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

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.