31 research outputs found
Real points of coarse moduli schemes of vector bundles on a real algebraic curve
We examine a moduli problem for real and quaternionic vector bundles on a
smooth complex projective curve with a fixed real structure, and we give a
gauge-theoretic construction of moduli spaces for semi-stable such bundles with
fixed topological type. These spaces embed onto connected subsets of real
points inside a complex projective variety. We relate our point of view to
previous work by Biswas, Huisman and Hurtubise (arxiv:0901.3071), and we use
this to study the Galois action induced on moduli varieties of stable
holomorphic bundles on a complex curve by a given real structure on the curve.
We show in particular a Harnack-type theorem, bounding the number of connected
components of the fixed-point set of that action by , where is the
genus of the curve. In fact, taking into account all the topological invariants
of the real structure, we give an exact count of the number of connected
components, thus generalising to rank the results of Gross and Harris
on the Picard scheme of a real algebraic curve.Comment: 24 pages, 1 figur
Lectures on Klein surfaces and their fundamental group
The goal of these lectures is to give an introduction to the study of the
fundamental group of a Klein surface. We start by reviewing the topological
classification of Klein surfaces and by explaining the relation with real
algebraic curves. Then we introduce the fundamental group of a Klein surface
and present its main basic properties. Finally, we study the variety of unitary
representations of this group and relate it to the representation variety of
the topological fundamental group of the underlying Riemann surface.Comment: To appear in the collection Advanced Courses in Mathematics - CRM
Barcelon
Quasi-Hamiltonian quotients as disjoint unions of symplectic manifolds
We show that the quotient associated to a quasi-Hamiltonian space has a
symplectic structure even when 1 is not a regular value of the momentum map: it
is a disjoint union of symplectic manifolds of possibly different dimensions,
which generalizes a result of Alekseev, Malkin and Meinrenken. We illustrate
this theorem with the example of representation spaces of surface groups. As an
intermediary step, we show that the isotropy submanifolds of a
quasi-Hamiltonian space are quasi-Hamiltonian spaces themselves
Rational points of quiver moduli spaces
For a perfect field , we study actions of the absolute Galois group of
on the -valued points of moduli spaces of quiver representations over
; the fixed locus is the set of -rational points and we obtain a
decomposition of this fixed locus indexed by elements in the Brauer group of
. We provide a modular interpretation of this decomposition using quiver
representations over division algebras, and we reinterpret this description
using twisted quiver representations. We also see that moduli spaces of twisted
quiver representations give different forms of the moduli space of quiver
representations.Comment: This paper is a revised and extended version of parts of
arXiv:1612.06593v1, which has now been split into two papers. This version is
an expanded version of the accepted publication (longer introduction
The Yang-Mills equations over Klein surfaces
Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed
topological type admit a presentation as Lagrangian quotients, and can be
embedded into the symplectic quotient corresponding to the moduli variety of
semi-stable holomorphic vector bundles of fixed rank and degree on a smooth
complex projective curve. From the algebraic point of view, these Lagrangian
quotients are connected sets of real points inside a complex moduli variety
endowed with a real structure; when the rank and the degree are coprime, they
are in fact the connected components of the fixed-point set of the real
structure. This presentation as a quotient enables us to generalize the methods
of Atiyah and Bott to a setting with involutions, and compute the mod 2
Poincare polynomials of these moduli spaces in the coprime case. We also
compute the mod 2 Poincare series of moduli stacks of all real and quaternionic
vector bundles of a fixed topological type. As an application of our
computations, we give new examples of maximal real algebraic varieties.Comment: Final version, 72 pages; formulae in the quaternionic, n>0 case
corrected; proof of Theorem 1.3 revised; references adde