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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