106,961 research outputs found
On the equivariant algebraic Jacobian for curves of genus two
We present a treatment of the algebraic description of the Jacobian of a
generic genus two plane curve which exploits an SL2(k) equivariance and
clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The
treatment is also applied to the Kummer variety
Algebraic and hamiltonian approaches to isostokes deformations
We study a generalization of the isomonodromic deformation to the case of
connections with irregular singularities. We call this generalization Isostokes
Deformation. A new deformation parameter arises: one can deform the formal
normal forms of connections at irregular points. We study this part of the
deformation, giving an algebraic description. Then we show how to use loop
groups and hypercohomology to write explicit hamiltonians. We work on an
arbitrary complete algebraic curve, the structure group is an arbitrary
semisimiple group.Comment: 23 pages, minor corrections in the introduction, references expande
Fibrations of low genus, I
In the present paper we consider fibrations f: S \ra B of an algebraic
surface onto a curve , with general fibre a curve of genus . Our main
results are:
1) A structure theorem for such fibrations in the case
2) A structure theorem for such fibrations in the case and general
fibre nonhyperelliptic
3) A theorem giving a complete description of the moduli space of minimal
surfaces of general type with , showing in particular
that it has four unirational connected components
4) some other applications of the two structure theorems.Comment: 50 pages, to appear on Annales Scientifiques de l'Ecole Normale
Superieur
Spectral Curves, Opers and Integrable Systems
We establish a general link between integrable systems in algebraic geometry
(expressed as Jacobian flows on spectral curves) and soliton equations
(expressed as evolution equations on flat connections). Our main result is a
natural isomorphism between a moduli space of spectral data and a moduli space
of differential data, each equipped with an infinite collection of commuting
flows. The spectral data are principal G-bundles on an algebraic curve,
equipped with an abelian reduction near one point. The flows on the spectral
side come from the action of a Heisenberg subgroup of the loop group. The
differential data are flat connections known as opers. The flows on the
differential side come from a generalized Drinfeld-Sokolov hierarchy. Our
isomorphism between the two sides provides a geometric description of the
entire phase space of the Drinfeld-Sokolov hierarchy. It extends the Krichever
construction of special algebro-geometric solutions of the n-th KdV hierarchy,
corresponding to G=SL(n).
An interesting feature is the appearance of formal spectral curves, replacing
the projective spectral curves of the classical approach. The geometry of these
(usually singular) curves reflects the fine structure of loop groups, in
particular the detailed classification of their Cartan subgroups. To each such
curve corresponds a homogeneous space of the loop group and a soliton system.
Moreover the flows of the system have interpretations in terms of Jacobians of
formal curves.Comment: 64 pages, Latex, final version to appear in Publications IHE
Higgs bundles, abelian gerbes and cameral data
We study the Hitchin map for -Higgs bundles on a smooth
curve, where is a quasi-split real form of a complex reductive
algebraic group . By looking at the moduli stack of regular
-Higgs bundles, we prove it induces a banded gerbe structure on
a slightly larger stack, whose band is given by sheaves of tori. This
characterization yields a cocyclic description of the fibres of the
corresponding Hitchin map by means of cameral data. According to this, fibres
of the Hitchin map are categories of principal torus bundles on the cameral
cover. The corresponding points inside the stack of -Higgs bundles are
contained in the substack of points fixed by an involution induced by the
Cartan involution of . We determine this substack of fixed
points and prove that stable points are in correspondence with stable
-Higgs bundles.Comment: 34 page
Dual generators of the fundamental group and the moduli space of flat connections
We define the dual of a set of generators of the fundamental group of an
oriented two-surface of genus with punctures and the
associated surface with a disc removed. This dual is
another set of generators related to the original generators via an involution
and has the properties of a dual graph. In particular, it provides an algebraic
prescription for determining the intersection points of a curve representing a
general element of the fundamental group with the
representatives of the generators and the order in which these intersection
points occur on the generators.We apply this dual to the moduli space of flat
connections on and show that when expressed in terms both, the
holonomies along a set of generators and their duals, the Poisson structure on
the moduli space takes a particularly simple form. Using this description of
the Poisson structure, we derive explicit expressions for the Poisson brackets
of general Wilson loop observables associated to closed, embedded curves on the
surface and determine the associated flows on phase space. We demonstrate that
the observables constructed from the pairing in the Chern-Simons action
generate of infinitesimal Dehn twists and show that the mapping class group
acts by Poisson isomorphisms.Comment: 54 pages, 13 .eps figure
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