Prelectura de tesis

Automorphisms of Higgs bundle Moduli spaces for real groups.

Manuel Jesus Perez Garca.

Advisor:  Oscar Garca Prada.

30 de Mayo, 16h., Sala 520, Módulo 17, Dpto. de Matemáticas

Abstract: Let G be a connected real form of a complex semisimple Lie
group GC with Lie algebra g. Let H be a maximal compact subgroup of
G and let  be a Cartan involution of g such that it induces a decompo-
sition into 1-eigenspaces g = h  m, where h is the Lie algebra of H. A
(G; )-Higgs bundle over a compact Riemann surface X is a pair consisting
on a holomorphic principal HC-bundle E and a holomorphic section ' of
E(mC) K where E(mC) is the bundle associated to E via the isotropy
representation C : HC ! GL(mC) and K is the canonical bundle over X.
Consider the moduli space M(G; ) of isomorphism classes of polystable
(G; )-Higgs bundles over X. In this talk we study the action of nite or-
der automorphisms of M(G) dened by combining the multiplication of
the Higgs eld by an nth-root of unity and the action of an element in
(H1(X;Z(HC) Ker(C)) o Out(g; ))n, where Out(G; ) is the group of
outer automorphisms of G that commute with . In addition, we describe its
xed points subvarieties and, through non-abelian Hodge correspondence,
we translate these results to the moduli