On the motives of moduli of chains and Higgs bundles

García-Prada, Oscar, Heinloth, Jochen, and Schmitt, Alexander. "On the motives of moduli of chains and Higgs bundles." Journal of the European Mathematical Society 016.12 (2014): 2617-2668. <http://eudml.org/doc/277446>.

@article{García2014,
abstract = {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 Rodríguez-Villegas. Along the way we find explicit recursion formulas for the motives of several types of moduli spaces of stable chains.},
author = {García-Prada, Oscar, Heinloth, Jochen, Schmitt, Alexander},
journal = {Journal of the European Mathematical Society},
keywords = {Higgs bundle; holomorphic chain; moduli stack; motive; Poincaré polynomial; Grothendieck ring of varieties; chains; Higgs bundles; stacks; Grothendieck ring of varieties; chains},
language = {eng},
number = {12},
pages = {2617-2668},
publisher = {European Mathematical Society Publishing House},
title = {On the motives of moduli of chains and Higgs bundles},
url = {http://eudml.org/doc/277446},
volume = {016},
year = {2014},
}

TY - JOUR
AU - García-Prada, Oscar
AU - Heinloth, Jochen
AU - Schmitt, Alexander
TI - On the motives of moduli of chains and Higgs bundles
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 12
SP - 2617
EP - 2668
AB - 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 Rodríguez-Villegas. Along the way we find explicit recursion formulas for the motives of several types of moduli spaces of stable chains.
LA - eng
KW - Higgs bundle; holomorphic chain; moduli stack; motive; Poincaré polynomial; Grothendieck ring of varieties; chains; Higgs bundles; stacks; Grothendieck ring of varieties; chains
UR - http://eudml.org/doc/277446
ER -