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

D. Alfaya, A. Oliveira

Journal of Geometry and Physics Vol. 201, pp. 105195-1 - 105195-55

Resumen:

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 K_x(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.

Resumen divulgativo:

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.

 

Palabras Clave: Lie algebroid connections; Higgs bundles; Moduli space; Motive; Hodge structure; E-polynomial

Índice de impacto JCR y cuartil WoS: 1,600 - Q1 (2023)

Referencia DOI: https://doi.org/10.1016/j.geomphys.2024.105195

Publicado en papel: Julio 2024.

Publicado on-line: Abril 2024.

Cita:
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, Julio 2024. [Online: Abril 2024]