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On the holomorphic maps from riemann surfaces to grassmannians

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Differential Geometry and Topology

Part of the book series: Lecture Notes in Mathematics ((2803,volume 1369))

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Abstract

With the recent results of Atiyah-Bott [2] about the Yang-Mills connections over the Riemann surface, and those of Narasimha- Seshadri [6] and Donaldson [3] about the stable holomorphic vector bundles, we have proved the following:

Theorem Let M be a compact Riemann surface with genus g(g ≥ 2), Hol d (M, G r (N)) the set of all full, indecomposable holomorphic maps with degree d from M to G r (N), (see §2 for the detailed definitions of the degree and full property). Then we have

$$\dim Hol_d (M,G_r (N)) = N(d + r - rg) + r^2 (g - 1)$$

, if d≥r(r-1)(3g-2)+2rg

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References

  1. M.F.Atiyah, Vector bundles over an elliptic curve, Proc. Lond. Math. Soc., 7, 414–452.

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Authors and Affiliations

  1. Dept. of Math., Fudan Univ., Shanghai, China

    Chun-li Shen

Authors
  1. Chun-li Shen
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Boju Jiang Chia-Kuei Peng Zixin Hou

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© 1989 Springer-Verlag

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Shen, Cl. (1989). On the holomorphic maps from riemann surfaces to grassmannians. In: Jiang, B., Peng, CK., Hou, Z. (eds) Differential Geometry and Topology. Lecture Notes in Mathematics, vol 1369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087536

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  • DOI: https://doi.org/10.1007/BFb0087536

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51037-6

  • Online ISBN: 978-3-540-46137-1

  • eBook Packages: Springer Book Archive

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