Torelli theorems for moduli of logarithmic connections and parabolic bundles

Sebastian, Ronnie

doi:10.1007/s00229-011-0446-9

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…In [5], the moduli space of rank n logarithmic connections singular exactly over one point has been considered and several properties, like algebraic functions, compactification and computation of Picard group have been studied. Also, the moduli space of rank one logarithmic connections singular over finitely many points with fixed residues has been considered in [15], and it is proved that it has a natural symplectic structure and there is no non-constant algebraic functions on it.…”

Section: Introductionmentioning

confidence: 99%

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Singh¹

2019

Preprint

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Let S be a finite subset of a compact connected Riemann surface X of genus g ≥ 2. Let M lc (n, d) denote the moduli space of pairs (E, D), where E is a holomorphic vector bundle over X and D is a logarithmic connection on E singular over S, with fixed residues in the centre of gl(n, C), where n and d are mutually corpime. Let L denote a fixed line bundle with a logarithmic connection D L singular over S. Let M ′ lc (n, d) and M lc (n, L) be the moduli spaces parametrising all pairs (E, D) such that underlying vector bundle E is stable andbe the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of M ′ lc (n, d) and M ′ lc (n, L) and compute their Picard groups. We also show that M ′ lc (n, L) and hence M lc (n, L) do not have any nonconstant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over S, with arbitrary residues.

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Section: Introductionmentioning

confidence: 99%

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Singh¹

2019

Preprint

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“…The moduli space of logarithmic connections over a complex projective variety singular over a smooth normal crossing divisor has been constructed in [33]. Several algebro-geometric invariants like Picard group, algebraic functions of the moduli space of holomorphic and logarithmic connections have been studied, see [7], [9], [10], [36] [27], and [26].…”

Section: Introduction and Statements Of The Resultsmentioning

confidence: 99%

Moduli space of rank one logarithmic connections over a compact Riemann surface

Singh¹

2020

Comptes Rendus. Mathématique

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Let X be a compact Riemann surface of genus g ≥ 3. We consider the moduli space of holomorphic connections over X and the moduli space of logarithmic connections singular over a finite subset of X with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces. Contents 1. Introduction and statements of the results 1 2. Preliminaries 4 2.1. Moduli spaces of holomorphic and logarithmic connections 4 3. Chow group of the moduli spaces 9 4. Differential operators on the moduli spaces 14 5. Torelli type theorem for the moduli spaces 21 6. Rational connectedness of the moduli spaces 28 References 29

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“…In [5], the moduli space of rank n logarithmic connections singular exactly over one point has been considered and several properties, like algebraic functions, compactification and computation of Picard group have been studied. Also, the moduli space of rank one logarithmic connections singular over finitely many points with fixed residues has been considered in [15], [16], and it is proved that it has a natural symplectic structure and there are no non-constant algebraic functions on it.…”

Section: Introductionmentioning

confidence: 99%

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Singh

2021

Mathematical Research Letters

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Let S be a finite subset of a compact connected Riemann surface X of genus g ≥ 2. Let M lc (n, d) denote the moduli space of pairs (E, D), where E is a holomorphic vector bundle over X and D is a logarithmic connection on E singular over S, with fixed residues in the centre of gl(n, C), where n and d are mutually corpime. Let L denote a fixed line bundle with a logarithmic connection D L singular over S. Let M ′ lc (n, d) and M lc (n, L) be the moduli spaces parametrising all pairs (E, D) such that underlying vector bundle E is stable and (be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of M ′ lc (n, d) and M ′ lc (n, L) and compute their Picard groups. We also show that M ′ lc (n, L) and hence M lc (n, L) do not have any non-constant algebraic functions but they admit non-constant holomorphic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over S, with arbitrary residues.

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Torelli theorems for moduli of logarithmic connections and parabolic bundles

Cited by 5 publications

References 15 publications

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Moduli space of rank one logarithmic connections over a compact Riemann surface

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

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