Motivic classes of irregular Higgs bundles and irregular connections on a curve

TL;DR

This paper computes motivic classes of moduli stacks of irregular Higgs bundles and connections on a curve, providing new criteria for their existence and developing homological algebra tools for irregular cases.

Contribution

It introduces formulas for motivic classes of irregular Higgs bundles and connections, and develops homological algebra methods for these irregular objects.

Findings

Motivic classes of irregular Higgs bundles are explicitly calculated.

A criterion for the existence of connections on higher level parabolic bundles is established.

Homological algebra tools for irregular connections are developed.

Abstract

Let $X$ be a smooth projective curve over a field of characteristic zero and let $\mathcal D$ be an effective divisor on $X$. We calculate motivic classes of various moduli stacks of parabolic vector bundles with irregular connections on $X$ and of irregular parabolic Higgs bundles on $X$ with poles bounded by $\mathcal D$ and with fully or partially fixed formal normal forms. Along the way, we obtain several results about irregular connections and irregular parabolic Higgs bundles. In particular, we give a criterion for the existence of a connection on a higher level parabolic bundle and also develop homological algebra for irregular connections and irregular parabolic Higgs bundles. We also simplify our previous results in the regular case by re-writing the formulas for motivic classes in terms of the HLV generating function.