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.
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.
Reviewer: André Oliveira (Vila Real)
Citations:
Zbl 1213.14020References:
| [1] | Álvarez-Cónsul, L., García-Prada, O.: Dimensional reduction, SL(2, C)-equivariant bundles and stable holomorphic chains. Int. J. Math. 12, 159-201 (2001) · Zbl 1110.32305 · doi:10.1142/S0129167X01000745 |
| [2] | Álvarez-Cónsul, L., García-Prada, O., Schmitt, A.: On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces. Int. Math. Res. Papers 2006, art. ID 73597, 82 pp. · Zbl 1111.32012 · doi:10.1155/IMRP/2006/73597 |
| [3] | Artin, M., Grothendieck, A., Verdier, J. L.: Théorie des topos et cohomologie étale des schémas. Tome 3 (SGA 4). Avec la collaboration de P. Deligne et B. Saint-Donat. Lecture Notes in Math. 305, Springer, Berlin (1973) |
| [4] | Atiyah, M. F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London A 308, 523-615 (1983) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017 |
| [5] | Behrend, K., Dhillon, A.: On the motivic class of the stack of bundles. Adv. Math. 212, 617-644 (2007) · Zbl 1138.14014 · doi:10.1016/j.aim.2006.11.003 |
| [6] | Białynicki-Birula, A.: Theorems on actions of algebraic groups. Ann. of Math. 98, 480-497 (1973) · Zbl 0275.14007 · doi:10.2307/1970915 |
| [7] | Bifet, E., Ghione, F., Letizia, M.: On the Abel-Jacobi map for divisors of higher rank on a curve. Math. Ann. 299, 641-672 (1994) · Zbl 0840.14003 · doi:10.1007/BF01459804 |
| [8] | Biswas, I., Ramanan, S.: An infinitesimal study of the moduli of Hitchin pairs. J. London Math. Soc. (2) 49, 219-231 (1994) · Zbl 0819.58007 · doi:10.1112/jlms/49.2.219 |
| [9] | Bradlow, S. B., García-Prada, O., Gothen, P.: Moduli spaces of holomorphic triples over compact Riemann surfaces. Math. Ann. 328, 299-351 (2004) · Zbl 1041.32008 · doi:10.1007/s00208-003-0484-z |
| [10] | Chuang, W., Diaconescu, D. E., Pan, G.: Wallcrossing and cohomology of the moduli space of Hitchin pairs. Comm. Number Theory Phys. 5, 1-56 (2011) · Zbl 1259.14009 · doi:10.4310/CNTP.2011.v5.n1.a1 |
| [11] | Del Baño, S.: On the motive of moduli spaces of rank two vector bundles over a curve. Compos. Math. 131, 1-30 (2002) · Zbl 1060.14050 · doi:10.1023/A:1014756205008 |
| [12] | Ekedahl, T.: The Grothendieck group of algebraic stacks. · Zbl 0591.14010 |
| [13] | García-Prada, O., Heinloth, J.: The y-genus of the moduli space of PGLn-Higgs bundles on a curve (for degree coprime to n). Duke Math. J. 162, 2731-2749 (2013) · Zbl 1300.14013 · doi:10.1215/00127094-2381369 |
| [14] | Gothen, P.: The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface. Int. J. Math. 5, 861-875 (1994) · Zbl 0860.14030 · doi:10.1142/S0129167X94000449 |
| [15] | Gothen, P., King, A.: Homological algebra of twisted quiver bundles. J. London Math. Soc. (2) 71, 85-99 (2005) · Zbl 1095.14012 · doi:10.1112/S0024610704005952 |
| [16] | Göttsche, L.: On the motive of the Hilbert scheme of points on a surface. Math. Res. Lett. 8, 613-627 (2001) · Zbl 1079.14500 · doi:10.4310/MRL.2001.v8.n5.a3 |
| [17] | Harder, G., Narasimhan, M. S.: On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212, 215-48 (1975) · Zbl 0324.14006 · doi:10.1007/BF01357141 |
| [18] | Hausel, T.: Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve, In: Geometric Methods in Algebra and Number Theory, Progr. Math. 235, Birkhäuser Boston, 193-217 (2005) · Zbl 1099.14026 |
| [19] | Hausel, T., Rodríguez-Villegas, F.: Mixed Hodge polynomials of character varieties (with an · Zbl 1213.14020 · doi:10.1007/s00222-008-0142-x |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
