A complex abelian variety AA of even dimension 2m2m whose algebra of rational endomorphisms contains an imaginary quadratic field QQ, in such a way that the tangent space at the origin is a free Q⊗CQ\otimes\mathbb{C}-module, is said to be of Weil type. For every polarization λ\lambda of (A,Q)(A,Q), the middle cohomology H2m(A,Q)(m)H^{2m}(A,\mathbb{Q})(m) contains a canonically defined one-dimensional QQ-vector space W(A,Q)W(A,Q) of Hodge classes, the Weil classes, and the pair (A,Q,λ)(A,Q,\lambda) carries a discrete determinant invariant in Q×/Nm(Q×)\mathbb{Q}^{\times}/\mathrm{Nm}(Q^{\times}). Weil (1977) predicted that these classes are always algebraic, making them the standard test case for the Hodge conjecture on abelian varieties; Milne's study of the Tate and standard conjectures for certain abelian varieties reduces broad classes of those conjectures to exactly this algebraicity problem. As of 2026, algebraicity is proved for all abelian fourfolds of Weil type (all QQ, all determinants; Markman 2025, building on Schoen, Koike, and his own determinant-1 theorem, with independent proofs of the determinant-1 case by Floccari–Fu and van Geemen–Rapagnetta) and for abelian sixfolds of Weil type with determinant −1-1; consequently the Hodge conjecture is now known for all abelian varieties of dimension at most 5. The problem remains open for sixfolds of arbitrary other determinant, for all dimensions 2m≥82m\geq 8, and for the extension from imaginary quadratic fields to general CM fields, where it becomes equivalent to the Hodge conjecture for CM abelian varieties and hence, by Milne's theorems, implies the Tate and standard conjectures for all abelian varieties over finite fields.

Background

On a smooth complex projective variety, a rational cohomology class of degree 2p2p is called a Hodge class if it has Hodge type (p,p)(p,p). The Hodge conjecture predicts that every Hodge class is a rational linear combination of classes of algebraic subvarieties of codimension pp; it is known only for p=1p=1, where it reduces to the Lefschetz theorem on (1,1)(1,1)-classes, i.e. to divisor classes. An abelian variety is a projective complex torus; it has complex multiplication (CM) when its algebra of rational endomorphisms is unusually large. In the 1960s, Mumford and Tate tried to prove the Hodge conjecture for abelian varieties by showing that all their Hodge classes are generated by divisor classes; Mumford then found a simple abelian fourfold where this fails. Weil (1977) isolated the essential structure behind such examples: an action of an imaginary quadratic field QQ (a degree-2 subfield of C\mathbb{C} of the form Q(−d)\mathbb{Q}(\sqrt{-d})).

A 2m2m-dimensional complex abelian variety AA with an embedding Q↪End0(A)Q\hookrightarrow\mathrm{End}^{0}(A) of QQ into its rational endomorphism algebra is of Weil type when each eigenspace of −d\sqrt{-d} on the tangent space at the origin is mm-dimensional. In that case the middle cohomology contains a canonical line of Hodge classes: the Weil classes

W(A,Q)=(⋀Q2mH1(A,Q))(m)⊂H2m(A,Q)(m),W(A,Q)=\Big(\bigwedge\nolimits_{Q}^{2m}H^{1}(A,\mathbb{Q})\Big)(m)\subset H^{2m}(A,\mathbb{Q})(m),

a one-dimensional QQ-vector space, consisting of classes that are QQ-balanced with respect to the QQ-action. A polarization λ\lambda of (A,Q)(A,Q) is an ample line bundle whose induced Rosati involution (the adjoint involution on endomorphisms determined by a polarization) restricts to complex conjugation on QQ; it determines a QQ-valued hermitian form on H1(A,Q)H_{1}(A,\mathbb{Q}) whose determinant, taken modulo norms Nm(Q×)\mathrm{Nm}(Q^{\times}), is a discrete invariant of the triple (A,Q,λ)(A,Q,\lambda) (Markman calls this same class in Q×/Nm(Q×)\mathbb{Q}^{\times}/\mathrm{Nm}(Q^{\times}) the discriminant det⁡H\det H); since the hermitian form attached to a polarization has signature (m,m)(m,m), the admissible determinant classes have sign (−1)m(-1)^{m}, so for sixfolds (m=3m=3) every determinant class is represented by a negative rational number. For fixed (m,Q,det⁡)(m,Q,\det), the polarized varieties of Weil type form an m2m^{2}-dimensional family, a Shimura variety (a moduli space of Hodge structures of arithmetic origin, in this case for the corresponding special unitary group, i.e. the group of determinant-one linear maps preserving the hermitian form), and the triple (m,Q,det⁡)(m,Q,\det) determines the family up to isogeny. Weil showed that, for the general member of such a family, the Weil classes are exotic, meaning they are not generated by divisor classes; they are the first obstruction to the Mumford–Tate strategy, and Weil proposed their algebraicity as a good test case for the Hodge conjecture.

Milne's paper "The Tate and Standard Conjectures for Certain Abelian Varieties" situates this problem in arithmetic: it proves that the algebraicity of the Weil classes for an auxiliary Weil-type pair A×BmA\times B^{m} implies the Hodge conjecture for the powers of ACA_{\mathbb{C}} and the Tate conjecture and Grothendieck's standard conjectures (the conjectural semisimplicity and positivity properties of algebraic cycles underlying the motivic picture) for the powers of the reduction A0A_{0} over the finite field: the proof verifies a certain intersection condition (∗)(*) — asking roughly that the Tate classes of the reduction (the finite-field analogue of Hodge classes, i.e. rational classes invariant under a sufficiently large power of the Frobenius endomorphism of the finite base field) arise from Hodge classes of the lift — and applies his Theorem 3.1, which deduces the Tate and standard conjectures for the reduction from (∗)(*) together with the Hodge conjecture for the powers of the lift. The condition compares the Lefschetz group of the reduction (introduced by Murty: the group of symplectic automorphisms of the first homology commuting with all rational endomorphisms, an arithmetic analogue of the Mumford–Tate group, which is the group of automorphisms of the cohomology of all powers fixing every Hodge class) with the Mumford–Tate group of the lift. Thus, determining for which polarized Weil-type pairs the Weil classes are algebraic would convert a large family of conditional statements into unconditional theorems. In a wider sense, QQ may be replaced by any CM field (a totally imaginary quadratic extension of a totally real field); Deligne proved that algebraicity of all Weil classes in this generalized sense is equivalent to the Hodge conjecture for all CM abelian varieties, a reduction recorded in modern form by Milne (2020) and in Markman's ICM 2026 survey. By Milne's earlier theorems, the Hodge conjecture for CM abelian varieties in turn implies the standard conjectures and the Tate conjecture for all abelian varieties over finite fields.

Before 2025, the algebraicity of the Weil classes was known only in scattered special cases: for Q=Q(−3)Q=\mathbb{Q}(\sqrt{-3}) by work of Schoen (fourfolds with arbitrary determinant, sixfolds with determinant −1-1), for Q=Q(i)Q=\mathbb{Q}(i) by Koike, and — the result quoted in Milne's paper as Theorem 1.5 — Markman's theorem for polarized abelian fourfolds of Weil type with determinant 11 and arbitrary imaginary quadratic QQ, proved via the deformation theory of hyperkähler manifolds (compact simply connected complex manifolds with a unique holomorphic symplectic 22-form, here of generalized Kummer type, a class built from abelian surfaces, whose intermediate Jacobians — complex tori constructed from the middle odd-degree cohomology — furnish complete families of Weil-type fourfolds, an observation due to O'Grady). In 2025 Markman removed the determinant restriction for fourfolds and settled sixfolds of determinant −1-1, as detailed in the Problem Statement below; the cases of sixfolds with other determinants, of all dimensions 2m≥82m\geq 8, and of general CM fields remain open.

Problem Statement

Determine for which polarized abelian varieties of Weil type (A,Q,λ)(A,Q,\lambda) over C\mathbb{C} the Weil classes W(A,Q)⊂H2m(A,Q)(m)W(A,Q)\subset H^{2m}(A,\mathbb{Q})(m) are algebraic, i.e. spanned over Q\mathbb{Q} by cohomology classes of algebraic cycles on AA. Weil's conjecture (1977) predicts that they are algebraic for every such triple, for every imaginary quadratic field QQ, every dimension 2m2m, and every determinant class in Q×/Nm(Q×)\mathbb{Q}^{\times}/\mathrm{Nm}(Q^{\times}); the problem, as isolated in Milne's study of the Tate and standard conjectures, is to prove this (or to exhibit a counterexample), thereby converting Milne's conditional Theorem 0.1 and the closing expectations of that paper into unconditional results on the Hodge, Tate, and standard conjectures. At present the conjecture is proved for all fourfolds of Weil type and for sixfolds of determinant −1-1 (Markman 2025), so the surviving open core is: (i) sixfolds of Weil type with determinant other than −1-1 modulo norms, for arbitrary imaginary quadratic QQ; (ii) all Weil-type dimensions 2m≥82m\geq 8, for arbitrary imaginary quadratic QQ and arbitrary determinant; and, as a generalization beyond Weil's original conjecture, (iii) the extension in which QQ is replaced by a general CM field KK acting on AA with H1(A,Q)H_{1}(A,\mathbb{Q}) a free KK-module (the split Weil classes), which is equivalent to the Hodge conjecture for all CM abelian varieties. Thus what remains is to determine, for every moduli component (m,Q,det⁡)(m,Q,\det) with QQ imaginary quadratic and det⁡∈Q×/Nm(Q×)\det\in\mathbb{Q}^{\times}/\mathrm{Nm}(Q^{\times}) not already covered — that is, the cases (i) and (ii) — whether the Weil classes are algebraic, or to refute Weil's conjecture by an explicit counterexample; the CM-field extension (iii), which Milne records only as a generalization (Aside 1.7), goes beyond the original conjecture.

References

  1. James S. Milne, The Tate and Standard Conjectures for Certain Abelian Varieties, arXiv:2112.12815 (2021), https://arxiv.org/abs/2112.12815
  2. André Weil, Abelian varieties and the Hodge ring (1977), in Œuvres Scientifiques Collected Papers, Volume III, Springer, 1979, DOI 10.1007/978-1-4757-1705-1_123, https://doi.org/10.1007/978-1-4757-1705-1_123
  3. Eyal Markman, Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds, arXiv:2502.03415 (2025), https://arxiv.org/abs/2502.03415
  4. Eyal Markman, The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians, Journal of the European Mathematical Society 25 (2023), no. 1, 231–321, DOI 10.4171/jems/1199, arXiv:1805.11574, https://arxiv.org/abs/1805.11574
  5. Eyal Markman, Secant sheaves and Weil classes on abelian varieties (contribution to the ICM 2026), arXiv:2509.23403 (2025), https://arxiv.org/abs/2509.23403
  6. Eyal Markman, Secant sheaves on abelian n-folds with real multiplication and Weil classes on abelian 2n-folds with complex multiplication, arXiv:2509.23079 (2025), https://arxiv.org/abs/2509.23079
  7. Salvatore Floccari and Lie Fu, The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties, Journal de Mathématiques Pures et Appliquées 210 (2026), article 103876, DOI 10.1016/j.matpur.2026.103876, arXiv:2504.13607, https://arxiv.org/abs/2504.13607
  8. Bert van Geemen, Weil classes and decomposable abelian fourfolds, Symmetry, Integrability and Geometry: Methods and Applications 18 (2022), 097, DOI 10.3842/SIGMA.2022.097, arXiv:2108.02087, https://arxiv.org/abs/2108.02087
  9. Bert van Geemen and Antonio Rapagnetta, Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class, arXiv:2607.18341 (2026), https://arxiv.org/abs/2607.18341
  10. B. J. J. Moonen and Yu. G. Zarhin, Weil classes on abelian varieties, Journal für die reine und angewandte Mathematik 496 (1998), 83–92, DOI 10.1515/crll.1998.034, arXiv:alg-geom/9612017, https://arxiv.org/abs/alg-geom/9612017
  11. B. J. J. Moonen and Yu. G. Zarhin, Hodge classes on abelian varieties of low dimension, Mathematische Annalen 315 (1999), no. 4, 711–733, DOI 10.1007/s002080050333, https://doi.org/10.1007/s002080050333
  12. Chad Schoen, Hodge classes on self-products of a variety with an automorphism, Compositio Mathematica 65 (1988), no. 1, 3–32, https://www.numdam.org/item/?id=CM_1988__65_1_3_0
  13. Chad Schoen, Addendum to: Hodge classes on self-products of a variety with an automorphism, Compositio Mathematica 114 (1998), no. 3, 321–328, DOI 10.1023/A:1000566205021, https://doi.org/10.1023/A:1000566205021
  14. Kenji Koike, Algebraicity of some Weil Hodge classes, Canadian Mathematical Bulletin 47 (2004), no. 4, 566–572, DOI 10.4153/CMB-2004-055-x, https://doi.org/10.4153/CMB-2004-055-x
  15. James S. Milne, Hodge classes on abelian varieties, arXiv:2010.08857 (2020), https://arxiv.org/abs/2010.08857
  16. Giuseppe Ancona, Standard conjectures for abelian fourfolds, Inventiones Mathematicae 223 (2021), no. 1, 149–212, DOI 10.1007/s00222-020-00990-7, arXiv:1806.03216, https://doi.org/10.1007/s00222-020-00990-7
  17. Pierre Deligne, Hodge cycles on abelian varieties, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer, 1982, 9–100, DOI 10.1007/bfb0091629, https://doi.org/10.1007/bfb0091629
  18. Amir Mostaed, McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds, arXiv:2603.20268 (2026), https://arxiv.org/abs/2603.20268