The rational Hodge conjecture for CM abelian varieties

1 · The essentials

The main manuscript concerns complex geometric objects called abelian varieties, with a special kind of extra symmetry called complex multiplication. It asks whether certain features detected by mathematical measurements of their shape come from pieces defined by algebraic equations. The claimed answer is yes: each such feature can be expressed by combining those pieces with rational-number weights. The claim covers every dimension and every size of piece within this special class of objects. The manuscript also claims consequences for related questions, including questions about abelian varieties over finite fields. Companion manuscripts address further families, sometimes under additional assumptions and sometimes only at suitably general members of a family. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a description of a mosaic given through measurements rather than through its individual tiles. The claim says that every measurement pattern of the specified kind can be reconstructed from actual tiles, allowing rational weights. Here the “tiles” are algebraic pieces, and the image does not capture the precise restrictions on which patterns qualify.

Why it matters

The claimed result connects rational Hodge classes with algebraic cycles across the entire class of complex CM abelian varieties, without a dimension or codimension bound. The repository summary attributes further consequences to Milne’s theorems: the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. The companion manuscripts indicate how this result enters other Hodge-conjecture questions, while retaining important geometric and genericity hypotheses.

2 · Going further

The precise main claim is that every rational Hodge class on a complex abelian variety with complex multiplication is a rational linear combination of algebraic cycle classes. This is the rational Hodge conjecture for CM abelian varieties, asserted in every dimension and codimension. The main abstract also claims the generalized Hodge conjecture for these varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic. The repository summary identifies Milne’s theorems as the bridge to the latter two consequences. Neither the main abstract nor the summary supplies the construction or argument establishing the central algebraicity claim, so its proof strategy cannot be reconstructed from this source.

The K3 companion claims the rational and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space admits a rational isometric embedding into $U_{Q}^{\oplus 2} ⊥ ⟨ - 1 ⟩^{4}$ . This includes every ample $P$ -polarized K3 surface for $P = U \oplus D_{8} (- 1) \oplus D_{4} (- 1)$ , including Picard jumps. Its stated mechanism is the construction of algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor on this locus. More generally, it claims that algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power of any projective complex K3 surface.

The abelian-powers companion explicitly uses the CM result. It claims the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature $(3, 3)$ . It also claims the same conclusion for complex abelian varieties of dimension at most five admitting an imaginary-quadratic action, including nonsimple varieties and special periods with additional endomorphisms. The covers companion claims the conjecture for every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves, in every base genus and compatible branching pattern. On the same locus it includes every self-power of the product of that Jacobian with any CM abelian variety; separately, it treats every self-power of a very general member of the full smooth labelled family of diagonal complete intersections defined by at most two equations of a common degree, in every dimension and every degree at least two.

The conditional Kuga–Satake manuscript isolates a missing input rather than asserting its existence. For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, it claims that one algebraic correspondence inducing a nonzero map to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. It also gives an equivalent condition using holomorphic one-forms on a generically finite surface cover. If the input holds very generally in a polarized component, the claimed conclusion extends throughout that component for every choice of standard data on the transcendental part; the initial correspondence remains a hypothesis.

Useful notions

Complex abelian varieties and complex multiplication
Rational cohomology and Hodge classes
Algebraic cycles and their classes
K3 surfaces and algebraic correspondences
Self-products and genericity conditions

What was known before / what is new

Before

The source frames the questions through the rational and generalized Hodge conjectures, the Tate conjecture, and the Hodge standard conjecture. It cites Milne’s theorems as providing implications from the main result to the finite-field Tate and Hodge standard conclusions. It does not give a historical account or describe which special cases were previously established.

New

The central claimed advance is the rational Hodge conjecture for every complex CM abelian variety, in all dimensions and codimensions. The companion manuscripts claim results for specified K3 surfaces, abelian varieties with imaginary-quadratic actions, Jacobians of abelian covers, and diagonal complete intersections. They also identify a conditional route to full Kuga–Satake correspondences from a single initial algebraic correspondence.

Limits and assumptions

The main claim concerns rational classes on complex abelian varieties with complex multiplication; it is not a claim of the rational Hodge conjecture for all complex varieties. The companion results retain their stated restrictions: a quadratic embedding condition for the unconditional K3 locus, dimension and Hermitian-form conditions in the imaginary-quadratic setting, tensor Hodge-generic points for the Jacobian families, and very general members for the stated diagonal complete intersections. The conditional Kuga–Satake manuscript leaves existence of the initial algebraic correspondence as a hypothesis. The supplied abstracts do not give the main argument or establish that this missing input exists generally. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

CM abelian variety: An abelian variety with complex multiplication, the extra endomorphism structure specifying the class treated by the main manuscript.
Rational Hodge class: A rational cohomology class of the Hodge type considered by the rational Hodge conjecture.
Algebraic cycle class: A cohomology class associated with a combination of algebraic subvarieties.
Self-power: A product of a variety with copies of itself.
Algebraic correspondence: An algebraic cycle on a product of varieties used to induce a map between their cohomology groups.

Keywords : Hodge conjecture · CM abelian varieties · Kuga–Satake correspondences

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

10/09/2026

To be verified

For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover. If this input holds very generally in a polarized component, the prescribed correspondence is algebraic throughout that component, for every choice of standard data on the transcendental part. The existence of the initial correspondence remains a hypothesis.

PDF GitHub folder