Algebraicity of Weil classes on polarized abelian varieties of Weil type

The Hodge conjecture predicts that every rational $(p,p)$-cohomology class on a smooth complex projective variety is a rational combination of classes of algebraic subvarieties. A polarized abelian variety of Weil type is a triple $(A,\eta,h)$ in which $A$ is an abelian variety of even dimension carrying an action of a CM field $K$ whose eigenspaces on $H^{1}$ are balanced with respect to the Hodge decomposition; André Weil, and later Deligne in generality, associated to such a triple a distinguished space $HW(A,\eta)$ of Hodge classes, the Weil classes, which for a generic triple lie outside the algebra generated by divisor classes, so their algebraicity is exactly the new content of the Hodge conjecture for these varieties. The problem is to determine for which triples $(A,\eta,h)$ the Weil classes are algebraic. Work of Markman (2025) settled all abelian fourfolds of Weil type with $K$ imaginary quadratic and sixfolds of discriminant $-1$, which yields the Hodge conjecture for abelian fourfolds (indeed for all abelian varieties of dimension at most five, by reductions of Moonen–Zarhin and others); but sixfolds of nontrivial discriminant, all dimensions $\ge 8$, and all CM fields of degree $>2$ remain open. A full positive answer would imply the Hodge conjecture for all CM abelian varieties, Grothendieck's standard conjectures for all abelian varieties, and the Tate conjecture for abelian varieties over finite fields; a counterexample would refute the Hodge conjecture itself.

Background

A rational cohomology class of a smooth complex projective variety is called a Hodge class if it has type $(p,p)$ in the Hodge decomposition; the Hodge conjecture predicts that every Hodge class is algebraic, i.e. a rational linear combination of cohomology classes of complex algebraic subvarieties of codimension $p$. For $p=1$ this is the Lefschetz $(1,1)$ theorem, which identifies divisor classes with the rational classes of type $(1,1)$, but already for abelian varieties of dimension $\ge 2$ and $p=2$ it is wide open in general. On an abelian variety $A$ the cohomology ring is the exterior algebra of $H^{1}(A,\mathbb{Q})$, and a polarization (the class $h$ of an ample divisor) singles out divisor classes in $H^{1,1}$ and induces the Rosati involution on the endomorphism algebra $\mathrm{End}^{0}(A)=\mathrm{End}(A)\otimes\mathbb{Q}$: the anti-involution $f\mapsto f^{\dagger}$ determined by $h$, in the same way a Hermitian form determines adjoint maps.

Let $K$ be a CM field, that is, a totally imaginary quadratic extension $K/F$ of a totally real number field $F$, and set $e=[K:\mathbb{Q}]$. A pair $(A,\eta)$ consisting of an abelian variety and an algebra embedding $\eta\colon K\to\mathrm{End}^{0}(A)$ is of Weil type if $H^{1}(A,\mathbb{Q})$ is free of rank $d$ over $K$ and the eigenspace decomposition of $H^{1}(A,\mathbb{C})$ under $\eta(K)$ is balanced: for every embedding $\sigma\colon K\hookrightarrow\mathbb{C}$ one has $\dim H^{1,0}_{\sigma}=\dim H^{0,1}_{\sigma}=d/2$. Then $\dim A=ed/2$ is even, say $2n$. A polarization $h$ is compatible if the Rosati involution acts on $\eta(K)$ as complex conjugation; such an $h$ defines a $K$-valued Hermitian form $H$ on $H^{1}(A,\mathbb{Q})$, whose determinant class in $\mathbb{Q}^{\times}/\mathrm{Nm}(K^{\times})$ is the discriminant, a discrete invariant labelling the components of the moduli space of polarized Weil-type triples. The triple is of split Weil type if $H$ admits an isotropic subspace of half the dimension; for $\dim A=2n$, the trivial (split) discriminant is the coset of $(-1)^{n}$, that is, $1$ for fourfolds and $-1$ for sixfolds.

The space of Weil classes is

$$HW(A,\eta)=\wedge^{d}_{K}H^{1}(A,\mathbb{Q})\;\subset\;H^{d/2,d/2}(A,\mathbb{Q}),$$

a $K$-line, hence an $e$-dimensional $\mathbb{Q}$-vector space (two-dimensional over $\mathbb{Q}$ in Weil's classical imaginary-quadratic case). The balanced-eigenspace condition forces these classes to be of type $(d/2,d/2)$, and they are absolute Hodge, meaning they remain Hodge classes under every embedding of the coefficient field into $\mathbb{C}$ and correspond compatibly to étale classes - a rigidity property strictly weaker than algebraicity. Moonen and Zarhin proved that the Weil classes are exceptional: for a generic polarized triple of Weil type with $d\ge 4$, the Hodge ring (the graded algebra of rational Hodge classes) is generated by divisor classes together with $HW(A,\eta)$, and $HW(A,\eta)$ meets the subalgebra generated by divisor classes trivially. Their algebraicity is thus precisely the essential new content of the Hodge conjecture for abelian varieties.

The construction originates in Weil's 1977 note 'Abelian varieties and the Hodge ring', written after Mumford found abelian varieties whose rational $(p,p)$-classes are not generated by divisor classes; in a 1965 letter Weil wrote of the resulting 'abnormal classes' that he would 'rather attempt to show there is no such cycle'. Deligne generalized the construction to arbitrary CM fields, and Moonen–Zarhin systematized the theory, proving in particular that Weil classes on Weil-type varieties account for all exceptional Hodge (and Tate) classes on abelian varieties of dimension $\le 5$ up to divisor classes.

The significance of the problem is structural. André proved that every Hodge class on an abelian variety of CM type is a sum of pullbacks of Weil classes on abelian varieties of split Weil type, so the Hodge conjecture for all CM abelian varieties reduces to algebraicity of split Weil classes. Milne proved that, conditional on Weil-class algebraicity, the Tate conjecture for divisors (the statement that $\ell$-adic Tate classes on an abelian variety over a finite field are generated by algebraic cycles) and Grothendieck's standard conjectures (algebraicity of the Künneth components and of the Lefschetz operator's inverse) follow for large classes of abelian varieties, including products with CM elliptic factors and their reductions.

Algebraicity is known in an expanding family of cases: Schoen proved it for fourfolds of Weil type with $K=\mathbb{Q}(\sqrt{-3})$ and, in an addendum, for arbitrary discriminant with that $K$ and for sixfolds of trivial discriminant; Koike treated $K=\mathbb{Q}(\sqrt{-1})$; Markman settled the generic fourfold of discriminant $1$ for all imaginary quadratic $K$ using algebraic cycles on the intermediate Jacobians (complex tori built from a variety's odd-degree cohomology, which receive classes of algebraic cycles) of hyper-Kähler varieties - simply connected projective varieties with trivial canonical bundle and, up to scalar, a unique holomorphic symplectic two-form - of generalized Kummer type, i.e. deformation-equivalent to a fibre of the summation map from a Hilbert scheme of points on an abelian surface; and in 2025 Markman proved the sixfold case of discriminant $-1$ for every imaginary quadratic $K$, implying every abelian fourfold of Weil type, for all discriminants, via a degeneration argument of Schoen. Floccari–Fu and van Geemen–Rapagnetta gave independent alternative proofs via hyper-Kähler geometry, the former for all fourfolds of trivial discriminant and the latter for the general fourfold of trivial discriminant. The current frontier is Markman's secant-sheaf strategy: when an abelian variety $X$ has real multiplication by $F$, the even cohomology of $X$ realizes the half-spin representation (one of the two irreducible spin representations of the spin group of a quadratic space) and contains the spinorial variety (a component of the Grassmannian of maximal isotropic subspaces); a rational $K$-secant plane to it determines a Weil-type structure on $X\times\hat{X}$, and the normalized Chern character (the characteristic class summarizing the Chern classes of a coherent sheaf) of a suitable derived transform - an integral functor between derived categories of coherent sheaves - of sheaves with Chern characters in the secant plane remains of Hodge type under all Weil-type deformations. If the transformed object is semiregular (a condition ensuring that it deforms along with the Weil-type data), algebraicity of all Weil classes on the entire moduli component follows; this semiregularity has been verified only for $\dim X\le 3$, which forces $K$ imaginary quadratic.

Problem Statement

Let $K$ be a CM field, i.e. a totally imaginary quadratic extension of a totally real field $F$, and set $e=[K:\mathbb{Q}]$. A polarized abelian variety of Weil type is a triple $(A,\eta,h)$ consisting of a complex abelian variety $A$, an algebra embedding $\eta\colon K\to\mathrm{End}^{0}(A)$ such that the rank-$d$ $K$-module $H^{1}(A,\mathbb{Q})$ is balanced ($\dim H^{1,0}_{\sigma}=\dim H^{0,1}_{\sigma}=d/2$ for every embedding $\sigma\colon K\hookrightarrow\mathbb{C}$), and an ample class $h$ compatible with $\eta$ in that the Rosati involution acts on $\eta(K)$ as complex conjugation. The space of Weil classes

$$HW(A,\eta)=\wedge^{d}_{K}H^{1}(A,\mathbb{Q})\;\subset\;H^{d/2,d/2}(A,\mathbb{Q})$$

is a $K$-line, $e$-dimensional over $\mathbb{Q}$, and consists of Hodge (indeed absolute Hodge) classes.

Determine for which polarized abelian varieties of Weil type the Weil classes are algebraic, i.e. spanned over $\mathbb{Q}$ by cohomology classes of algebraic cycles. Equivalently, settle Question 1.1 of Markman's survey (arXiv:2509.23403): does $HW(A,\eta)$ consist of algebraic classes for every triple $(A,\eta,h)$ of Weil type? The problem is posed in its natural generality - arbitrary CM field, arbitrary even dimension, arbitrary discriminant. The established boundary cases are: algebraicity is already proved for every abelian fourfold of Weil type with $K$ imaginary quadratic (all discriminants) and for abelian sixfolds of Weil type of discriminant $-1$ with $K$ imaginary quadratic, while sixfolds of nontrivial discriminant, all dimensions $\ge 8$, and all CM fields with $[K:\mathbb{Q}]>2$ are open.

References

1. A. Weil, Abelian varieties and the Hodge ring (1977), in Œuvres scientifiques / Collected Papers, Vol. III, Springer, 1979. DOI: 10.1007/978-1-4757-1705-1_123, https://doi.org/10.1007/978-1-4757-1705-1_123
2. P. Deligne (notes by J. S. Milne), Hodge cycles on abelian varieties, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer, 1982, pp. 9–100. DOI: 10.1007/BFb0091629, https://doi.org/10.1007/BFb0091629
3. B. J. J. Moonen, 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
4. B. J. J. Moonen, Yu. G. Zarhin, Hodge classes and Tate classes on simple abelian fourfolds, Duke Mathematical Journal 77 (1995), no. 3. DOI: 10.1215/s0012-7094-95-07717-5, https://doi.org/10.1215/s0012-7094-95-07717-5
5. B. J. J. Moonen, Yu. G. Zarhin, Hodge classes on abelian varieties of low dimension, Mathematische Annalen 315 (1999), no. 4, 711–733. DOI: 10.1007/s002080050333, arXiv: math/9901113, https://arxiv.org/abs/math/9901113
6. C. Schoen, Hodge classes on self-products of a variety with an automorphism, Compositio Mathematica 65 (1988), no. 1, 3–32. EuDML: https://eudml.org/doc/89882, NumDAM: http://www.numdam.org/item?id=CM_1988__65_1_3_0
7. C. 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
8. K. 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
9. Y. André, Une remarque à propos des cycles de Hodge de type CM, in Séminaire de Théorie des Nombres, Paris 1989–90, Progress in Mathematics 102, Birkhäuser, 1992, pp. 1–7. DOI: 10.1007/978-1-4757-4269-5_1, https://doi.org/10.1007/978-1-4757-4269-5_1
10. E. 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
11. E. Markman, Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds, arXiv:2502.03415 (2025, v2 June 2025). arXiv: 2502.03415, https://arxiv.org/abs/2502.03415
12. E. Markman, Secant sheaves on abelian n-folds with real multiplication and Weil classes on abelian 2n-folds with complex multiplication, arXiv:2509.23079 (2025). arXiv: 2509.23079, https://arxiv.org/abs/2509.23079
13. E. Markman, Secant sheaves and Weil classes on abelian varieties, arXiv:2509.23403 (2025; v2 February 2026). arXiv: 2509.23403, https://arxiv.org/abs/2509.23403
14. S. Floccari, L. Fu, The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties, arXiv:2504.13607 (2025). arXiv: 2504.13607, https://arxiv.org/abs/2504.13607
15. B. van Geemen, Weil Classes and Decomposable Abelian Fourfolds, arXiv:2108.02087 (2021). arXiv: 2108.02087, https://arxiv.org/abs/2108.02087
16. B. van Geemen, A. Rapagnetta, Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class, arXiv:2607.18341 (2026). arXiv: 2607.18341, https://arxiv.org/abs/2607.18341
17. J. S. Milne, Hodge classes on abelian varieties, arXiv:2010.08857 (2020). arXiv: 2010.08857, https://arxiv.org/abs/2010.08857
18. J. S. Milne, The Tate and Standard Conjectures for Certain Abelian Varieties, arXiv:2112.12815 (2021; v2 2022). arXiv: 2112.12815, https://arxiv.org/abs/2112.12815
19. A. Mostaed, McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds, arXiv:2603.20268 (2026). arXiv: 2603.20268, https://arxiv.org/abs/2603.20268
20. I. Zharkov, Tropical Abelian varieties, Weil classes and the Hodge Conjecture, arXiv:2002.02347 (2020). arXiv: 2002.02347, https://arxiv.org/abs/2002.02347