A support theorem for\break the Hitchin fibration: The case of GL n and K C

We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for GL n {\mathrm{GL}_{n}} over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every n ≥ 2 {n\geq{2}} . A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over versal deformations of spectral curves.

Endoscopic decompositions and the Hausel–Thaddeus conjecture

We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.

Semi-abelian Spectral Data for Singular Fibres of the 𝖲𝖫(2,ℂ)-Hitchin System

We describe spectral data for singular fibres of the $\textsf{SL}(2,{\mathbb{C}})$-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 $\textsf{SL}(2,{\mathbb{R}})$-Higgs bundles in terms of these semi-abelian spectral data.

An Explicit View of the Hitchin Fibration on the Betti Side for ℙ1 minus Five Points

The dual complex of the divisor at infinity of the character variety of local systems on P1−{t1,…,t5} with monodromies in prescribed conjugacy classes Ci⊂SL2(C), was shown by Komyo to be the sphere S3. This chapter compares in some detail the projection from a tubular neighbourhood to this dual complex, with the corresponding Hitchin fibration at infinity.

HIGGS BUNDLES OVER ELLIPTIC CURVES FOR COMPLEX REDUCTIVE GROUPS

We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalisation of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.

A support theorem for the Hitchin fibration: the case of

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.