arXiv subjects

Johannes Horn

Publications and source records attributed to Johannes Horn.

7 recordsLinked to original sources

arXiv2026

Fourier-Mukai transforms and normalisation of nodal curves

We study Arinkin's Poincaré sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $Σ\to C$. We prove that the Poincaré sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincaré sheaf $\mathcal{P}_Σ$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_Σ$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems.

math.AG↗

arXiv2026

Visible Lagrangians for Hitchin Systems and Pillowcase Covers

We study complex Lagrangians in Hitchin systems that factor through a proper subvariety of the Hitchin base non-trivially intersecting the regular locus. This gives a general framework for several examples in the literature. We compute the fiber-wise Fourier-Mukai transform of flat line bundles on visible Lagrangians. This proposes a construction of mirror dual branes to visible Lagrangians. Finally, we study a new example of visible Lagrangians in detail. Such visible Lagrangian exists whenever the underlying Riemann surface is a pillowcase cover. The proposed mirror dual brane turns out to be closely related to Hausel's toy model.

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arXiv2025

Wall-crossing formulas via spectral networks

We give a self-contained proof of the Kontsevich-Soibelman wall-crossing formula entirely in the scope of quadratic differentials without relying on input from DT theory. Our approach is based on path-lifting rules for spectral networks introduced by Gaiotto, Moore and Neitzke. We provide a framework to justify the convergence of the path liftings, including the cases with spiral domains. In particular, we define path lifting rules for spectral networks associated to holomorphic quadratic differentials. As an intermediate step in the proof of the wall-crossing formula, we show that upon extending the path lifting rules to $\mathcal{A}_0$-laminations we generate the hat-homology lattice.

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arXiv2025

The asymptotics of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin metric on the singular locus: subintegrable systems

We study the asymptotic hyperkähler geometry of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin moduli space over the singular fibers of the Hitchin fibration. We extend the previously known exponential convergence results for solutions to the Hitchin equation to the class of locally fiducial Higgs bundles defined by a special local description at the singularities of the spectral curve. This condition is satisfied by the Higgs bundles contained in certain subintegrable systems introduced by Hitchin. We prove that the restriction of the hyperkähler metric to the subintegrable system converges exponentially fast to the corresponding semi-flat metric along a ray $(\mathcal{E},tφ)$. This answers a question posed by Hitchin in \cite{Hitchin2021subintegrable_special_Kaehler}. More generally, we prove that for each stratum of quadratic differentials there is a closed subset of the corresponding Hitchin fibers, such that the restricted hyperkähler metric converges to a generalized semi-flat metric.

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arXiv2024

Compactifying the rank two Hitchin system via spectral data on semistable curves

We study resolutions of the rational map to the moduli space of stable curves that associates with a point in the Hitchin base the spectral curve. In the rank two case the answer is given in terms of the space of quadratic multi-scale differentials introduced in [BCGGM3]. This space defines a compactification (of the projectivization) of the regular locus of the $\mathrm{GL}(2,\mathbb{C})$-Hitchin base and provides a compactification of the Hitchin system by compactified Jacobians of pointed stable curves. We show how the classical $\mathrm{GL}(2,\mathbb{C})$- and $\mathrm{SL}(2,\mathbb{C})$-spectral correspondence extend to the compactified Hitchin system by a correspondence along an admissible cover between torsion-free rank $1$ sheaves and (multi-scale) Higgs pairs of rank $2$.

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arXiv2021

$\mathfrak{sl}(2)$-type singular fibres of the symplectic and odd orthogonal Hitchin system

We define and parametrise so-called $\mathfrak{sl}(2)$-type fibres of the $\mathsf{Sp}(2n,\mathbb{C})$- and $\mathsf{SO}(2n+1,\mathbb{C})$-Hitchin system. These are (singular) Hitchin fibres, where the spectral curve induces a two-sheeted covering of a second Riemann surface $Y$. This identifies the $\mathfrak{sl}(2)$-type Hitchin fibres with fibres of an $\mathsf{SL}(2,\mathbb{C})$- respectively $\mathsf{PSL}(2,\mathbb{C})$-Hitchin map on $Y$. We give a stratification of these singular spaces by semi-abelian spectral data, study their irreducible components and obtain a global description of the first degenerations. Comparing the semi-abelian spectral data of $\mathfrak{sl}(2)$-type Hitchin fibres for the two Langlands dual groups, we extend the well-known Langlands duality of regular Hitchin fibres to $\mathfrak{sl}(2)$-type Hitchin fibres. Finally, we construct solutions to the decoupled Hitchin equation for Higgs bundles of $\mathfrak{sl}(2)$-type. We conjecture these to be limiting metrics along rays to the ends of the moduli space.

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arXiv2020

Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system

We describe spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibres and give a description of $\mathsf{SL}(2,\mathbb{R})$-Higgs bundles in terms of these semi-abelian spectral data.

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