A criterion for holomorphic Lie algebroid connections

2025/06/12 by David Alfaya, Alfaya, David, Indranil Biswas +5
Mathematics · #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2506.10514

Abstract

Given a holomorphic Lie algebroid (V, ϕ) on a compact connected Riemann surface X, we give a necessary and sufficient condition for a holomorphic vector bundle E on X to admit a holomorphic Lie algebroid connection. If (V, ϕ) is nonsplit, then every holomorphic vector bundle on X admits a holomorphic Lie algebroid connection for (V, ϕ). If (V, ϕ) is split, then a holomorphic vector bundle E on X admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of E is zero.

Citations

• Parabolic vector bundles and Lie algebroid connections
• A criterion for Lie algebroid connections on a compact Riemann surface
• The Self-Duality Equations on a Riemann Surface

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