Logarithmic spectral correspondence for $V$--twisted Higgs bundles on punctured curves

TL;DR

This paper develops a logarithmic spectral correspondence for $V$-twisted Higgs bundles on punctured curves, classifying them via spectral data and extending previous compactification results to the punctured setting.

Contribution

It introduces a classification of logarithmic $V$-twisted Higgs bundles on punctured curves through spectral data, extending the spectral correspondence to the punctured case with explicit local conditions.

Findings

Unique lift of fields under local Hecke conditions

Integrability of the lift when fields commute

Moduli stack equivalence to $ ext{Pic}^d(Y) imes A_Z$ on the line-bundle locus

Abstract

Let $X$ be a smooth projective complex curve, $P\subset X$ a reduced effective divisor, and $X^{0}=X\setminus P$. We study logarithmic $V$-twisted Higgs bundles arising from a logarithmic Hecke compactification of a rank-two bundle on $X^{0}$. We show that a pair of induced logarithmic line-twisted fields lifts uniquely exactly under explicit local Hecke conditions, and that the lift is integrable precisely when the fields commute. Fixing the compactified spectral curve $Y$, we classify such Higgs bundles by pairs $(F,\,\vartheta)$, where $F$ is a rank-one torsion-free sheaf on $Y$ and $\vartheta$ satisfies a marked spectral condition on a finite subscheme $Z\subset Y$. This gives a logarithmic extension of the compact rank-two spectral correspondence of~\cite{ABK} to the punctured case. On the line-bundle locus, the moduli stack is canonically equivalent to $\mathrm{Pic}^{d}(Y)\times A_Z$.