Abstract
Abstract 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.
Highlights
By [1, 2], the cohomology ∗ N, C is generated by the tautological classes – the Künneth factors of the Chern characters of a universal family
The purpose of this paper is to study the Γ-action on the cohomology of the moduli space of stable SL -Higgs bundles from the viewpoint of the Hausel–Thaddeus conjecture [22]
A central question concerning the cohomological aspect of nonabelian Hodge theory is the P=W conjecture formulated by de Cataldo, Hausel and Migliorini [7], connecting the perverse filtration associated with the Hitchin fibration h to the weight filtration
Summary
Throughout, we work over the complex numbers C. We study several Hitchin-type moduli spaces relevant to Theorems 0.2 and 0.3
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