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On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles. (English) Zbl 0625.14026
The paper contains two parts. In the first one, the following result is proved: “Let M be a Riemann surface of genus \(g,\) G(n,m) the Grassmann manifold of n-dimensional quotients of \({\mathbb{C}}^ m\) and k a positive integer with \(k\geq n-2g\). Then the inclusion \(Hol_ d(M,G(n,m))\to Map_ d(M,G(n,m))\) induces cohomology isomorphisms up to dimension \(k- 2m^ 2g\), provided that \(d\geq d_ 0(k,n,g)\). - Here \(Hol_ d\) (resp. \(Map_ d)\) stands for holomorphic (resp. continuous) maps of degree \(d.\) The bound \(d_ 0\) is done explicitly. For maps in the projective space, the result was shown by G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] who conjectured more general results, as the above ones. The proof is done by induction, starting with Segal’s result, and uses extensively holomorphic bundles on M.
One uses all this, and previous technique of the author [“Cohomology of quotients in symplectic and algebraic geometry”, Math. Notes 31 (1984; Zbl 0553.14020)], to rederive in the second part of the paper the formula given by M. F. Atiyah and R. Bott [Philos. Trans. R. Soc. Lond., A 308, 523-615 (1982; Zbl 0509.14014)] for the cohomology of the moduli spaces of holomorphic bundles on curves.
Reviewer: C.Bănică

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L24 Geometric invariant theory
14M17 Homogeneous spaces and generalizations
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