Geom. 33, No. 3, 787–902 (1991; Zbl 0697.53061)] provide a better understanding of Verlinde’s arguments.

MSC:

32Q20 Kähler-Einstein manifolds
14H60 Vector bundles on curves and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
81T13 Yang-Mills and other gauge theories in quantum field theory
14M12 Determinantal varieties
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