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Sunday, July 19, 2020 | History

6 edition of Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces found in the catalog.

Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces

William Mark Goldman

# Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces

## by William Mark Goldman

Written in English

Subjects:
• Surfaces, Deformation of,
• Riemann surfaces,
• Geometry, Differential,
• Geometry, Algebraic

• Edition Notes

Classifications The Physical Object Statement William M. Goldman, Eugene Z. Xia. Series Memoirs of the American Mathematical Society -- no. 904 Contributions Xia, Eugene Zhu, 1963- LC Classifications QA648 .G65 2008 Pagination p. cm. Open Library OL18285306M ISBN 10 9780821841365 LC Control Number 2008060005

I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. structure and its reinterpretations as a space of orbifold Higgs bundles or SL2(C)-representations of (a central extension of) the orbifold fundamental group. We follow Hitchin’s original paper for (ordinary) Riemann surfaces [14] quite closely but there are many novelties in the orbifold situation. (There is some overlap with.

We associate to each stable Higgs pair (A 0, Φ 0) on a compact Riemann surface X a singular limiting configuration (A ∞, Φ ∞), assuming that det Φ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions (A t, t Φ t) to Hitchin’s equations, which converge to this limiting configuration as t → ∞.This provides a new proof, via gluing. to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface (a surface group) and the moduli space of Higgs bundles. a rank one Higgs bundle is a pair (L,φ), where L → X is a It is due to Donaldson [6] (in the case of rank 2 bundles on a Riemann surface) and Corlette [5] (for base.

HIGGS BUNDLES AND NON-ABELIAN HODGE THEORY 5 FIGURE Depiction of a skyscraper sheaf. We denote the sections of ˇover an open subset U ˆX by S(U). Example If G is an abelian group with the discrete topology, then M G .   We study the deformations of twisted harmonic maps $$f$$ with respect to the representation $$\rho$$.After constructing a continuous “universal” twisted harmonic map, we give a construction of every first order deformation of $$f$$ in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical .

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### Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces by William Mark Goldman Download PDF EPUB FB2

Get this from a library. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces. [William Mark Goldman; Eugene Zhu Xia]. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces / Material Type: Document, Internet resource: Document Type: Internet Resource, Computer File: All Authors / Contributors: William Mark Goldman; Eugene Zhu Xia.

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $$X$$ and their relation to representations of the fundamental group of $$X$$.

The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. This expository article details the theory of rank one Higgs bundles over a closed Riemann surface X and their relation to representations of the fundamental group of X.

We construct an equivalence between the deformation theories of flat connections and Higgs pairs. Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces Article (PDF Available) in Memoirs of the American Mathematical Society () February with 19 Reads.

Abstract: This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X.

We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X.

We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. Memoirs of the American Mathematical Society: Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces.

/ Goldman, William M.; Xia, Eugene Zhu. American Mathematical Society, 69 p. Research output: Book/Report › BookAuthor: William M. Goldman, Eugene Zhu Xia. Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces William M. Goldman Eugene Z. Xia Author address: Department of Mathematics, University of Maryland, College Park, MD USA E-mail address: [email protected] DepartmentofMathematics, NationalChengKungUni-versity, TainanTaiwan E-mail.

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces 作者: William M. Goldman / Eugene Z. Xia 出版社: Amer Mathematical Society 出版年: 页数: 69 定价: USD 装帧: Paperback 丛书: memoirs of the american mathematical society. History.

It was proven by M. Narasimhan and C. Seshadri in that stable vector bundles on a compact Riemann surface correspond to irreducible projective unitary representations of the fundamental group.

This theorem was phrased in a new light in the work of Simon Donaldson inwho showed that stable vector bundles correspond to. Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces @inproceedings{HitchinIntegrableST, title={Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces}, author={Nigel J.

Hitchin and Graeme Segal and Richard Samuel Ward}, year={} } Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces About this Title. William M. Goldman and Eugene Z. Xia. Publication: Memoirs of the American Mathematical Society Publication Year VolumeNumber ISBNs: (print); (online).

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces. By William M. Goldman and Eugene Z. Xia. Download PDF ( KB) Abstract. This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X.

RANK ONE HIGGS BUNDLES 3 Introduction The set of equivalence classes of representations of the fundamental group πof a closed Riemann surface Xinto a Lie group Gis a basic object naturally associated to πand G. Powerful analytic techniques have been employed by Hitchin, Simpson, Corlette and Donaldson et.

Representations of the fundamental group 3 2. Abelian groups and rank one Higgs bundles 5 3. Stable vector bundles and Higgs bundles 6 4. Hyperbolic geometry: G= PSL(2,R) 8 5. Moduli of hyperbolic structures and representations 13 6. Rank two Higgs bundles 19 7.

Split R-forms and Hitchin’s Teichmu¨ller component 21 8. Hermitian symmetric. Abstract. This expository article details the theory of rank one Higgs bundles over a closed Riemann surface X and their relation to representations of the fundamental group of X.

We construct an equivalence between the deformation theories of flat connections and Higgs pairs. In the well-known paper [2], Hitchin introduced Higgs bundles, and established a one-to-one correspondence between equivalence classes of irreducible GL(2,C) representations of the fundamental group of a compact Riemann surface and isomorphism classes of rank two stable Higgs of degree zero.

In [7], Simpson deﬁned parabolic Higgs bundles. A Higgs bundle is a holomorphic vector bundle together with a Higgs field. Such objects first emerged twenty years ago in Nigel Hitchin's study of the self-duality equations on a Riemann. 2 Vector Bundles on Riemann Surfaces.

3 Higgs Bundles on Riemann Surfaces. 4 Representations of the Fundamental Group. 5 Nonabelian Hodge Theory. 6 Representations in Upq and Higgs Bundles.

7 Moment Maps and Geometry of Moduli Spaces. Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): (external link) http.The Hitchin-Kobayashi-Simpson correspondence for Higgs bundles on Riemann surfaces.

The Corlette-Donaldson theorem relating the moduli spaces of Higgs bundles and semisim-ple representations of the fundamental group.

A description of the oper moduli space and its relationship to systems of holomorphic.A flat connections is equivalent to a local system and the parallel transport of this connection in a loop only depends on the loop, so gives you a represenation of the fundamental group. A representation rep of the fundamental group of X defines a complex vector bundle of rank r via X x C^{r}/~ (fibers identified by rep).