A remark on the moduli space of Lie algebroid λ-connections

Abstract

Let X be a compact Riemann surface of genus $g\ge 3$. Let $\mathcal{L}=(L,[.,.],♯)$ be a holomorphic Lie algebroid over X of rank one and degree $(L)<2-2g$. We consider the moduli space of holomorphic ${\mathcal{L}}_{\lambda }$-connections over X, where $\lambda \in \mathbb{C}$. We compute the Picard group of the moduli space of ${\mathcal{L}}_{\lambda }$-connections by constructing an explicit smooth compactification of the moduli space of those ${\mathcal{L}}_{\lambda }$-connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of ${\mathcal{L}}_{\lambda }$-connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of $\mathcal{L}$-connections and we determine its Chow group.

Communicated by Manuel Reyes

KEYWORDS:

• Chow group
• Lie algebroid connections
• Higgs bundles
• moduli space
• Picard group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14D20
• 14C15
• 14C22

Acknowledgments

The authors would like to thank the referee for helpful comments. The first named author is supported in part by the INFOSYS scholarship. The second named author would like to thank Harish-Chandra Research Institute (HRI), Prayagraj for their hospitality where a part of work was carried out while he was visiting Prof. N. Raghavendra.

Additional information

Funding

The second named author is partially supported by SERB SRG Grant SRG/2023/001006