On the motives of moduli of chains and Higgs bundles

By O. Garc'ia-Prada, J. Heinloth, A. Schmitt. 2011

122 citations recorded by the source corpus.

We take another approach to Hitchin's strategy of computing the cohomology of moduli spaces of Higgs bundles by localization with respect to the circle-action. Our computation is done in the dimensional completion of the Grothendieck ring of varieties and starts by describing the classes of moduli stacks of chains rather than their coarse moduli spaces. As an application we show that the n-torsion of the Jacobian acts trivially on the middle dimensional cohomology of the moduli space of twisted SL_n-Higgs-bundles of degree coprime to n and we give an explicit formula for the motive of the moduli space of Higgs bundles of rank 4 and odd degree. This provides new evidence for a conjecture of Hausel and Rodr\'iguez-Villegas. Along the way we find explicit recursion formulas for the motives of several types of moduli spaces of stable chains.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8a7c2798c200814502ac791b5e5379d703aa552d

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Citations in the corpus

Global topology of the Hitchin system (2011, methodology): “iterature thus we can well hope that with this new point of view we will be able to progress our understanding of the problems surveyed in this paper. We nish by mentioning another promising new work [GHS] where they manage to extend the original Morse theory method of Hitchin for n= 2 and Gothen fo”
CO-HIGGS BUNDLES ON P 1 (2010, methodology): “Before the recent work of Garćıa-Prada, Heinloth, and Schmitt [8], the most formidable obstacle to computing the Betti numbers for the moduli space of ordinary rank-4 Higgs bundles was the existence of stable (2, 2) chains, which could not be directly attacked by Thaddeus’ treatment of chains of len”
Hodge polynomials of SL (2,C)-character varieties for curves of small genus (2013, background): “The fibration defines a local system Hkc → B whose fibres are the cohomology groups Hkc (Fb), b ∈ B, Fb := π−1(b).”