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Lectures on the Moduli Stack of Vector Bundles on a Curve

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Book cover Affine Flag Manifolds and Principal Bundles

Part of the book series: Trends in Mathematics ((TM))

Abstract

These are lecture notes of a short course on the moduli stack of vector bundles on an algebraic curve. The aim of the course was to use this example to introduce the notion of algebraic stacks and to illustrate how one can work with these objects. Applications given are the (non-)existence of universal families on coarse moduli spaces and the computation of the cohomology of the moduli stack.

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References

  1. M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189.

    Article  MATH  MathSciNet  Google Scholar 

  2. M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1983), 523–615.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Beauville, Sur la cohomologie de certains espaces de modules de fibrés vectoriels in Geometry and Analysis (Bombay, 1992), 37–40, Tata Inst. Fund. Res., Bombay, 1995.

    Google Scholar 

  4. K.A. Behrend, The Lefschetz Trace Formula for the Moduli Space of Principal Bundles, PhD thesis, Berkeley, 1990, 96 pp., available at http://www.math.ubc.ca/∼behrend/thesis.html.

    Google Scholar 

  5. K.A. Behrend, The Lefschetz trace formula for algebraic stacks, Invent. Math. 112 (1993), 127 49.

    Article  MATH  MathSciNet  Google Scholar 

  6. K.A. Behrend, Derived ℓ-adic categories for algebraic stacks, Mem. Amer. Math. Soc. 163 (2003), no. 774, viii+93 pp.

    MathSciNet  Google Scholar 

  7. K.A. Behrend, A. Dhillon, On the motive of the stack of bundles, Adv. Math. 212 (2007), 617–644.

    Article  MATH  MathSciNet  Google Scholar 

  8. K.A. Behrend, A. Dhillon, Connected components of moduli stacks of torsors via Tamagawa numbers, Can. J. Math. 61 (2009), 3–28.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Bifet, F. Ghione, M. Letizia, On the Abel-Jacobi map for divisors of higher rank on a curve, Math. Ann. 299 (1994), 641–672.

    Article  MATH  MathSciNet  Google Scholar 

  10. I. Biswas, N. Hoffmann, The line bundles on moduli stacks of principal bundles on a curve, preprint, arXiv:0805.2915.

    Google Scholar 

  11. P. Deligne, Cohomologie Étale (SGA 41/2), Lecture Notes in Mathematics 569, Springer Verlag 1977.

    Google Scholar 

  12. A. Dhillon, On the cohomology of moduli of vector bundles and the Tamagawa number of SLn, Can. J. Math. 58 (2006), 1000–1025.

    Article  MATH  MathSciNet  Google Scholar 

  13. J.-M. Drezet, M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. Fantechi, Stacks for everybody, European Congress of Mathematics, Vol. I (Barcelona, 2000), 349–359, Progr. Math., 201, Birkhäuser, Basel, 2001.

    Google Scholar 

  15. E.M. Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies, 104, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1982, vii+190 pp.

    Google Scholar 

  16. R. Friedman, J.W. Morgan, On the converse to a theorem of Atiyah and Bott, J. Algebr. Geom. 11 (2002), 257–292.

    MATH  MathSciNet  Google Scholar 

  17. W. Fulton, Intersection theory, second edition, Springer Verlag, 1998.

    Google Scholar 

  18. T. Gómez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 1, 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Grothendieck et. al., Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris), 3, Société Mathématique de France, Paris, 2003, xviii+327 pp.

    Google Scholar 

  20. G. Harder, M.S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215–48.

    Article  MATH  MathSciNet  Google Scholar 

  21. G. Hein, Faltings’ construction of the moduli space, this volume.

    Google Scholar 

  22. J. Heinloth, Über den Modulstack der Vektorbündel auf Kurven, Diploma thesis, Bonn, 1998, 64 pp, available at http://www.uni-essen.de/∼hm0002/.

    Google Scholar 

  23. J. Heinloth, A.H.W. Schmitt, The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, preprint, available at http://www.uni-essen.de/∼hm0002/.

    Google Scholar 

  24. G. Laumon, L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 39, Springer-Verlag, Berlin, 2000, xii+208 pp.

    Google Scholar 

  25. G. Laumon, M. Rapoport, The Langlands lemma and the Betti numbers of stacks of G-bundles on a curve, Intern. J. Math. 7 (1996), 29–45.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Lieblich, Moduli of twisted sheaves, Duke Math. J. 138 (2007), no. 1, 23–118.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Olsson, Sheaves on Artin stacks, J. reine angew. Math. 603 (2007), 55–112.

    MATH  MathSciNet  Google Scholar 

  28. S. Ramanan, The moduli space of vector bundles on an algebraic curve, Math. Ann. 200 (1973), 69–84.

    Article  MATH  MathSciNet  Google Scholar 

  29. C.S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982).

    Google Scholar 

  30. D. Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., 9, Bar-Ilan Univ., Ramat Gan, 1996, 445–462.

    Google Scholar 

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Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, NL-1018 TV, Amsterdam, The Netherlands

    Jochen Heinloth

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  1. Jochen Heinloth
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Editors and Affiliations

  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Alexander Schmitt

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Heinloth, J. (2010). Lectures on the Moduli Stack of Vector Bundles on a Curve. In: Schmitt, A. (eds) Affine Flag Manifolds and Principal Bundles. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0346-0288-4_4

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