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On the motives of moduli of chains and Higgs bundles. (English) Zbl 1316.14060

A Higgs bundle over a smooth, projective curve \(C\) is a pair \((\mathcal{E},\theta)\) where \(\mathcal{E}\) is a vector bundle and \(\theta\) — the Higgs field — is a global section of \(\mathrm{End}(\mathcal{E})\otimes\Omega_C\), with \(\Omega_C\) being the sheaf of differentials of \(C\). In this paper the authors implement a new method to compute the cohomology of the moduli space \(M_n^d\) of stable rank \(n\) and degree \(d\) Higgs bundles, whenever \((n,d)=1\).
It is studied the class (or motive) of \(M_n^d\) in the dimensional completion \(\widehat{K_0}(Var)\) of the Grothendieck ring of varieties. It is known that this class can be written in terms of the classes of the subvarieties of \(M_n^d\), defined as the fixed point locus of the usual action of \(\mathbb{G}_m\) by multiplication of scalars on the Higgs field. These subvarieties are moduli spaces of chains, which are objects consisting by a collection of vector bundles \((\mathcal{E}_i)_i\) together with a collection of maps \((\phi_i)_i\), with \(\phi_i:\mathcal{E}_i\to\mathcal{E}_{i-1}\otimes \Omega_C\). Briefly, the motive of the moduli spaces of \(\alpha\)-semistable chains (the semistability condition of chains depends on a parameter \(\alpha\) and there is a specific one corresponding to the semistability of Higgs bundles) is computed by first computing the motive of the whole stack of chains and then studying its stratification according to the different types of canonical destabilising subchains — the Harder-Narasimhan stratification. These Harder-Narasimhan strata are fibered over spaces of semistable chains of lower rank, for which the motive is known by induction.
As an application of this approach, it is proved that the group of \(n\)-torsion points of the Jacobian of \(C\) acts trivially on the middle-dimensional cohomology of \(M_n^L\), where \(M_n^L\) is the moduli space of rank \(n\) Higgs bundles \((\mathcal{E},\theta)\) with trivial determinant i.e. such that \(\mathrm{det}(\mathcal{E})\cong L\) and \(\mathrm{tr}(\theta)=0\) and where \((n,\mathrm{deg}(L))=1\).
An explicit formula for the motive of \(M_4^d\), with \(d\) odd, is obtained, providing new evidence for a conjecture of T. Hausel and F. Rodriguez-Villegas [Invent. Math. 174, No. 3, 555–624 (2008; Zbl 1213.14020)] on the Poincaré polynomial of this space.
Along the way, several explicit recursive formulas for the motives of several types of spaces of \(\alpha\)-semistable chains are obtained.

MSC:

14H60 Vector bundles on curves and their moduli
14D23 Stacks and moduli problems

Citations:

Zbl 1213.14020
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References:

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