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Parabolic vector bundles and Lie algebroid connections

David Alfaya, Indranil Biswas, Pradip Kumar, Anoop Singh

TL;DR

The paper develops a theory of parabolic Lie algebroid connections on parabolic vector bundles over marked curves, including a parabolic Atiyah-type exact sequence and a criterion for existence under stability conditions. It introduces quasi-parabolic and parabolic Lie algebroid connections, and identifies the residue-like map $\mathcal{S}_x$ that governs compatibility with parabolic structures. A key result is a two-case existence criterion depending on the map $\widetilde{\phi}^*$, along with a holomorphic splitting interpretation via the generalized Atiyah sequence. Finally, the work links integrable parabolic connections to parabolic $\Lambda$-modules, and constructs moduli spaces of semistable integrable parabolic connections, providing nonemptiness conditions and a moduli-theoretic framework for these objects.

Abstract

Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection.

Parabolic vector bundles and Lie algebroid connections

TL;DR

The paper develops a theory of parabolic Lie algebroid connections on parabolic vector bundles over marked curves, including a parabolic Atiyah-type exact sequence and a criterion for existence under stability conditions. It introduces quasi-parabolic and parabolic Lie algebroid connections, and identifies the residue-like map that governs compatibility with parabolic structures. A key result is a two-case existence criterion depending on the map , along with a holomorphic splitting interpretation via the generalized Atiyah sequence. Finally, the work links integrable parabolic connections to parabolic -modules, and constructs moduli spaces of semistable integrable parabolic connections, providing nonemptiness conditions and a moduli-theoretic framework for these objects.

Abstract

Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection.
Paper Structure (10 sections, 12 theorems, 133 equations)

This paper contains 10 sections, 12 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Lie algebroid connections and parabolic bundles
  3. Lie algebroid connection
  4. Parabolic bundles
  5. Connections on parabolic bundles
  6. Quasi-parabolic Lie algebroid connections
  7. Parabolic Lie algebroid connections
  8. The Atiyah exact sequence
  9. A criterion for parabolic Lie algebroid connections
  10. Integrable parabolic connections and parabolic $\Lambda$-modules

Key Result

Proposition 1.1

Let $(V,\, \phi)$ be a Lie algebroid and $E_*$ a parabolic vector bundle. A Lie algebroid connection $D\,:\, E\,\longrightarrow \, E\otimes V^*$ on the vector bundle $E$ underlying $E_*$ gives a quasi-parabolic Lie algebroid connection on the parabolic vector bundle $E_*$ if and only if the followin

Theorems & Definitions (26)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 16 more