Lie algebroid Connections, Moduli of $\mathcal{L}$--twisted Principal Objects and motives

TL;DR

This paper classifies Lie algebroids on complex projective varieties, introduces associated Higgs bundles and moduli spaces, and establishes a motivic non-abelian Hodge correspondence using a Tannakian framework.

Contribution

It develops a Tannakian approach to study moduli of Lie algebroid connections and Higgs bundles, and constructs their moduli spaces with new geometric and motivic insights.

Findings

Classified integrable transitive algebraic Lie algebroids on X.

Constructed moduli spaces of \\mathcal{L}-twisted Higgs bundles and principal G-bundles.

Established a motivic non-abelian Hodge correspondence for these moduli spaces.

Abstract

Let \(X\) be an irreducible smooth complex projective variety, and let \(G\) be a connected reductive linear algebraic group over \(\mathbb{C}\). In this paper, we first classify integrable transitive algebraic Lie algebroids on $X$. We then introduce Higgs bundles associated to a Lie algebroid and study their moduli spaces. In particular, we show that the category of vector bundles equipped with integrable \(\mathcal{L}\)-connections and the category of \(\mathcal{L}\)-twisted Higgs bundles of semiharmonic type on \(X\) are neutral Tannakian categories, provided that \(\mathcal{L}\) is a transitive Lie algebroid. Using this Tannakian framework, we obtain a characterization of principal \(G\)-bundles with integrable \(\mathcal{L}\)-connections and \(\mathcal{L}\)-twisted principal \(G\)-Higgs bundles of semiharmonic type on \(X\), and construct their moduli spaces via Mumford's geometric invariant theory. We further introduce the notion of the \(\mathcal{L}\)-Hodge moduli space for principal \(G\)-bundles and prove that the moduli spaces of principal \(G\)-bundles with integrable \(\mathcal{L}\)-connections, \(\mathcal{L}\)-twisted principal \(G\)-Higgs bundles of harmonic type, and the associated \(\mathcal{L}\)-Hodge moduli spaces are semiprojective varieties. Finally, using the semiprojectivity of the \(\mathcal{L}\)-Hodge moduli spaces for principal \(G\)-bundles, we obtain a description of smooth locus of these moduli spaces in the Grothendieck ring of varieties and establish a motivic non-abelian Hodge correspondence type theorem.