Summary:
Let ℒ =(L,[⋅,⋅],δ) be an algebraic Lie algebroid over a smooth projective curve X of genus g≥2 such that L is a line bundle whose degree is less than 2−2g. Let r and d be coprime numbers. We prove that the motivic class of the moduli space of ℒ -connections of rank r and degree d over X does not depend on the Lie algebroid structure [⋅,⋅] and δ of ℒ and neither on the line bundle L itself, but only on the degree of L (and of course on r, d and X). In particular it is equal to the motivic class of the moduli space of Kx(D)-twisted Higgs bundles of rank r and degree d, for D any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding E-polynomials. Some applications of these results are then deduced.
Spanish layman's summary:
Demostramos que el motivo de los espacios de moduli de conexiones sobre algebroides de Lie es independiente de la estructura de algebroide y, por lo tanto, es siempre igual al motivo de un moduli de Higgs logarítmico. Esto nos permite deducir varias propiedades geométricas de estos espacios.
English layman's summary:
We prove that the motive of the moduli space of connections on Lie algebroids is independent from the structure of the algebroid and, therefore, that it always coincides with the motive of a moduli of logarithmic Higgs bundles. Several geometric properties for theses spaces are derived from this.
Keywords: Lie algebroid connections; Higgs bundles; Moduli space; Motive; Hodge structure; E-polynomial
JCR Impact Factor and WoS quartile: 1,500 - Q1 (2022)
DOI reference:
https://doi.org/10.1016/j.geomphys.2024.105195
Published on paper: July 2024.
Published on-line: April 2024.
Citation:
D. Alfaya, A. Oliveira, Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces. Journal of Geometry and Physics. Vol. 201, pp. 105195-1 - 105195-55, July 2024. [Online: April 2024]
Download this paper