Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces (Q6542418)

\textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 79, 47--129 (1994; Zbl 0891.14005)] proved that the de Rham moduli space of rank r and degree 0 holomorphic flat connections and the Dolbeault moduli space, of rank r and degree 0 Higgs bundles over a Riemann surface of genus \(g\geq 2\) are particular fibers over \(\mathbb{C}\), over \(0\) and \(z \neq 0\) respectively, of the Hodge moduli space of \(\lambda\)-connections, and this can be generalized to degree \(d\). These fibers, being singular and algebraically non-isomorphic, share a lot of geometry in common through the so-called non-abelian Hodge correspondence, in particular being homeomorphic and having pure Hodge structures, the same E-polynomials and Voevodsky motives, among others.\N\NThe de Rham moduli space can be generalized to the moduli space \(\mathcal{M}_{\mathcal{L}}(r,d)\) of flat \(\mathcal{L}\)-connections: a rank \(1\) Lie algebroid \(\mathcal{L}=(L , [\cdot,\cdot], \delta)\) is given by a degree \(d\) line bundle \(L\rightarrow X\) plus a \(\mathbb{C}\)-linear Lie bracket \([\cdot,\cdot] : L \otimes_{\mathbb{C}} L \rightarrow L\) and an anchor map \(\delta : L \rightarrow T_X\) to the tangent bundle of \(X\); and a flat \(\mathcal{L}\)-connection is \((E, \nabla)\), where \(E\rightarrow X\) is a rank \(r\) vector bundle over \(X\) and the map \(\nabla : E \rightarrow E \otimes L^{\ast}\) satisfies properties generalizing those of a connection. For \(L=T_X\), the bracket being the Lie bracket of vector fields and \(\delta\) being the identity we get the Lie algebroid \(\mathcal{T}_X\), and a \(\mathcal{T}_X\)-connection is just a classical connection on \(X\).\N\NThen this article generalizes several results on the Hodge moduli space to this Lie algebroid setting by showing that \(\mathcal{M}_{\mathcal{L}}(r,d)\) is the \(z\neq 0\) fiber degenerating to the twisted Dolbeault moduli space of \(L^{-1}\)-twisted Higgs bundles of rank \(r\) and degree \(d\) over \(X\), for \(z=0\), forming the \(\mathcal{L}\)-Hodge moduli space, and use its semiprojectivity to deduce geometric properties. Theorem 1.2 (section 3) complety classifies isomorphism classes of Lie algebroid structures on line bundles to obtain non-emptiness results on these moduli. Theorem 1.4 (section 4) proves the smoothness of the \(\mathcal{L}\)-Hodge moduli space and the equivariancy of its map to \(\mathbb{C}\). In sections 5 and 6, several equalities for motives, E-polynomials and mixed Hodge structures are proved (see Theorems 1.1 and 1.6 for precise statements), showing that the cohomological information depends on \(d\) and not on the twisting \(L\) itself. In section 8 several explicit formulas for the motives and E-polynomials of the moduli spaces in ranks 2 and 3 are provided.