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Counts of maps to Grassmannians and intersections on the moduli space of bundles. (English) Zbl 1126.14044

Let \(\mathcal{N}\) denote the moduli space of stable bundles of a fixed determinant on a projective curve \(C\), where the rank \(r\) and degree \(d\) are assumed to be coprime.
The authors present a simple and entirely finite dimensional algebro-geometric derivation of the Witten-Szenes-Jeffrey-Kirwan residue formulas, which express the intersection numbers of the normalized characteristic classes of \(\mathcal{N}\) in terms of iterated residues. These include the Verlinda formula as a special case.
The main idea in the proof is to exploit the connection between the intersection theory of \(\mathcal{N}\) and that of a compactification Quot of the scheme of degree \(d\) morphisms from \(C\) to the Grassmannian \(G(r, N)\). In turn, the authors apply this connection to establish a vanishing result about intersections on Quot.

MSC:

14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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