For a smooth cubic hypersurface X⊂Pn+1X\subset\mathbb{P}^{n+1} of dimension n≥3n\geq 3 over a field kk, the Fano variety of lines F(X)F(X) is a smooth projective variety of dimension 2(n−2)2(n-2). Humberto A. Diaz (preprint, 2017) proved that Grothendieck's standard conjectures (Lefschetz standard, Künneth, and homological equals numerical equivalence) hold for F(X)F(X) when k=Ck=\mathbb{C} and cohomology is singular cohomology with Q\mathbb{Q}-coefficients. The problem is the extension left unresolved by that work: do the standard conjectures hold for F(X)F(X) over an arbitrary base field — in particular over algebraically closed fields of positive characteristic — and with respect to an arbitrary Weil cohomology theory, such as ℓ\ell-adic or crystalline cohomology? No published work addresses this extension; the problem is open.

Background

Let X⊂Pn+1X\subset\mathbb{P}^{n+1} be a smooth cubic hypersurface of dimension n≥3n\geq 3 over a field kk. The Fano variety of lines F(X)F(X) parametrizes lines contained in XX; work of Altman and Kleiman shows it is a smooth projective variety of dimension 2(n−2)2(n-2). It is a central auxiliary object in the geometry of cubic hypersurfaces: for n=3n=3 its Albanese variety (the abelian variety canonically attached to a smooth projective variety) is, via the Abel–Jacobi map, the intermediate Jacobian (the complex torus built from H3(X)H^{3}(X)) used by Clemens and Griffiths to prove irrationality of the cubic threefold, and for n=4n=4 Beauville and Donagi showed that F(X)F(X) is a hyper-Kähler (irreducible holomorphic symplectic) fourfold, deformation equivalent to the Hilbert scheme S[2]S^{[2]} of two points on a K3 surface (a simply connected smooth projective surface with trivial canonical class; S[2]S^{[2]} parametrizes length-two subschemes of SS).

A Weil cohomology theory assigns to each smooth projective variety over kk a graded anti-commutative algebra with coefficients in a field of characteristic zero, satisfying finite-dimensionality, Poincaré duality, Künneth formulas, cycle class maps for algebraic cycles, the weak Lefschetz theorem, and the hard Lefschetz theorem; the main examples are singular cohomology with Q\mathbb{Q}-coefficients over k=Ck=\mathbb{C}, ℓ\ell-adic cohomology, de Rham cohomology, and crystalline cohomology in positive characteristic. Grothendieck's standard conjectures predict that, with respect to any such theory, three fundamental homological operations on a smooth projective variety YY of dimension dYd_Y come from algebraic cycles (correspondences on Y×YY\times Y, i.e., elements of the Chow groups of Y×YY\times Y with Q\mathbb{Q}-coefficients, the Chow groups being the groups of algebraic cycles modulo rational equivalence): (1) the Lefschetz standard conjecture — for an ample divisor class hh, the inverse of the Lefschetz isomorphism Hj(Y)→⋅hdY−jH2dY−j(Y)H^{j}(Y)\xrightarrow{\cdot h^{d_{Y}-j}}H^{2d_{Y}-j}(Y), which is an isomorphism by the Hard Lefschetz theorem, is induced by an algebraic correspondence; (2) the Künneth conjecture — the Künneth components δj\delta_{j} of the diagonal in the decomposition H2dY(Y×Y)≅⨁jHj(Y)⊗H2dY−j(Y)H^{2d_{Y}}(Y\times Y)\cong\bigoplus_{j}H^{j}(Y)\otimes H^{2d_{Y}-j}(Y) are algebraic; (3) homological equivalence (having zero cohomology class) coincides with numerical equivalence (pairing to zero against every cycle of complementary dimension) for cycles on YY. The conjectures are known for varieties whose cycle class map is an isomorphism (e.g., cellular varieties — varieties with a stratification by affine spaces — such as Grassmannians, the projective varieties parametrizing linear subspaces of a vector space), for abelian varieties, and are stable under hyperplane sections, products, and projective bundles (Kleiman). Arapura proved them for uniruled threefolds (varieties covered by rational curves), unirational fourfolds (varieties dominated by rational maps from projective space), moduli spaces of vector bundles on curves, and Hilbert schemes of points on smooth projective surfaces, and Charles and Markman proved them for all complex smooth projective varieties deformation equivalent to a Hilbert scheme of points on a K3 surface — which, by the Beauville–Donagi result, covers F(X)F(X) for cubic fourfolds over C\mathbb{C}.

Diaz's preprint extends the known cases to F(X)F(X) for cubic hypersurfaces of every dimension n≥3n\geq 3 over C\mathbb{C}. The proof strategy combines Galkin and Shinder's relation [X[2]]=[X][Pn+1]+L2[F][X^{[2]}]=[X][\mathbb{P}^{n+1}]+\mathbb{L}^{2}[F] in the Grothendieck ring of varieties (the ring generated by isomorphism classes of varieties modulo the scissors relations [Y]=[Z]+[Y∖Z][Y]=[Z]+[Y\setminus Z]; here X[2]X^{[2]} is the Hilbert scheme of two points on XX and L\mathbb{L}, the Lefschetz motive, is the class of the affine line), whose Hodge-theoretic realization gives an explicit decomposition of H∗(F)H^{*}(F) into a summand pulled back from the Grassmannian of lines in Pn+1\mathbb{P}^{n+1} plus Tate twists (degree-shifting operations, tensoring with powers of the class of a point) of copies of the primitive cohomology Hprimn(X)H^{n}_{prim}(X) (the part of Hn(X)H^{n}(X) annihilated by cup-product with the hyperplane class), together with the cylinder correspondence between XX and FF induced by the universal line over FF. One then constructs explicit algebraic correspondences Γk∈CHk(F×F)\Gamma_{k}\in CH^{k}(F\times F) inducing isomorphisms H4(n−2)−k(F)→Hk(F)H^{4(n-2)-k}(F)\to H^{k}(F) for 0≤k≤2(n−2)0\leq k\leq 2(n-2), and a criterion of Kleiman converts these isomorphisms into the standard conjectures. The argument uses tools specific to k=Ck=\mathbb{C}: the Hodge-theoretic decomposition above, Hard Lefschetz and Poincaré duality for singular cohomology, and Hodge-theoretic orthogonality. The note's conventions state that "most of the results hold over any field with any choice of Weil cohomology", but Theorem 0.1 itself is stated and proved only over C\mathbb{C} with singular cohomology, and the note (which was never published in a journal and has remained a single 2017 arXiv version) does not carry out the extension.

Problem Statement

Let X⊂Pn+1X\subset\mathbb{P}^{n+1} (n≥3n\geq 3) be a smooth cubic hypersurface over a field kk, and let F(X)F(X) be its Fano variety of lines, a smooth projective variety of dimension 2(n−2)2(n-2). Diaz's theorem covers only the case k=Ck=\mathbb{C} with H∗H^{*} equal to singular cohomology with Q\mathbb{Q}-coefficients. Determine whether Grothendieck's standard conjectures hold for F(X)F(X) over an arbitrary base field kk and with respect to an arbitrary Weil cohomology theory H∗H^{*}; that is, over arbitrary kk (at minimum every algebraically closed field, including positive characteristic) and for any Weil cohomology (e.g., ℓ\ell-adic or crystalline cohomology), prove or disprove that: (1) (Lefschetz standard) for an ample divisor class hh on F(X)F(X), the inverse of the Lefschetz isomorphism Hj(F)→⋅h2(n−2)−jH4(n−2)−j(F)H^{j}(F)\xrightarrow{\cdot h^{2(n-2)-j}}H^{4(n-2)-j}(F) is induced by an algebraic correspondence Λj∈CHj(F×F)\Lambda_{j}\in CH^{j}(F\times F) for each 0≤j≤2(n−2)0\leq j\leq 2(n-2); (2) (Künneth) the Künneth components of the diagonal of F(X)F(X) are algebraic; (3) homological and numerical equivalence coincide for cycles on F(X)F(X). A complete answer supplies the statements for arbitrary kk and arbitrary Weil cohomology at the same level of generality as the complex case, without restricting the dimension, the characteristic, or the class of cubics.

References

  1. Humberto A. Diaz, The standard conjectures for the variety of lines on a cubic hypersurface, preprint, arXiv:1706.06683 (2017). URL: https://arxiv.org/abs/1706.06683
  2. Sergey Galkin, Evgeny Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, preprint, arXiv:1405.5154 (2014). URL: https://arxiv.org/abs/1405.5154
  3. François Charles, Eyal Markman, The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces, Compositio Mathematica 149 (2013), no. 3, 481–494. DOI: 10.1112/S0010437X12000607, URL: https://doi.org/10.1112/S0010437X12000607 (also arXiv:1009.0413)
  4. Donu Arapura, Motivation for Hodge cycles, Advances in Mathematics 207 (2006), no. 2, 762–781. DOI: 10.1016/j.aim.2006.01.005, URL: https://doi.org/10.1016/j.aim.2006.01.005
  5. Steven L. Kleiman, The standard conjectures, Proceedings of Symposia in Pure Mathematics 55, Part 1, American Mathematical Society (1994), 3–20. DOI: 10.1090/pspum/055.1/1265519, URL: https://doi.org/10.1090/pspum/055.1/1265519
  6. A. Grothendieck, Standard conjectures on algebraic cycles, in: Algebraic Geometry (International Colloquium, Tata Institute of Fundamental Research, Bombay, 1968), Oxford University Press (1969), 193–199. Zbl 0201.23301, URL: https://zbmath.org/3319220
  7. A. Altman, S. Kleiman, Foundations of the theory of Fano schemes, Compositio Mathematica 34 (1977), no. 1, 3–47. URL: http://www.numdam.org/item/CM_1977__34_1_3_0/
  8. Arnaud Beauville, Ron Donagi, La variété des droites d'une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706. Zbl 0602.14041, URL: https://zbmath.org/3970926
  9. Humberto Diaz, The motive of the Fano surface of lines, C. R. Math. Acad. Sci. Paris 354 (2016), no. 9, 925–930. DOI: 10.1016/j.crma.2016.07.003, URL: https://doi.org/10.1016/j.crma.2016.07.003 (also arXiv:1602.06403)
  10. H. Anthony Diaz, The Chow ring of a cubic hypersurface, International Mathematics Research Notices 2021 (2021), no. 22, 17071–17090. DOI: 10.1093/imrn/rnz299, URL: https://doi.org/10.1093/imrn/rnz299
  11. Robert Laterveer, A remark on the motive of the Fano variety of lines of a cubic, Annales Mathématiques du Québec 41 (2017), no. 1, 141–154. DOI: 10.1007/s40316-016-0070-x, URL: https://doi.org/10.1007/s40316-016-0070-x (also arXiv:1611.08818)
  12. Lie Fu, Charles Vial, Cubic fourfolds, Kuznetsov components, and Chow motives, Documenta Mathematica 28 (2023), no. 4, 827–856. DOI: 10.4171/DM/925, URL: https://doi.org/10.4171/DM/925 (also arXiv:2009.13173)
  13. Giuseppe Ancona, Mattia Cavicchi, Robert Laterveer, Giulia Saccà, Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations, Journal of the London Mathematical Society 111 (2025), no. 4, article e70133. DOI: 10.1112/jlms.70133, URL: https://doi.org/10.1112/jlms.70133 (also arXiv:2304.00978)
  14. Lie Fu, Ben Moonen, Algebraic cycles on Gushel–Mukai varieties, Épijournal de Géométrie Algébrique, special issue in honour of Claire Voisin (2024). DOI: 10.46298/epiga.2024.9815, URL: https://doi.org/10.46298/epiga.2024.9815 (also arXiv:2207.01118)
  15. Renjie Lyu, Cylinder maps of algebraic cycles on cubic hypersurfaces, Bulletin of the London Mathematical Society 57 (2025), no. 12, 3949–3968. DOI: 10.1112/blms.70200, URL: https://doi.org/10.1112/blms.70200 (also arXiv:1810.12394)