Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

Maulik, Davesh¹ ; Shen, Junliang²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
² Department of Mathematics, Yale University, New Haven, CT, United States

Geometry & topology, Tome 27 (2023) no. 4, pp. 1539-1586.

Résumé

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $\chi$–independence conjecture by Bousseau for ${\mathbb{P}}^2$, and verify Toda’s conjecture for Gopakumar–Vafa invariants for certain local curves and local surfaces.

For the proof, we combine a generalized version of Ngô’s support theorem, a dimension estimate for the stacky Hilbert–Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

Détail

DOI : 10.2140/gt.2023.27.1539

Keywords: moduli of sheaves, perverse sheaves, support theorem, Gopakumar–Vafa invariants

Bibliographie

[1] A A Beĭlinson, J Bernstein, P Deligne, Faisceaux pervers, from: "Analysis and topology on singular spaces, I", Astérisque 100, Soc. Math. France (1982) 5

[2] P Bousseau, A proof of N Takahashi’s conjecture for (P2,E) and a refined sheaves/Gromov–Witten correspondence, preprint (2020)

[3] P Bousseau, Scattering diagrams, stability conditions, and coherent sheaves on P2, J. Algebraic Geom. 31 (2022) 593 | DOI

[4] M A De Cataldo, A support theorem for the Hitchin fibration : the case of SLn, Compos. Math. 153 (2017) 1316 | DOI

[5] M A A De Cataldo, T Hausel, L Migliorini, Topology of Hitchin systems and Hodge theory of character varieties : the case A1, Ann. of Math. 175 (2012) 1329 | DOI

[6] M A De Cataldo, D Maulik, J Shen, Hitchin fibrations, abelian surfaces, and the P = W conjecture, J. Amer. Math. Soc. 35 (2022) 911 | DOI

[7] M A A De Cataldo, A Rapagnetta, G Saccà, The Hodge numbers of O’Grady 10 via Ngô strings, J. Math. Pures Appl. 156 (2021) 125 | DOI

[8] P H Chaudouard, G Laumon, Un théorème du support pour la fibration de Hitchin, Ann. Inst. Fourier Grenoble 66 (2016) 711 | DOI

[9] W Y Chuang, D E Diaconescu, G Pan, BPS states and the P = W conjecture, from: "Moduli spaces" (editors L Brambila-Paz, O García-Prada, P Newstead, R P Thomas), London Math. Soc. Lecture Note Ser. 411, Cambridge Univ. Press (2014) 132

[10] B Davison, Cohomological Hall algebras and character varieties, Internat. J. Math. 27 (2016) | DOI

[11] B Davison, The integrality conjecture and the cohomology of preprojective stacks, preprint (2017)

[12] B Davison, Refined invariants of finite-dimensional Jacobi algebras, preprint (2019)

[13] B Davison, BPS Lie algebras and the less perverse filtration on the preprojective CoHA, preprint (2020)

[14] B Davison, S Meinhardt, Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras, Invent. Math. 221 (2020) 777 | DOI

[15] M Demazure, Surfaces de del Pezzo, I–V, from: "Séminaire sur les singularités des surfaces" (editors M Demazure, H C Pinkham, B Teissier), Lecture Notes in Math. 777, Springer (1980)

[16] S Di Rocco, k–very ample line bundles on del Pezzo surfaces, Math. Nachr. 179 (1996) 47 | DOI

[17] B Fantechi, L Göttsche, D Van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999) 115

[18] V Ginzburg, The global nilpotent variety is Lagrangian, Duke Math. J. 109 (2001) 511 | DOI

[19] M Groechenig, D Wyss, P Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via p–adic integration, Invent. Math. 221 (2020) 505 | DOI

[20] G Harder, M S Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975) 215 | DOI

[21] N J Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59 | DOI

[22] N Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91 | DOI

[23] B Iversen, Cohomology of sheaves, Springer (1986) | DOI

[24] D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, 1020, Amer. Math. Soc. (2012) | DOI

[25] A D King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. 45 (1994) 515 | DOI

[26] Y Laszlo, M Olsson, The six operations for sheaves on Artin stacks, I : Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008) 109 | DOI

[27] Y Laszlo, M Olsson, The six operations for sheaves on Artin stacks, II : Adic coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008) 169 | DOI

[28] Y Laszlo, M Olsson, Perverse t–structure on Artin stacks, Math. Z. 261 (2009) 737 | DOI

[29] J Le Potier, Faisceaux semi-stables de dimension 1 sur le plan projectif, Rev. Roumaine Math. Pures Appl. 38 (1993) 635

[30] J Manschot, S Mozgovoy, Intersection cohomology of moduli spaces of sheaves on surfaces, Selecta Math. 24 (2018) 3889 | DOI

[31] D Maulik, J Shen, Endoscopic decompositions and the Hausel–Thaddeus conjecture, Forum Math. Pi 9 (2021) | DOI

[32] D Maulik, Y Toda, Gopakumar–Vafa invariants via vanishing cycles, Invent. Math. 213 (2018) 1017 | DOI

[33] D Maulik, Z Yun, Macdonald formula for curves with planar singularities, J. Reine Angew. Math. 694 (2014) 27 | DOI

[34] S Meinhardt, Donaldson–Thomas invariants vs intersection cohomology for categories of homological dimension one, preprint (2015)

[35] S Meinhardt, M Reineke, Donaldson–Thomas invariants versus intersection cohomology of quiver moduli, J. Reine Angew. Math. 754 (2019) 143 | DOI

[36] A Mellit, Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures), Invent. Math. 221 (2020) 301 | DOI

[37] L Migliorini, V Shende, A support theorem for Hilbert schemes of planar curves, J. Eur. Math. Soc. 15 (2013) 2353 | DOI

[38] S Mozgovoy, M Reineke, Intersection cohomology of moduli spaces of vector bundles over curves, preprint (2015)

[39] S Mozgovoy, O Schiffmann, Counting Higgs bundles and type A quiver bundles, Compos. Math. 156 (2020) 744 | DOI

[40] B C Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010) 1 | DOI

[41] R Pandharipande, R P Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009) 407 | DOI

[42] M Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990) 221 | DOI

[43] O Schiffmann, Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, Ann. of Math. 183 (2016) 297 | DOI

[44] O G Schiffmann, Kac polynomials and Lie algebras associated to quivers and curves, from: "Proceedings of the International Congress of Mathematicians, II : Invited lectures" (editors B Sirakov, P N de Souza, M Viana), World Sci. (2018) 1393

[45] J Shen, Q Yin, Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds, Duke Math. J. 171 (2022) 209 | DOI

[46] S Sun, Generic base change, Artin’s comparison theorem, and the decomposition theorem for complex Artin stacks, J. Algebraic Geom. 26 (2017) 513 | DOI

[47] Y Toda, Stability conditions and curve counting invariants on Calabi–Yau 3–folds, Kyoto J. Math. 52 (2012) 1 | DOI

[48] Y Toda, Gopakumar–Vafa invariants and wall-crossing, J. Differential Geom. 123 (2023) 141 | DOI

[49] Z Yun, Langlands duality and global Springer theory, Compos. Math. 148 (2012) 835 | DOI

[50] Z Yun, W Zhang, Shtukas and the Taylor expansion of L–functions, Ann. of Math. 186 (2017) 767 | DOI

[51] X Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, from: "Geometry of moduli spaces and representation theory" (editors R Bezrukavnikov, A Braverman, Z Yun), IAS/Park City Math. Ser. 24, Amer. Math. Soc. (2017) 59