1,668 research outputs found

### A weight two phenomenon for the moduli of rank one local systems on open varieties

The twistor space of representations on an open variety maps to a weight two
space of local monodromy transformations around a divisor component at infinty.
The space of $\sigma$-invariant sections of this slope-two bundle over the
twistor line is a real 3 dimensional space whose parameters correspond to the
complex residue of the Higgs field, and the real parabolic weight of a harmonic
bundle

### The dual boundary complex of the $SL_2$ character variety of a punctured sphere

Suppose $C_1,\ldots , C_k$ are generic conjugacy classes in $SL_2({\mathbb
C})$. Consider the character variety of local systems on ${\mathbb P}^1-\{
y_1,\ldots , y_k\}$ whose monodromy transformations around the punctures $y_i$
are in the respective conjugacy classes $C_i$. We show that the dual boundary
complex of this character variety is homotopy equivalent to a sphere of
dimension $2(k-3)-1$

### Poisson varieties from Riemann surfaces

Short survey based on talk at the Poisson 2012 conference. The main aim is to
describe and give some examples of wild character varieties (naturally
generalising the character varieties of Riemann surfaces by allowing more
complicated behaviour at the boundary), their Poisson/symplectic structures
(generalising both the Atiyah-Bott approach and the quasi-Hamiltonian
approach), and the wild mapping class groups.Comment: 33 pages, 3 figure

### Rigidity of Spreadings and Fields of Definition

Varieties without deformations are defined over a number field. Several old
and new examples of this phenomenon are discussed such as Bely\u \i\ curves and
Shimura varieties. Rigidity is related to maximal Higgs fields which come from
variations of Hodge structure. Basic properties for these due to P. Griffiths,
W. Schmid, C. Simpson and, on the arithmetic side, to Y. Andr\'e and I. Satake
all play a role. This note tries to give a largely self-contained exposition of
these manifold ideas and techniques, presenting, where possible, short new
proofs for key results.Comment: Accepted for the EMS Surveys in Mathematical Science

### On the classification of rank two representations of quasiprojective fundamental groups

Suppose $X$ is a smooth quasiprojective variety over \cc and \rho : \pi
_1(X,x) \to SL(2,\cc) is a Zariski-dense representation with quasiunipotent
monodromy at infinity. Then $\rho$ factors through a map $X\to Y$ with $Y$
either a DM-curve or a Shimura modular stack.Comment: minor changes in exposition, citation

### Sheaves on Toric Varieties for Physics

In this paper we give an inherently toric description of a special class of
sheaves (known as equivariant sheaves) over toric varieties, due in part to A.
A. Klyachko. We apply this technology to heterotic compactifications, in
particular to the (0,2) models of Distler, Kachru, and also discuss how
knowledge of equivariant sheaves can be used to reconstruct information about
an entire moduli space of sheaves. Many results relevant to heterotic
compactifications previously known only to mathematicians are collected here --
for example, results concerning whether the restriction of a stable sheaf to a
Calabi-Yau hypersurface remains stable are stated. We also describe
substructure in the Kahler cone, in which moduli spaces of sheaves are
independent of Kahler class only within any one subcone. We study F theory
compactifications in light of this fact, and also discuss how it can be seen in
the context of equivariant sheaves on toric varieties. Finally we briefly
speculate on the application of these results to (0,2) mirror symmetry.Comment: 83 pages, LaTeX, 4 figures, must run LaTeX 3 times, numerous minor
cosmetic upgrade

### Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated
to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs
bundles on a Riemann surface C agrees with the weight filtration on the
cohomology of the twisted G character variety of C, when the cohomologies are
identified via non-Abelian Hodge theory. The proof is accomplished by means of
a study of the topology of the Hitchin map over the locus of integral spectral
curves.Comment: 67 pages, arguments streamlined, to appear in Annals of Mathematic

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