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2 results

  • 7 - Moduli Space of Semistable G-Bundles Over a Smooth Curve

  • Shrawan Kumar, University of North Carolina, Chapel Hill
  • Book:
    Conformal Blocks, Generalized Theta Functions and the Verlinde Formula
    Published online:
    19 November 2021
    Print publication:
    25 November 2021, pp 277-327

  • Summary

    Using some general result of Grothendieck on the existence of quot schemes, we construct the coarse moduli space M(r, d) for rank-r and degree-d vector bundles on a smooth projective curve ?, which consists of S-equivalence classes of semistable vector bundles of rank r and degree d. The construction proceeds via the Geometric Invariant Theory. The moduli space M(r, d) is an irreducible, normal projective variety with rational singularities. Moreover, the subset consisting of stable vector bundles is an open subset, which bijectively parameterizes the isomorphism classes of stable vector bundles. This subset provides the coarse moduli space of stable vector bundles. We extend the above results for vector bundles to G-bundles over? and even more generally to equivariant G-bundles. It is achieved by taking an embedding of G into the general linear group GL(r) and realizing a G-bundle as a rank-r vector bundle together with a reduction of the structure group to G.

  • Moduli spaces of stable quotients and wall-crossing phenomena

  • Part of
  • Yukinobu Toda
  • Journal:
    Compositio Mathematica / Volume 147 / Issue 5 / September 2011
    Published online by Cambridge University Press:
    31 May 2011, pp. 1479-1518
    Print publication:
    September 2011
    • Article
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  • The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.