Line bundles on the moduli space of Lie algebroid connections over a curve

TL;DR

This paper studies the geometric and algebraic properties of the moduli space of Lie algebroid connections over a compact Riemann surface, including compactification, Picard group, and function theory.

Contribution

It constructs a smooth compactification of the moduli space, computes its Picard group, and analyzes the space of algebraic functions, revealing new geometric insights.

Findings

Constructed a smooth compactification with a divisor boundary

Computed the Picard group of the moduli space

Showed the space of certain Lie algebroid connections has only constant regular functions

Abstract

We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid ($ \mathcal{L} $) connections on a compact Riemann surface $X$ of genus $g \,\geq\, 3$. A smooth compactification of the moduli space of $\mathcal{L}$-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of $\mathcal{L}$-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored.