Publication: On character varieties of singular manifolds
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2020-11-09
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Abstract
In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of G-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group G, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case G = SL2(k), the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios.
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[1] M. Atiyah. Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. , (68):175–186 (1989), 1988.
[2] U. Bhosle, I. Biswas, and J. Hurtubise. Grassmannian-framed bundles and generalized parabolic structures.
International Journal of Mathematics, 24(12):1350090, 2013.
[3] U. N. Bhosle. Representations of the fundamental group and vector bundles. Mathematische Annalen, 302(1):601–608, 1995.
[4] U. N. Bhosle, A. Parameswaran, and S. K. Singh. Hitchin pairs on an integral curve. Bulletin des Sciences Matématiques, 138(1):41–62, 2014.
[5] L. A. Borisov. The class of the affine line is a zero divisor in the Grothendieck ring. Journal of Algebraic
Geometry, 27(2):203–209, 2018.
[6] R. Brown. Groupoids and van Kampen’s theorem. Proc. London Math. Soc. (3), 17:385–401, 1967.
[7] P. Candelas, P. S. Green, and T. Hübsch. Rolling among Calabi-Yau vacua. Nuclear Physics B, 330(1):49–102, 1990.
[8] E. Carlsson and F. Rodriguez Villegas. Vertex operators and character varieties. Adv. Math., 330:38–60, 2018.
[9] K. Corlette. Flat G-bundles with canonical metrics. J. Differential Geom., 28(3):361–382, 1988.
[10] D.-E. Diaconescu. Local curves, wild character varieties, and degenerations. Preprint arXiv:1705.05707, 2017.
[11] C. Florentino, A. Nozad, and A. Zamora. E-polynomials of SLn-and P GLn-character varieties of free groups. arXiv preprint arXiv:1912.05852, 2019.
[12] C. A. Florentino and J. A. Silva. Hodge-Deligne polynomials of abelian character varieties. arXiv preprint
arXiv:1711.07909, 2017.
[13] A. González-Prieto. Motivic theory of representation varieties via topological quantum field theories. Preprint
arXiv:1810.09714, 2018.
[14] A. González-Prieto. Stratification of algebraic quotients and character varieties. Preprint arXiv:1807.08540, 2018.
[15] A. González-Prieto. Virtual classes of parabolic SL 2(C)-character varieties. Advances in Mathematics, 368:107148, 2020.
[16] A. González-Prieto, M. Logares, and V. Muñoz. A lax monoidal Topological Quantum Field Theory for representation varieties. Bulletin des Sciences Mathématiques, page 102871, 2020.
[17] A. González-Prieto and V. Muñoz. Motive of the SL4-character variety of torus knots. Preprint arXiv:2006.01810, 2020.
[18] M. Hablicsek and J. Vogel. Virtual classes of representation varieties of upper triangular matrices via topological quantum field theories. arXiv preprint arXiv:2008.06679, 2020.
[19] T. Hausel, E. Letellier, and F. Rodriguez-Villegas. Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J., 160(2):323–400, 2011.
[20] T. Hausel, E. Letellier, and F. Rodriguez-Villegas. Arithmetic harmonic analysis on character and quiver
varieties II. Adv. Math., 234:85–128, 2013.
[21] T. Hausel and F. Rodriguez-Villegas. Mixed Hodge polynomials of character varieties. Invent. Math., 174(3):555–624, 2008. With an appendix by Nicholas M. Katz.
[22] M. Heusener, V. Munoz, and J. Porti. The SL(3, C)-character variety of the figure eight knot. Illinois Journal
of Mathematics, 60(1):55–98, 2016.
[23] N. J. Hitchin. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3), 55(1):59–126, 1987.
[24] N. J. Hitchin. Hyper-kähler manifolds. Séminaire Bourbaki, 1991:92, 1992.
[25] M. Logares. On Higgs bundles on nodal curves. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 15:024, 2019.
[26] M. Logares and V. Muñoz. Hodge polynomials of the SL(2, C)-character variety of an elliptic curve with two marked points. Internat. J. Math., 25(14):1450125, 22, 2014.
[27] M. Logares, V. Muñoz, and P. E. Newstead. Hodge polynomials of SL(2, C)-character varieties for curves of
small genus. Rev. Mat. Complut., 26(2):635–703, 2013.
[28] J. Martínez and V. Muñoz. E-polynomials of the SL(2, C)-character varieties of surface groups. Int. Math. Res. Not. IMRN, (3):926–961, 2016.
[29] J. Martínez and V. Muñoz. e-polynomials of the SL(2, C)-character varieties of surface groups. International Mathematics Research Notices, 2016(3):926–961, 2016.
[30] A. Mellit. Poincar´e polynomials of character varieties, macdonald polynomials and affine springer fibers. Annals of Mathematics, 192(1):165–228, 2020.
[31] M. Mereb. On the E-polynomials of a family of SLn-character varieties. Math. Ann., 363(3-4):857–892, 2015.
[32] J. Milnor. Singular Points of Complex Hypersurfaces.(AM-61), Volume 61, volume 61. Princeton University Press, 2016.
[33] S. Mozgovoy. Solutions of the motivic ADHM recursion formula. Int. Math. Res. Not. IMRN, (18):4218–4244,
2012.
[34] V. Muñoz. The SL(2, C)-character varieties of torus knots. Rev. Mat. Complut., 22(2):489–497, 2009.
[35] V. Muñoz and J. Porti. Geometry of the SL(3, C) character variety of torus knots. Algebr. Geom. Topol., 16(1):397–426, 2016.
[36] M. S. Narasimhan and C. S. Seshadri. Stable and unitary vector bundles on a compact riemann surface. Annals of Mathematics, pages 540–567, 1965.
[37] C. T. Simpson. Harmonic bundles on noncompact curves. J. Amer. Math. Soc., 3(3):713–770, 1990.
[38] C. T. Simpson. Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. , (75):5–95, 1992.
[39] C. T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. I. Inst.
Hautes Études Sci. Publ. Math. , (79):47–129, 1994.
[40] C. T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. II. Inst.
Hautes Études Sci. Publ. Math. , (80):5–79 (1995), 1994.
[41] J. R. Smith. Complements of codimension-two submanifolds. The fundamental group. Illinois J. Math.,
22(2):232–239, 1978.
[42] J. Vogel. Representation varieties of non-orientable surfaces via topological quantum field theories. arXiv
preprint arXiv:2009.12310, 2020.