On generalized Beauville decompositions

• Younghan Bae ^[1] ; Davesh Maulik ^[2] ; Junliang Shen ^[3] ; Qizheng Yin ^[4]
  1. [1] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
  2. [2] Massachusetts Institute of Technology

     Massachusetts Institute of Technology

     City of Cambridge, Estados Unidos

     
  3. [3] Yale University

     Yale University

     Town of New Haven, Estados Unidos

     
  4. [4] Peking University

     Peking University

     China

     

  Mostrar afiliaciones +
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 32, Nº. 3, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Motivated by the Beauville decomposition of an abelian scheme and the “Perverse = Chern” phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. On the one hand, we prove for the Beauville–Mukai system associated with an irreducible curve class on a K3 surface the existence of a Fourier-stable multiplicative splitting of the perverse filtration, which extends the Beauville decomposition for the nonsingular fibers.

    Our approach is to construct a Lefschetz decomposition associated with a Fourierconjugate sl2-triple, which relies heavily on recent work concerning the interaction between derived equivalences and LLV algebras for hyper-Kähler varieties. Motivic lifting and connections to the Beauville–Voisin conjectures are also discussed. On the other hand, we construct for any g ≥ 2 a compactified Jacobian fibration of genus g curves such that each curve is integral with at worst simple nodes and the (multiplicative) perverse filtration does not admit a multiplicative splitting. Our argument relies on the recently established universal double ramification cycle relations. This shows that in general an extension of the Beauville decomposition cannot exist for compactified Jacobian fibrations even when the simplest singular point appears.

• Referencias bibliográficas
  • Ancona, G., Cavicchi, M., Laterveer, R., Saccà, G.: Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations, arXiv:2304.00978
  • Ancona, G., Huber, A., Pepin Lehalleur, S.: On the relative motive of a commutative group scheme. Algebr. Geom. 3(2), 150–178 (2016)
  • Arinkin, D.: Cohomology of line bundles on compactified Jacobians. Math. Res. Lett. 18(6), 1215–1226 (2011)
  • Arinkin, D.: Autoduality of compactified Jacobians for curves with plane singularities. J. Algebraic Geom. 22(2), 363–388 (2013)
  • Bae, Y., Holmes, D., Pandharipande, R., Schmitt, J., Schwarz, R.: Pixton’s formula and Abel-Jacobi theory on the Picard stack. Acta Math....
  • Beauville, A.: Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, Algebraic geometry (Tokyo/Kyoto,...
  • Beauville, A.: Variétés kählériennes dont la première classe de Chern est nulle. J. Differential Geom. 18(4), 755–782 (1983)
  • Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273(4), 647–651 (1986)
  • Beauville, A.: Systèmes hamiltoniens complètement intégrables associés aux surfaces , Problems in the theory of surfaces and their classification...
  • Beauville, A.: On the splitting of the Bloch–Beilinson filtration, Algebraic cycles and motives. Vol. 2, 38–53, London Math. Soc. Lecture...
  • Beauville, A., Voisin, C.: On the Chow ring of a surface. J. Algebraic Geom. 13(3), 417–426 (2004)
  • Beckmann, T.: Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure. Compos. Math. 159(1), 109–152 (2023)
  • Buryak, A., Shadrin, S., Zvonkine, D.: Top tautological group of . J. Eur. Math. Soc. (JEMS) 18(12), 2925–2951 (2016)
  • de Cataldo, M.A., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties: the case ,. Ann. of Math.(2)...
  • de Cataldo, M.A., Migliorini, L.: The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup. (4) 38(5), 693–750 (2005)
  • Corti, A., Hanamura, M.: Motivic decomposition and intersection Chow groups. I, Duke Math. J. 103(3), 459–522 (2000)
  • Deninger, C., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991)
  • Ebert, J., Randal-Williams, O.: Stable cohomology of the universal Picard varieties and the extended mapping class group. Doc. Math. 17, 417–450...
  • Esteves, E., Pacini, M.: Semistable modifications of families of curves and compactified Jacobians. Arkiv. Mat. 54(1), 55–83 (2016)
  • Faber, C.: Chow rings of moduli spaces of curves. I. The Chow ring of , Ann. of Math. (2) 132 (2), 331–419 (1990)
  • Fulton, W.: Intersection theory. Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in...
  • Gordon, B.B., Hanamura, M., Murre, J.P.: Relative Chow-Künneth projectors for modular varieties. J. Reine Angew. Math. 558, 1–14 (2003)
  • Hausel, T., Mellit, A., Minets, A., Schiffmann, O.: via , arXiv:2209.05429v1
  • Hoskins, V.: Two proofs of the conjecture [after Maulik–Shen and Hausel–Mellit–Minets–Schiffmann], Séminaire Bourbaki, (2023)
  • Huybrechts, D.: Chow groups of surfaces and spherical objects. J. Eur. Math. Soc. (JEMS) 12(6), 1533–1551 (2010)
  • Kononov, Y., Pi, W., Shen, J.: Perverse filtrations, Chern filtrations, and refined BPS invariants for local . Adv. Math. 433, 109294 (2023)
  • Kretschmer, A.: The Chow ring of hyperkähler varieties of -type via Lefschetz actions. Math. Z. 300(2), 2069–2090 (2022)
  • Künnemann, K.: A Lefschetz decomposition for Chow motives of abelian schemes. Invent. Math. 113(1), 85–102 (1993)
  • Laza, R., Saccà, G., Voisin, C.: A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic -fold. Acta...
  • Looijenga, E.: Cohomology of and , Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), 205–228,...
  • Looijenga, E., Lunts, V.A.: A Lie algebra attached to a projective variety. Invent. Math. 129(2), 361–412 (1997)
  • Markman, E.: Stable vector bundles on a hyper-Kähler manifold with a rank obstruction map are modular. Kyoto J. Math. 64(3), 635–742 (2024)
  • Maulik, D., Shen, J.: The conjecture for , Ann. of Math. (2) 200 (2), 529–556 (2024)
  • Maulik, D., Shen, J., Yin, Q.: Perverse filtrations and Fourier transforms, arXiv:2308.13160
  • Moonen, B., Yin, Q.: On a question of O’Grady about modified diagonals, arXiv:1311.1185
  • Mukai, S.: Duality between and with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)
  • Mumford, D.: On the Kodaira dimension of the Siegel modular variety, Algebraic geometry—open problems (Ravello, 1982), 348–375, Lecture Notes...
  • Mumford, D.: Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, 271–328, Progr. Math., 36, Birkhäuser...
  • Murre, J. P., Nagel, J., Peters, C. A. M.: Lectures on the theory of pure motives, Univ. Lecture Ser., 61, Amer. Math. Soc., Providence, RI,...
  • Neguţ, A., Oberdieck, G., Yin, Q.: Motivic decompositions for the Hilbert scheme of points of a surface. J. Reine Angew. Math. 778, 65–95...
  • Oberdieck, G.: A Lie algebra action on the Chow ring of the Hilbert scheme of points of a surface. Comment. Math. Helv. 96(1), 65–77 (2021)
  • Rieß, U.: On Beauville’s conjectural weak splitting property, Int. Math. Res. Not. IMRN (20), 6133–6150 (2016)
  • Shen, J., Yin, Q.: Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds, with Appendix B by Claire Voisin. Duke Math....
  • Shen, J., Yin, Q., Zhao, X.: Derived categories of surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties. Compos....
  • Shende, V.: The weights of the tautological classes of character varieties. Int. Math. Res. Not. IMRN 22, 6832–6840 (2017)
  • Taelman, L.: Derived equivalences of hyperkähler varieties. Geom. Topol. 27(7), 2649–2693 (2023)
  • Verbitsky, M.: Cohomology of compact hyperkaehler manifolds, Thesis (Ph.D.)–Harvard University, 89 pp (1995)
  • Verbitsky, M.: Mirror symmetry for hyper-Kähler manifolds, Mirror symmetry, III (Montreal, PQ, 1995), 115–156, AMS/IP Stud. Adv. Math., 10,...
  • Voisin, C.: On the Chow ring of certain algebraic hyper-Kähler manifolds. Pure Appl. Math. Q. 4(3), 613–649 (2008)
  • Voisin, C.: On the Lefschetz standard conjecture for Lagrangian covered hyper-Kähler varieties. Adv. Math. 396, 108108 (2022)
  • Voisin, C.: On Chern classes of Lagrangian fibered hyper-Kähler manifolds. Pure Appl. Math. Q. 21(3), 1393–1435 (2025)
  • Zhang, Z.: Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces, II. Trans. Amer. Math. Soc. 374(12), 8573–8602...