Irregular Hodge numbers for rigid $G_2$-connections

TL;DR

This paper investigates the irregular Hodge numbers of specific rigid $G_2$-connections, revealing their relation to local systems on punctured projective lines and computing their Hodge filtration properties.

Contribution

It extends the understanding of irregular Hodge filtrations to new $G_2$-connections using pullbacks, Fourier transforms, and the methods inspired by Katz, Sabbah, and Yu.

Findings

Computed jumping indices for irregular Hodge filtrations.

Determined dimensions of the Hodge filtrations.

Established relations between $G_2$-connections and local systems.

Abstract

Certain rigid irregular $G_2$-connections constructed by the first-named author are related via pullbacks along a finite covering and Fourier transform to rigid local systems on a punctured projective line. This kind of property was first observed by Katz for hypergeometric connections and used by Sabbah and Yu to compute irregular Hodge filtrations for hypergeometric connections. This strategy can also be applied to the aforementioned $G_2$-connections and we compute jumping indices and dimensions for their irregular Hodge filtrations.