Logarithmic spectral correspondence for $V$--twisted Higgs bundles on punctured curves
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912968730476544 |
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| author | Kumar, Pradip |
| author_facet | Kumar, Pradip |
| contents | 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 $\mathrm{Pic}^{d}(Y)\times A_Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15496 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Logarithmic spectral correspondence for $V$--twisted Higgs bundles on punctured curves Kumar, Pradip Algebraic Geometry Differential Geometry 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 $\mathrm{Pic}^{d}(Y)\times A_Z$. |
| title | Logarithmic spectral correspondence for $V$--twisted Higgs bundles on punctured curves |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2603.15496 |