Hodge–Weil Classes in CM Abelian Varieties

• Hodge–Weil classes are distinguished Hodge classes arising from abelian varieties with a compatible CM-field action, defined via K-linear exterior powers.
• They provide critical insights into algebraicity by distinguishing extra middle cohomology components not generated by standard divisor classes.
• Recent advances use secant sheaf constructions and rigid CM loci like McMullen’s curve to explore algebraicity and deformation obstructions in Weil type settings.

Searching arXiv for papers on Hodge–Weil classes, Weil type abelian varieties, and related algebraicity results. arXiv search query: "Hodge Weil classes abelian varieties Weil type secant sheaves Markman McMullen curve abelian sixfolds" Hodge–Weil classes are distinguished Hodge classes attached to abelian varieties carrying a compatible action of a CM-field, especially an imaginary quadratic field in the classical theory of abelian varieties of Weil type. In the general construction with $A=X\times \hat X$, $\dim X=g$, $K$ a CM-field, and $d:=4g/[K:\mathbb Q]$, one defines

$$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$

a $1$-dimensional $K$-vector space, hence a $[K:\mathbb Q]$-dimensional $\mathbb Q$-subspace, consisting of classes of type $(d/2,d/2)$ (Markman, 27 Sep 2025). In the classical imaginary quadratic case, Weil identified a $\dim X=g$0-dimensional space of rational classes of Hodge type $\dim X=g$1 in the middle cohomology of every $\dim X=g$2-dimensional abelian variety of Weil type; these are the Weil classes (Markman, 5 Feb 2025). In current usage, “Hodge–Weil classes” refers both to this classical $\dim X=g$3-dimensional situation and to its higher-degree analogues for general CM-fields, and these classes are typically exceptional in that they lie outside the subalgebra generated by divisor classes (Mostaed, 14 Mar 2026).

1. Definition through Weil type and $\dim X=g$4-exterior powers

An abelian variety of Weil type is a polarized abelian variety $\dim X=g$5 endowed with an embedding of a CM-field $\dim X=g$6 into $\dim X=g$7, such that the Rosati involution acts as complex conjugation on $\dim X=g$8, equivalently

$\dim X=g$9

for all $K$0, and such that for every embedding $K$1 the $K$2-isotypical components of $K$3 satisfy

$K$4

where $K$5 (Markman, 27 Sep 2025). In the imaginary quadratic case $K$6, this is the balanced signature condition underlying the classical theory of Weil classes (Markman, 5 Feb 2025).

The Hodge–Weil space is defined by taking the $K$7-linear exterior power of $K$8. In the general CM-field setting one has

$K$9

and for a generic polarized $d:=4g/[K:\mathbb Q]$0 of Weil type with $d:=4g/[K:\mathbb Q]$1,

$d:=4g/[K:\mathbb Q]$2

with $d:=4g/[K:\mathbb Q]$3 intersecting trivially the Lefschetz subalgebra generated by $d:=4g/[K:\mathbb Q]$4 (Markman, 27 Sep 2025). Thus Hodge–Weil classes are precisely the extra middle-degree Hodge classes contributed by the $d:=4g/[K:\mathbb Q]$5-action.

For abelian sixfolds of Weil type with $d:=4g/[K:\mathbb Q]$6, the paper on McMullen’s curve uses the notation

$d:=4g/[K:\mathbb Q]$7

which is a rank-$d:=4g/[K:\mathbb Q]$8 $d:=4g/[K:\mathbb Q]$9-submodule of $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$0, hence a $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$1-dimensional $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$2-subspace, and under the Weil signature condition its complexification lies in $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$3 (Mostaed, 14 Mar 2026).

2. Hodge-theoretic structure, Mumford–Tate groups, and split Weil classes

On abelian varieties, the Mumford–Tate group governs Hodge classes: Hodge classes on all powers of $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$4 are exactly the $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$5-invariants, and Deligne proved that Hodge classes on abelian varieties are absolute Hodge, meaning that they remain of type $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$6 under any field automorphism of $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$7 (Mostaed, 14 Mar 2026). Absolute Hodge does not imply algebraicity, and much of the modern theory of Hodge–Weil classes is organized around this distinction.

For CM-type abelian varieties, the cohomology admits a CM-eigenspace decomposition, and Pohlmann’s theorem characterizes degree-$$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$8 Hodge classes by a balanced condition on subsets $$HW(A,\eta):=\wedge^d_K H^1(A,\mathbb Q)\subset H^d(A,\mathbb Q),$$9: $1$0 Within this framework, if $1$1 is of Weil type for a CM-field $1$2, then

$1$3

consists of Hodge classes; moreover, André’s structural theorem, proved in the cited paper following Deligne and André, states that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes (Milne, 2020).

The split condition is imposed on the $1$4-Hermitian form attached to a compatible polarization. A polarized abelian variety $1$5 is of split Weil type if the Hermitian form admits a totally isotropic $1$6-subspace of half dimension (Markman, 27 Sep 2025). This notion is central because split Weil classes are the cases for which current algebraicity arguments are strongest. In Deligne’s language, split Weil classes are accessible; in the same source, accessibility of Hodge classes on abelian varieties is used together with a variational argument to show that Grothendieck’s standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties (Milne, 2020).

3. Secant sheaves, Spin representations, and algebraicity mechanisms

A major recent development is the secant-sheaf construction on $1$7. Starting with an abelian $1$8-fold $1$9 with real multiplication by a totally real field $K$0, an $K$1-bilinear polarization $K$2, and a CM-extension $K$3, one forms the pure spinor

$K$4

inside the half-spin module $K$5. This determines a maximal isotropic $K$6-subspace

$K$7

hence an embedding $K$8, and a rational secant subspace

$K$9

of dimension $[K:\mathbb Q]$0 generated by pure spinors associated to CM-types (Markman, 27 Sep 2025, Markman, 27 Sep 2025).

The derived-category input is Orlov’s equivalence

$[K:\mathbb Q]$1

Given coherent sheaves $[K:\mathbb Q]$2 on $[K:\mathbb Q]$3 with $[K:\mathbb Q]$4, one defines

$[K:\mathbb Q]$5

The class $[K:\mathbb Q]$6 is invariant under the relevant Spin subgroup and remains of Hodge type under deformations in the Weil-type period domain. A criterion phrased in terms of the $[K:\mathbb Q]$7-component of $[K:\mathbb Q]$8 ensures that the $[K:\mathbb Q]$9-part of $\mathbb Q$0 has nonzero projection to $\mathbb Q$1 (Markman, 27 Sep 2025).

For imaginary quadratic $\mathbb Q$2, this method is complete in the split cases emphasized by Markman. The paper “Secant sheaves and Weil classes on abelian varieties” proves algebraicity of the Weil classes on all polarized abelian sixfolds of split Weil type and on all abelian fourfolds of Weil type; it also states that the Hodge conjecture for abelian varieties of dimension at most $\mathbb Q$3 follows (Markman, 27 Sep 2025). In the more general CM-field setting, the strategy is established but semiregularity of the relevant object $\mathbb Q$4 is left open (Markman, 27 Sep 2025).

4. Fourfolds of Weil type

For abelian fourfolds, the geometry of Hodge–Weil classes is especially explicit. On a decomposable fourfold $\mathbb Q$5, where $\mathbb Q$6 is a general abelian surface, the space $\mathbb Q$7 is $\mathbb Q$8-dimensional and admits a concrete basis in terms of $\mathbb Q$9. The loci $(d/2,d/2)$0 arising from Weil-type structures sweep a determinantal cubic fourfold

$(d/2,d/2)$1

with singular locus a Veronese surface $(d/2,d/2)$2. A codimension-two Hodge class $(d/2,d/2)$3 deforms to a Hodge class in a family of abelian fourfolds of Weil type if and only if $(d/2,d/2)$4; if $(d/2,d/2)$5, then there is at most one Weil-type family realizing the deformation, while classes in $(d/2,d/2)$6 may deform in infinitely many families and are related to Markman’s Cayley classes (Geemen, 2021).

Algebraicity in dimension $(d/2,d/2)$7 is now established by several routes. One route uses singular OG6-varieties: the paper on discriminant $(d/2,d/2)$8 proves that the Hodge–Weil classes are algebraic for any abelian fourfold of Weil type with discriminant $(d/2,d/2)$9, and strengthens this to the Hodge conjecture for all powers of any such fourfold (Floccari et al., 18 Apr 2025). Another route passes through sixfolds: Markman proves that the Weil classes are algebraic for all abelian sixfolds of Weil type of discriminant $\dim X=g$00, for all imaginary quadratic number fields, and by a degeneration argument of C. Schoen deduces algebraicity for all abelian fourfolds of Weil type; the same paper states that the Hodge conjecture holds for abelian fourfolds (Markman, 5 Feb 2025). The secant-sheaf paper cited above reaches the same fourfold conclusion from the split sixfold construction (Markman, 27 Sep 2025).

5. Abelian sixfolds, the Weil locus, and McMullen’s curve

The most rigid recent setting for Hodge–Weil classes in dimension $\dim X=g$01 is the Hilbert modular sixfold

$\dim X=g$02

together with McMullen’s compact Kobayashi-geodesic curve $\dim X=g$03 arising from the triangle group $\dim X=g$04. The generic fiber $\dim X=g$05 along $\dim X=g$06 satisfies

$\dim X=g$07

so the generic point of $\dim X=g$08 carries no exceptional Hodge tensors beyond those generated by the polarization and the $\dim X=g$09-multiplication (Mostaed, 14 Mar 2026).

For a fixed imaginary quadratic field $\dim X=g$10, the Weil locus $\dim X=g$11 parametrizing abelian sixfolds of Weil type for $\dim X=g$12 has codimension $\dim X=g$13 and $\dim X=g$14 irreducible components, indexed by sign assignments with $\dim X=g$15. Since $\dim X=g$16, $\dim X=g$17, and $\dim X=g$18,

$\dim X=g$19

so any non-empty intersection is super-atypical in the sense of Zilber–Pink (Mostaed, 14 Mar 2026).

The main result is that $\dim X=g$20 is finite, possibly empty, and every intersection point is a CM point with

$\dim X=g$21

a CM field of degree $\dim X=g$22 satisfying

$\dim X=g$23

At such a point, the Hodge–Weil classes form a $\dim X=g$24-dimensional $\dim X=g$25-subspace in $\dim X=g$26 whose complexification lies in $\dim X=g$27; these classes are absolute Hodge, yet the paper states that they are inaccessible to all existing algebraicity theorems because of three independent obstructions: CM isolation, absence of a $\dim X=g$28-secant structure, and uncontrolled discriminant (Mostaed, 14 Mar 2026).

For $\dim X=g$29, so that $\dim X=g$30, the non-emptiness problem is reduced to a finite Hecke computation. With the prime $\dim X=g$31, one obtains $\dim X=g$32 candidate systems of algebraic equations, and the remaining open steps are to execute the $\dim X=g$33 computation, verify the Weil CM type of any resulting CM point, and prove algebraicity of the resulting Hodge–Weil classes (Mostaed, 14 Mar 2026).

6. Broader perspectives and open problems

The present landscape combines a strong structural theory with sharply delimited obstructions. On the one hand, for CM-type abelian varieties, Hodge classes can be expressed in terms of divisor classes and split Weil classes, and this gives a conceptual explanation for the centrality of split Weil type in current proofs (Milne, 2020). On the other hand, the McMullen-curve sixfolds show that absolute Hodge classes can occur in highly rigid CM-isolated situations where the secant-sheaf and semiregularity methods do not apply (Mostaed, 14 Mar 2026).

A separate line of inquiry uses tropical geometry. In a $\dim X=g$34-parameter family of tropical abelian fourfolds with an imaginary quadratic action of signature $\dim X=g$35, the paper on tropical abelian varieties writes down two explicit tropical Weil classes $\dim X=g$36 and formulates Kontsevich’s strategy for showing that such classes are not represented by tropical algebraic cycles. The current obstruction is that the proposed linear system can be solved only modulo the full lattice $\dim X=g$37, not modulo any proper sublattice, so the sought separation of tropical Hodge classes from tropical algebraic cycles has not been achieved (Zharkov, 2020).

This suggests a bifurcated research program. One branch seeks new algebraicity criteria, likely extending semiregularity or constructing secant sheaves for more general CM-fields and higher dimensions, as proposed in the general CM-field secant-sheaf framework (Markman, 27 Sep 2025). The other branch studies rigid arithmetic loci, such as $\dim X=g$38, where Hodge–Weil classes are explicit and absolute Hodge but remain outside the reach of current discriminant-sensitive, deformation-theoretic, and $\dim X=g$39-secant constructions (Mostaed, 14 Mar 2026). In both directions, Hodge–Weil classes remain a precise testing ground for the distinction between Hodge-theoretic existence and algebraic-cycle realization.