### Abstract

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky G-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino (Manuscr Math 105:69–83, 2001) to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.

Original language | English |
---|---|

Pages (from-to) | 379-409 |

Journal | Geometriae Dedicata |

Early online date | 18 Oct 2018 |

DOIs | |

Publication status | Published - 1 Aug 2019 |

### Fingerprint

### Keywords

- Character varieties
- Moduli spaces
- Principal bundles
- Representations of the fundamental group
- Riemann surfaces
- Schottky bundles
- Uniformization

### Cite this

*Geometriae Dedicata*, 379-409. https://doi.org/10.1007/s10711-018-0398-2

}

*Geometriae Dedicata*, pp. 379-409. https://doi.org/10.1007/s10711-018-0398-2

**Principal Schottky bundles over Riemann surfaces.** / Casimiro, A. C.; Ferreira, S.; Florentino, C.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Principal Schottky bundles over Riemann surfaces

AU - Casimiro, A. C.

AU - Ferreira, S.

AU - Florentino, C.

N1 - info:eu-repo/grantAgreement/FCT/3599-PPCDT/120411/PT# info:eu-repo/grantAgreement/FCT/3599-PPCDT/125240/PT# info:eu-repo/grantAgreement/FCT/3599-PPCDT/126662/PT# info:eu-repo/grantAgreement/FCT/5876/147204/PT# USA NSF Grants DMS 1107452, 1107263, 1107367

PY - 2019/8/1

Y1 - 2019/8/1

N2 - We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky G-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino (Manuscr Math 105:69–83, 2001) to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.

AB - We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky G-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino (Manuscr Math 105:69–83, 2001) to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.

KW - Character varieties

KW - Moduli spaces

KW - Principal bundles

KW - Representations of the fundamental group

KW - Riemann surfaces

KW - Schottky bundles

KW - Uniformization

UR - http://www.scopus.com/inward/record.url?scp=85055570909&partnerID=8YFLogxK

U2 - 10.1007/s10711-018-0398-2

DO - 10.1007/s10711-018-0398-2

M3 - Article

SP - 379

EP - 409

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

ER -