Resumen de Automorphism group of the moduli space of parabolic vector bundles over a curve

David Alfaya Sánchez

• The main objective of this thesis is the computation of the automorphism group of the moduli space of parabolic vector bundles over a smooth complex projective curve.

  We will start by de ning the notion of a parabolic -module { a module over a sheaf of rings of di erential operators with a parabolic structure at certain marked points { and building their moduli space. This will provide us a common theoretical framework that allows us to work with several kinds of moduli spaces of bundles with parabolic structure such parabolic vector bundles, parabolic (L-twisted) Higgs bundles, parabolic connections or parabolic -connections. As an application, we build the parabolic Hodge moduli space and the parabolic Deligne{Hitchin moduli space.

  Then, we will address the computation of the automorphism group of the moduli space of parabolic bundles. Let X and X0 be irreducible smooth complex projective curves with sets of marked points D X and D0 X0 and genus g 6 and g0 6 respectively. LetM(X; r; ; ) be the moduli space of rank r stable parabolic vector bundles on (X;D) with parabolic weights and determinant . We classify the possible isomorphisms : M(X; r; ; ) 􀀀���! M(X0; r0; 0; 0). First, a Torelli type theorem is proved, implying that for to exist it is necessary that (X;D) = (X0;D0) and r = r0. Then we prove that the possible isomorphisms are generated by automorphisms of the pointed curve (X;D), tensorization with suitable line bundles, dualization of parabolic vector bundles and Hecke transformations at the parabolic points. These results are extended to birational equivalences : M(X; r; ; ) 99K M(X0; r0; 0; 0) which are de ned over \big" open subsets. The particular case of \concentrated" weights (corresponding to \small" stability parameters) is studied further. In this case Hecke transformations give rise to birational morphisms that do not extend to automorphisms of the moduli space. Moreover, an analysis of the stability chambers for the weights allows us to determine an explicit computable presentation of the group of automorphisms of the moduli space for arbitrary generic weights.

  Finally, the automorphism group of the moduli space of framed bundles over a smooth complex projective curve X of genus g > 2 with a framing over a point x 2 X is also described. It is shown that this group is generated by pullbacks using automorphisms of the curve X that x the marked point x, tensorization with certain line bundles over X and the action of PGLr(C) by composition with the framing.