Automorphism group of the moduli space of parabolic vector bundles over a curve
Author
Alfaya Sánchez, DavidEntity
UAM. Departamento de MatemáticasDate
2018-10-26Subjects
Geometría algebraica - Tesis doctorales; Módulos (Álgebra) - Tesis doctorales; MatemáticasNote
Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 26-10-2018Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Abstract
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.
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