U(p, q)-HIGGS BUNDLES AND THE
HITCHIN–KOSTANT–RALLIS SECTION
´
ANA PEON-NIETO
Higgs bundles on Riemann surfaces are the holomorphic counterpart
to surface group representations. In this talk I will present the basic
theory of U(p, q)-Higgs bundles, from their definition to their relation
with representations.
The Hitchin map associates to a U(p, q)-Higgs bundle (E, φ), consisting of a vector bundle E and an endomorphism φ : E → E ⊗K twisted
by the canonical bundle (plus some extra conditions), the characteristic coefficients of φ. This induces a fibration of the moduli space of
U(p, q)-Higgs bundles onto a vector space :
h : HiggsX (G) → H 0 (X, ⊕qi=1 K 2i ),
whose study has been proved extremely useful in the understanding of
the geometry of moduli spaces of (Higgs) bundles.
A first step towards the analysis of the Hitchin map is the definition
of a section. I will explain this construction, generalising the one by
Hitchin, and some interesting properties, such as the topological type,
or the relation with Anosov representations.
´
This is joint work with Oscar
Garc´ıa-Prada and S. Ramanan.
¨ t, INF 288, 69129
Mathematisches Institut, Ruprecht–Karls Universita
Heideberg, Germany
E-mail address: [email protected]
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