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.


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