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

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 ${\mathrm{Pic}}^d(Y)\times A_Z$.

Commments are welcome