On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

Davesh Maulik; Junliang Shen

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arXiv · 2103.01285

On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

Davesh Maulik·Junliang Shen

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Abstract

We explore the cohomological structure for the (possibly singular) moduli of $\mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the $\mathrm{SL}_n$-Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ng\^{o}-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.

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Davesh Maulik, Junliang Shen. 2021-03-01. On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve. https://arxiv.org/abs/2103.01285

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