Conformal limits for parabolic SL(n,C)-Higgs bundles

TL;DR

This paper extends the conformal limit correspondence between Higgs bundles and holomorphic connections to the parabolic case, revealing new phenomena and establishing existence under generic conditions.

Contribution

It generalizes the conformal limit correspondence to parabolic Higgs bundles, providing a gauge theoretic construction and analyzing new phenomena in the parabolic setting.

Findings

Conformal limits exist under mild genericity assumptions.

The conformal limit defines holomorphic sections of parabolic lambda-connections.

Nonabelian Hodge correspondence behaves differently in the parabolic case.

Abstract

In this paper we generalize the conformal limit correspondence between Higgs bundles and holomorphic connections to the parabolic setting. Under mild genericity assumptions on the parabolic weights, we prove that the conformal limit always exists and that it defines holomorphic sections of the space of parabolic lambda-connections which preserve a natural stratification and foliate the moduli space. Along the way, we give a careful gauge theoretic construction of the moduli space of parabolic Higgs bundles with full flags which allows the eigenvalues of the residues of the Higgs field to vary. A number of new phenomena arise in the parabolic setting. In particular, in the generality we consider, unlike the nonparabolic case, the nonabelian Hodge correspondence does not define sections of the space of logarithmic lambda-connections, and the conformal limit does not define a one-parameter family in any given moduli space.