Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces

2025/11/14 by Biswas, Indranil
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.10994

Abstract

For a Γ--equivariant holomorphic Lie algebroid (V, ϕ), on a compact Riemann surface X equipped with an action of a finite group Γ, we investigate the equivariant holomorphic Lie algebroid connections on holomorphic principal G--bundles over X, where G is a connected affine complex reductive group. If (V, ϕ) is nonsplit, then it is proved that every holomorphic principal G--bundle admits an equivariant holomorphic Lie algebroid connection. If (V, ϕ) is split, then it is proved that the following four statements are equivalent: An equivariant principal G--bundle E_G admits an equivariant holomorphic Lie algebroid connection. The equivariant principal G--bundle E_G admits an equivariant holomorphic connection. The principal G--bundle E_G admits a holomorphic connection. For every triple (P, L(P), χ), where L(P) is a Levi subgroup of a parabolic subgroup P ⊂ G and χ is a holomorphic character of L(P), and every Γ--equivariant holomorphic reduction of structure group E_L(P) of E_G to L(P), the degree of the line bundle over X associated to E_L(P) for χ is zero. The correspondence between Γ--equivariant principal G--bundles over X and parabolic G--bundles on X/Γ translates the above result to the context of parabolic G--bundles.

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