2026-09-24 · article Hodge conjectureCM abelian varietiesalgebraic cycles

The Hodge conjecture — down to a single class

The Hodge conjecture is one of the seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000 (official problem page). A complex projective variety, a shape cut out by polynomial equations over the complex numbers, has cohomology, a way of measuring its shape by numbers, and inside it a distinguished part that respects the complex structure: the Hodge classes. The conjecture asks whether every Hodge class comes from algebraic subvarieties, the smaller shapes cut out inside it by further polynomial equations. Building a Hodge class from a subvariety is easy; no general way back is known. This page restricts the question to one special family, the CM abelian varieties, and counts which classes are at stake. It does not solve the conjecture.

Leanmachine-checked (Lean 4, no sorry, no native_decide; axioms a subset of propext, Classical.choice, Quot.sound; theorem names given) paperproved, not machine-checked computationchecked on this machine, within the range stated; not a claim made to the outside knowna known theorem, a restatement, or a check of the literature

Nothing here is a claim about the Hodge conjecture itself. This is combinatorics and inequalities; not one algebraic cycle (a formal sum of algebraic subvarieties, the object the conjecture says must exist) is constructed. "Not generated by divisors" does not mean "not algebraic".

On a smooth complex projective variety X, is every class in H2p(X, Q) of type (p,p) a rational combination of classes of algebraic subvarieties of codimension p?

The difficulty is one-sided: producing Hodge classes from subvarieties is easy, and there is no general procedure in the other direction. It is open for abelian varieties, but there is a family where the Hodge classes can be written down completely: abelian varieties with complex multiplication (CM), those with as many endomorphisms, maps from the variety to itself, as possible. The classes become finite combinatorics, so one can count which of them cannot be built from divisors. Algebraicity still lies outside the count. known

So on the CM side there is ample reason to expect algebraicity and no procedure to exhibit it. What follows is a search for such a procedure at the smallest place where the naive expectation breaks.

This section is heavy on notation. The point is that the question about Hodge classes becomes a count over finite sets.

Let A be a CM abelian variety: CM field E (a number field of degree 2g acting on A), Galois closure L, G = Gal(L/Q), H = Gal(L/E), complex conjugation c central in G, X = G/H the 2g embeddings, and a CM type Φ with Φ ⊔ cΦ = X, a choice of half of the embeddings (this choice fixes the complex structure of A).

Pohlmann's criterion. The Hodge classes in H2p(A, Q) are spanned by the classes eT of the subsets T ⊂ X with known

Call such a T admissible. Sets of the form {x, cx} are always admissible; those are the divisors. Hence:

• generated by products of divisors ⟺ T admits a perfect matching into admissible 2-element sets. An admissible set with no such matching is an exceptional Hodge class.
• In sign vectors, with s = 1T − 1cT and u = 1Φ − 1cΦ, admissibility reads ∀σ : ⟨s, σu⟩ = 0 — a purely integral linear condition. It is the same as the known criterion "divisors fail to generate ⟺ there is a nonzero annihilator with coefficients in {0, ±1}" (White 1993). known
• Degeneracy: a CM type is degenerate when the Hodge group Hg(A), the group governing the Hodge classes, has rank smaller than the dimension. dim A − rank Hg(A) equals the number of odd characters vanishing on the type (Kubota 1965 / Ribet 1980, in the form given by Hazama 2003). known
• Weil classes, the family of exceptional classes known longest: for a CM subfield F ⊂ E, WF = ∧[E:F]F H1, sitting in codimension p = [E:F]/2 = g/[F:Q] (Moonen–Zarhin). known

Only the action of G on X and the image of c enter these conditions. Since c is a fixed-point-free involution of X, all such are conjugate in S2g with centraliser C2 ≀ Sg, and centrality of c gives G ⊆ C2 ≀ Sg. For fixed g there are finitely many cases, and they can be enumerated. paper

Up to here the expectation holds. It is nevertheless false, and classically known to be false:

• Lenstra's theorem: over an abelian CM field, a degenerate simple CM type forces a sporadic Hodge class on A itself (White 1993, Thm 3 / Yanai 2015, §3). known
• Yanai's 2-power criterion: A is of Weil type (the type on which Weil classes appear) exactly when the order of the vanishing odd character is a power of 2. Hence for odd g no Weil type is possible. known

Put together: a degenerate simple CM type in odd dimension carries exceptional classes and no Weil classes at all. The smallest odd dimension is g = 9, occurring for both Z/9 and (Z/3)2 Leancomputation; from the latter one also gets a series at g = p2 for every odd prime p paper. The first case is Q(ζ19), the cyclotomic field obtained by adjoining a 19th root of unity to the rationals, and it is the setting for everything below.

K = Q(ζ19), G ≅ Z/18, complex conjugation 9; there are 29 = 512 CM types.

Each of the 54 degenerate primitive types (types not induced from a smaller CM field) gives a simple 9-dimensional A, one that does not split into a product of smaller abelian varieties, with exactly two vanishing odd characters (of order 6). Fix one such A.

(i) Down to codimension 3. Hazama's dominating dimension d(A), the minimal ℓ1 weight of the annihilator lattice, equals 3: in Z[x]/(x9+1) the annihilator is an explicit rank-2 family with norm 3(|a|+|b|+|a−b|) ≥ 6, and 3 is attained. For an N-dominated variety, the Hodge conjecture for all powers of A reduces to codimension ≤ N. Leanpaper

(ii) Six classes, explicitly. With F the sextic CM subfield and Ck its cosets, the admissible sets of size 6 number 90 = 84 products of divisors + 6 exceptional, and the exceptional ones are exactly Tk = Ck ⊔ Ck+1 (k ∈ Z/6). Of the 15 unordered pairs of cosets, 9 are admissible: distance 1 gives the 6 exceptional, distance 3 the 3 products of divisors, distance 2 is not admissible. Lean

(iii) The six are a cube of an elliptic curve. Seen on H1, the span WF has Hodge weights (2,1,2,1,2,1), hence level 1, and its type ΦW = {0,2,4} is the subgroup of order 3 in Z/6 — an imprimitive type induced from Q(√−19). As rational Hodge structures

with E the CM elliptic curve of Q(√−19) (an elliptic curve is an abelian variety of dimension 1; here j = −884736, y2+y = x3−38x+90, conductor 361). A non-degenerate type is the negative control, a comparison case that fails the condition: level 1 as well, but its type is not closed under the shift and is not induced. Leanpaper

(iv) One class on a 10-fold. On A × E, the Hodge classes in H4 number 51 = 36 + 9 + 6: the 36 in H4(A)⊗H0(E) are products of divisors, the 9 in H2(A)⊗H2(E) are divisor × point, and the 6 in H3(A)⊗H1(E) are exceptional. In codimension 1 the count is 9 + 0 + 1, the mixed term vanishing. The same counts hold for all 54 degenerate primitive types; the other 458 are negative controls. Lean The exceptional part HomHS(H1(E), WF(1)) has rank 1 over F, and F ⊂ End0(A) is realised by algebraic correspondences, so if one of the six is algebraic then all six are, and so are the six exceptional classes of codimension 3 on A. paper The problem is now a single class ξ.

(v) That one class is already placed. ξ is the pullback of a split Weil class (a Weil class of an especially tractable kind, shown by Deligne to be absolutely Hodge) on an explicit 36-dimensional CM variety Leanpaper; hence it is absolutely Hodge unconditionally (Deligne 1982) and algebraic under the Lefschetz standard conjecture (André / Abdulali) known. The shortest route is closed: no two of the four CM types involved are complementary, so this split Weil class cannot be written as a product of two divisor classes. Lean In geometric terms, ξ is algebraic ⟺ WF ⊆ N1H3(A, Q), which is exactly the WF part of GHC(1,3). paper

Everything below is negative: a divisor Y supporting WF (one from whose cohomology WF comes) cannot be of this shape. Each branch carries a hypothesis, and the complement of that hypothesis is not shown to be empty.

The tool. For A simple, Y a reduced effective divisor and f : Ỹ → Y a resolution,

with Alb(Ỹ) the Albanese variety of Ỹ (the abelian variety canonically built from it) and JY the adjoint ideal, which measures how bad the singularities of Y are; call the second term the defect. Reaching WF needs defect ≥ 3; all of GHC(1,3) needs ≥ 30. paper

• Theta divisors. A carries a principal polarisation (the smallest ample divisor class an abelian variety can carry), unique up to isomorphism papercomputation; simplicity makes the theta divisor Θ, which represents it, irreducible, and an irreducible theta divisor is normal with rational singularities, a mild kind of singularity (Ein–Lazarsfeld 1997). The adjoint ideal is then trivial, the defect 0, and the Gysin image lands in the 144-dimensional part that is unconditionally of coniveau ≥ 1. Whether Θ is singular is irrelevant to this. The same argument kills divisors whose singular locus has dimension ≤ g−4, isolated singularities included. paper
• Singular locus a complete intersection of three ample divisors (c = 3). Positivity of intersection numbers forces the polarisation to be a common multiple, so Y is cut by a form in three sections; then Hodd(P2) = 0 kills the (1,3) component of the cylinder map — for every polarisation and every multiplicity. The squeeze must be taken with max, not with a sum (the sum version holds for about 3% of ample triples: negative control). This branch cannot be closed by an index count alone — among 758 ample quadruples, 5 force a defect by dimension — so what closes it is positivity, a tool independent of the index. paperLeancomputation
• Two ample divisors (c = 2). Same conclusion, different argument: Serre duality sends the defect to h1 rather than h0, so no positivity inequality is available, and the Koszul complex with the index theorem takes over. A count of places bounds the concentration index by 2 (sharp), which is incompatible with the 6 the defect demands. The non-normal version with a 7-dimensional complete-intersection singular locus closes the same way. paperLean
• A single curve, and cyclic covers. One curve fails on dimension count. And A is not an isogeny factor of the Jacobian (the abelian variety built from a curve) of a cyclic degree-19 cover branched at three points: its type matches none of the 60 types of that family. Lean Descending to A/μ19 changes nothing: that quotient is non-Gorenstein but terminal (both are names for properties of singularities; all 18 Reid–Tai ages exceed 3), and a toric resolution leaves H3 containing WF. What survives is only a reduction: the search may be restricted to μ19-invariant objects. Leanpaper
• The constructions circle back. If J = Alb(Ỹ) ~ Am × R with no A factor in R, then NS(J)⊗Q has no mixed term, so divisor classes have even degree in both factors. This parity is closed under products and preserved by pushforward, so as long as α*[Ỹ] lies in the subring generated by NS(J), the (1,3) Künneth component vanishes. That kills, at one stroke, eightfold sums of a curve, symmetric products, complete intersections inside A, Thom–Porteous degeneracy loci, families in a linear system and families of translates. The surviving condition is "already possess a non-divisorial algebraic class", which is the statement to be proved. The route is circular. paper

Every closed branch uses one of: W cut out by ample divisors, Y cut from a linear system, α*[Ỹ] inside the subring generated by NS(J). Two branches lie outside all three.

Combining the standard conjecture with the defect inequality turns it into a concrete geometric prediction about A:

A carries a divisor with non-rational singularities, whose total failure of rationality is ≥ 3 (and ≥ 30 for all of GHC(1,3)).

Conversely, showing that every divisor on A is normal with rational singularities would refute the standard conjecture for A. Continuing to close (a) and (b) is therefore work pointed at a refutation, and neither is closed unconditionally, so no step in that direction has been taken.

What is in Lean 4 is the discrete core only. Namespace CMHodge; no sorry, no native_decide; every line of every axiom log is a subset of propext, Classical.choice, Quot.sound (all 26 theorems of CMHodgeLevelOne use propext alone, and seven CM-type lemmas in CMHodgeNormal depend on no axiom).

What this does and does not say. Lean holds finite enumerations, integral linear conditions, positivity of permanents and counts of real places. Hodge structures, algebraic cycles, adjoint ideals, Gysin maps, Serre duality, Kodaira vanishing, Grothendieck–Lefschetz and the Künneth decomposition are not formalised. The steps that use them are labelled "paper". So neither "ξ is algebraic" nor "ξ is not algebraic" is machine-checked; neither is claimed.

Known theorems used

Statements for which novelty is undetermined, but which are mathematically the case

These may well be known; nothing stronger than "not found in the literature searched" is claimed. None of them says anything about algebraicity.

Where the open items now stand is kept in what remains. Only the currently undecided is listed here.

• Every closed branch carries a hypothesis, and the complement is not shown to be empty. The generation degree of the ideal sheaf is an input of W, and there is no tool bounding it from above on the side of A.
• A necessary condition is only that. Neither "defect ≥ 3" nor "the pinch divisor is not ample" shows that no such object exists. Just as a positive Euler characteristic is no ground for existence, an inequality going through is no ground either — existence is settled in neither direction.
• Known theorems point the other way. ξ is absolutely Hodge unconditionally, and algebraic under the standard conjecture; concluding "does not reach" would require the list of branches to be complete, which it is not.
• If a counterexample is to be sought, the larger side comes first. GHC(1,3) asks the remaining 54 dimensions to be supported on a divisor as well — nine times larger (defect ≥ 27).
• Primary sources not obtained: Kubota 1965; Hazama 2000; the texts of Tankeev 1983 and Ribet 1983; Shioda 1981; the originals of André 1992, Deligne 1982 and Abdulali 1994 (used verbatim through a survey).

Why this road does not reach the conjecture itself

What is treated here is one class on one variety. Even were its algebraicity established, what follows is the Hodge conjecture for the powers of that A, by the reduction to codimension 3 — not for the CM family, and certainly not for projective varieties in general. And refuting it would refute the standard conjecture, which requires closing the two remaining branches unconditionally; both are closed only under hypotheses. What this page adds is closed roads, not open ones.

References

Range searched: degenerate CM types, exceptional Hodge classes, generalised and split Weil classes, adjoint ideals, the generalised Hodge conjecture, and the bibliographies of the papers above. Citation lists were not followed. Nothing here is called a first; the most that is said is "not found in the literature searched".