# Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

@inproceedings{Goldman2004RankOH, title={Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces}, author={William M. Goldman and Eugene Z. Xia}, year={2004} }

Introduction Equivalences of deformation theories The Betti and de Rham deformation theories and their moduli spaces The Dolbeault groupoid Equivalence of de Rham and Dolbeault groupoids Hyperkahler geometry on the moduli space The twistor space The moduli space and the Riemann period matrix Bibliography.

## 33 Citations

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

- Mathematics
- 2007

The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor com- ponent at infinty. The space of �-invariant sections of…

### Moduli Spaces of Higgs Bundles – Old and New

- MathematicsJahresbericht der Deutschen Mathematiker-Vereinigung
- 2021

We give an overview of the differential geometric and analytic aspects of Higgs bundles and their moduli spaces and highlight some of their interrelations with neighboring fields. We review various…

### Abelian and non-abelian cohomology

- Mathematics
- 2014

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable…

### Non-Abelian Hodge theory gives a real analytic isomorphism between two algebraically quite di erent varieties associated to a Riemann surface X : The Betti moduli

- Mathematics
- 2021

Non-Abelian Hodge theory gives a real analytic isomorphism between two algebraically quite di erent varieties associated to a Riemann surface X: The Betti moduli space, the moduli space of…

### Recent results and conjectures on the non abelian Hodge theory of curves

- Mathematics
- 2017

We give an introduction to non-abelian Hodge theory for curves with the aim of stating the $$P = W$$P=W conjecture both in its original cohomological version and in the more recent geometric one, and…

### On the geometric P=W conjecture

- MathematicsSelecta Mathematica
- 2022

We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we…

### Hitchin's equations and Fourier transform on curves

- Mathematics
- 2007

These are informal notes to my talks at the Lie algebras and moduli spaces seminar of the Eotvos Lorand University of Budapest in March 2007. First, we review some standard facts and constructions…

### Deligne pairings and families of rank one local systems on algebraic curves

- MathematicsJournal of Differential Geometry
- 2020

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting,…

### METRICS AND SPECIAL KÄHLER GEOMETRY ON THE MODULI SPACES OF HIGGS BUNDLES AND HITCHIN SYSTEMS

- MathematicsBulletin of the Australian Mathematical Society
- 2019

The notions of Hitchin systems and Higgs bundles (also called Higgs pairs) were introduced by N. Hitchin in 1987. They rapidly formed a subject lying on the crossroads of representation theory,…

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