The rational Hodge conjecture for CM abelian varieties

Introduction

For a smooth projective complex variety $X$, the rational Hodge conjecture asks whether every rational cohomology class of type $(p,p)$ is a rational linear combination of codimension-$p$ algebraic cycle classes. We prove this assertion for abelian varieties with complex multiplication, in every dimension and codimension.

A complex abelian variety $A$ is of CM type, or simply CM, if $\operatorname{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a commutative semisimple $\mathbb{Q}$-algebra of dimension $2\dim A$. We include the point by convention. Finite products and powers of CM abelian varieties are again CM.

Theorem 1.1. Let $A$ be a complex CM abelian variety. For every integer $p \ge0$, the Betti cycle-class map

is surjective, where $\operatorname{CH}^{p}(A)_{\mathbb{Q}} = \operatorname{CH}^{p}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ and the intersection is taken in $H^{2p}(A,\mathbb{C})$. In particular, the rational Hodge conjecture holds in every codimension on every finite product of complex CM abelian varieties and on every power of such a product.

The theorem also has consequences beyond the algebraicity of individual CM Hodge classes. The generalized Hodge conjecture predicts algebraic supports of the codimension allowed by the Hodge types of a cohomology substructure. We obtain it for CM abelian varieties by applying the theorem to correspondences from auxiliary CM varieties, following Hazama and Abdulali [?, ?]. Two theorems of Milne then give the Tate conjecture for all abelian varieties over finite fields, which asks for algebraic representatives of Galois-invariant $\ell$-adic classes, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic, which asserts positivity of the intersection form on primitive algebraic classes [?, ?]. Section 8 states these conclusions with their coefficient and support conventions and explains the separate implications.

Hodge classes on CM abelian varieties

The codimension-one case of the Hodge conjecture is the Lefschetz (1, 1) theorem. Hodge formulated the broader problem by asking for an extension of this algebraicity criterion to higher-dimensional cycles [?]. Cup products of divisor classes therefore give algebraic Hodge classes in higher degree. For an abelian variety, $H^{*}(A,\mathbb{Q})=\Lambda^{*}H^{1}(A,\mathbb{Q})$, but this description of its cohomology does not make every Hodge class a linear combination of divisor products. The problem is to construct cycles for the remaining classes.

Weil classes provide an important instance of this difficulty. Suppose an imaginary quadratic field $K$ acts on an abelian variety of dimension $2n$, with its two embeddings occurring with equal multiplicity in $H^{1,0}$. Then $\Lambda_{K}^{2n}H^{1}(A,\mathbb{Q})$ supplies a two-dimensional rational space of classes of type $(n,n)$. These determinant classes and their generalization to CM fields are discussed in [?]. Their Hodge type follows from the multiplicity condition; algebraicity requires a geometric construction. Weil classes need not lie in the algebra generated by divisor classes; this occurs already on abelian fourfolds [?]. Recent work of Markman proves algebraicity of the Weil classes on polarized abelian sixfolds of Weil type for an imaginary quadratic field, with polarization discriminant $-1$, and deduces the Hodge conjecture for all abelian fourfolds [?]. The sixfold theorem assumes an imaginary quadratic action, without requiring the full CM condition of Theorem 1.1.

Deligne proved that every Hodge class on an abelian variety is absolute Hodge: after transport to a conjugate variety through de Rham cohomology, it remains a Hodge class [?]. This establishes an arithmetic compatibility shared by algebraic cycle classes, but does not by itself produce algebraic cycles. CM varieties play a central role in that argument. Their Hodge groups are tori, and their degree-one cohomology splits, after extending coefficients, into lines indexed by field embeddings. An embedding-line tensor belongs to the scalar extension of the rational Hodge classes exactly when every scalar conjugate has equally many holomorphic and antiholomorphic factors. This is Pohlmann’s balance criterion [?]; see [?] and [?].

The CM arguments of Deligne and André reduce these Hodge classes to Weil-type classes on auxiliary CM abelian varieties [?, ?]. More precisely, the formulation in [?], attributed there to André, expresses a CM Hodge class as a sum of pullbacks of split Weil classes. Here split refers to the Hermitian polarization form having a totally isotropic subspace of half its dimension. This reduction identifies classes whose algebraicity would suffice; it does not supply their algebraic representatives in all dimensions.

There is also a reduction to relations involving only four labels. Hazama’s arrangement method spans its rational balance kernel by antipodal pairs and four-label square relations [?]; his Corollary 7.14.1 gives the corresponding codimension-two algebraicity reduction. Gao and Ullmo give equivalent quadratic indicator relations and use them, after adjoining auxiliary CM factors, to generate Hodge-induced period relations [?]. Their conclusion concerns relations among periods. Our task is to construct algebraic representatives for the corresponding four-factor tensors and to compose those representatives as correspondences. This distinction matters: the reduction uses auxiliary factors and polarization contractions, rather than asserting that low-codimension classes generate the Hodge ring of a fixed power.

From balanced tensors to surface periods

Choose a Galois CM field $E$ containing the CM fields in $H^{1}(A,\mathbb{Q})$, and write $G_{E}=\operatorname{Gal}(E/\mathbb{Q})$ and $c$ for its complex conjugation. An odd sign function $v:G_{E}\to\{1,-1\}$, satisfying $v(c\tau)=-v(\tau)$, specifies a weight-one Hodge structure $U(v)$ on the rational vector space $E$. Its embedding line $U(v)_\lambda$, on which $a \in E$ acts as $\lambda(a)$, has type $(1, 0)$ or $(0, 1)$ according as $v(\lambda)$ is positive or negative. Section 2 realizes $U(v) = H^1(B(v), \mathbb{Q})$ for a CM abelian variety $B(v)$.

The elementary relation needed in the proof is

It makes every scalar conjugate of the four-factor line

have type $(2, 2)$. We prove that this line lies in the scalar extension of the rational cycle span. Section 2 then assembles arbitrary balanced tensors from these relations: two-row switches connect lists with the same sum, and algebraic polarization pairings contract their intermediate factors. Transfer back to the original abelian variety and descent of coefficients give Theorem 1.1.

The geometric construction of a four-factor class takes place on a surface. Suppose there are rational Hodge maps $h_i: U(v_i) \to H^1(S,\mathbb{Q})$, for a smooth projective surface $S$, and vectors $\alpha_i$ on the four selected embedding lines such that

Degree-one Hodge maps are induced, up to rational multiples, by morphisms $S \to B(v_i)$. The image of $S$ in their product therefore gives an algebraic cycle whose pairing with the desired line is nonzero. Section 3 shows how four algebraic divisor kernels convert this functional into a vector on that exact line. The remaining problem is thus to find four degree-one classes with both a nonzero surface period and the prescribed rational CM sources.

Theta forms and their CM sources

Section 4 constructs the period on a compact arithmetic quotient of the complex two-ball. Its unitary group has signature $(2, 1)$ at one distinguished embedding and is definite at all the other real places. After the elementary normalization in Section 2, all four lines $U(v_i)_1$ are holomorphic at that embedding. To each type the construction assigns a family of holomorphic theta one-forms. Their values span a cotangent fiber, so two such families supply a wedge that is a nonzero holomorphic two-form.

To compare the two sides of the four-factor relation, we realize each pair of types by an orthogonal decomposition of the same Hermitian plane. The balance relation says that the two decompositions have the same local sign multiplicities. Weak approximation produces a rational second decomposition whose compact-place data are as close as desired to those of the first. At the distinguished embedding, the total period data depend only on the plane and remain unchanged.

A rank-two theta tensor packages the wedges into one total two-form. We keep its finite theta input and the compact quotient fixed while varying the compact-place data. The total form varies continuously, so its pairing with the initial nonzero form stays close to the nonzero self-pairing. At the two rational endpoints it is a finite sum of wedges of the required one-forms. Factoring these endpoints and passing to a common finite cover therefore gives a nonzero mixed period with two holomorphic and two antiholomorphic factors.

The construction uses Weil’s rational theta covariance and the unitary splittings of the Weil representation [37, ?, ?]. Kudla and Millson developed differential-form theta kernels in this unitary setting [?] (Introduction and Theorem 3). The projection turning the rank-two tensor into wedges has a close predecessor in the pure-cocycle product calculation of Bergeron, Millson, and Mœglin [?] (Lemmas 5.7 and 5.11). We derive the required tensor identity in coordinates and use the fixed-input variation to obtain the mixed period.

These are initially complex one-form classes. The surface criterion also requires that they come from specified rational CM Hodge structures. Sections 5 and 6 establish this arithmetic input. First we realize the ball quotients in a projective moduli space and construct algebraic Hecke endomorphisms of the product of its components’ Albanese varieties. Counting ordinary isogeny neighbors gives a polynomial relation between these operators and Frobenius. We then compute the Hecke parameters of the theta forms. The Frobenius relation and the degree-one Weil weight force slope zero or one on the relevant Hecke summands, with the choice determined by the prescribed CM signs.

The Hecke projectors preserve the Hodge filtration, so weak admissibility turns these extreme slopes into pure Hodge types. Applying the calculation to every scalar conjugate identifies the complete CM type of each summand. Scalar descent then expresses every relevant theta class as a complex linear combination of images of the selected CM line under rational Hodge maps. Section 7 puts the finite families on one projective surface, expands the nonzero period, and applies the surface criterion. This completes the four-factor construction and hence the proof of the main theorem.

CM tensors and four-factor relations

This section reduces Theorem 1.1 to the algebraicity of a specified family of tensors in the fourth cohomology of products of four CM abelian varieties. The reduction changes the CM types of individual factors while preserving their total Hodge type; its intermediate factors may therefore be absent from the original variety. We first record how to work with individual embedding lines and recover rationality after extending coefficients. We then construct the auxiliary CM varieties and assemble arbitrary balanced tensors from four-factor relations by algebraic correspondences.

Scalar coefficients and degree-one maps

For a smooth projective complex variety $X$, write

Cohomology is Betti cohomology unless another theory is specified. We fix $\overline{\mathbb{Q}}\subset\mathbb{C}$, and write $T_k=T\otimes_{\mathbb{Q}}k$ for a rational vector space $T$. A class with coefficients in $\overline{\mathbb{Q}}$ or $\mathbb{C}$ is called algebraic if it belongs to the corresponding scalar extension of $Z^{2p}(X)$. Scalar conjugation therefore preserves algebraicity on the same complex variety: it fixes each rational cycle class and changes only its coefficient. We use the compatibility of cycle classes with exterior products, cup products, pullback between smooth varieties, and proper pushforward.

Our Tate convention is $V(r)^{a,b}=V^{a+r,b+r}$, so $V(r)$ has weight $\operatorname{wt}(V)-2r$.

Lemma 2.1 (Scalar descent). Let $T$ be a finite-dimensional rational vector space and $P\subset T_{\mathbb{C}}$ a complex-linear subspace. If $t\in T_{\overline{\mathbb{Q}}}$ and $\sigma t\in P$ for every $\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, then

The same assertion applies to the subspace of maps of any fixed Hodge type in a Hom space.

Proof. Choose a finite Galois extension $K/\mathbb{Q}$ containing the coefficients of $t$. For $a\in K$, the vector

is rational. Each automorphism of $K$ extends to $\overline{\mathbb{Q}}$, so every summand belongs to $P$; hence $T_a\in T\cap P$. If $b_1,\ldots,b_r$ is a rational basis of $K$ and $b_1^*,\ldots,b_r^*$ is its trace-dual basis, write $t=\sum_j t_jb_j$ with $t_j\in T$. Then $T_{b_j^*}=t_j$, which proves the assertion. The condition that a complex-linear map have a fixed Hodge type is complex-linear, so the same argument applies in a Hom space.

We will also use

Indeed, extend a rational basis of $Z^{2p}(X)$ to one of $H^{2p}(X,\mathbb{Q})$ and compare coefficients.

Lemma 2.2 (Realization in degree one). For complex abelian varieties $A$ and $B$, pullback induces an isomorphism

If $S$ is a smooth connected projective complex variety and $h:H^1(B,\mathbb{Q})\to H^1(S,\mathbb{Q})$ is a rational Hodge map, then there are an integer $m\geq1$ and a morphism $f:S\to B$ such that $f^*=mh$ on $H^1(B,\mathbb{Q})$.

Proof. Dualizing a rational Hodge map gives a rational map on homology commuting with the complex structures. Multiplication by a positive integer makes it carry the integral lattice of $A$ into that of $B$. It consequently induces a holomorphic homomorphism of complex tori $A^{\mathrm{an}}\to B^{\mathrm{an}}$, which is algebraic. Conversely, an algebraic homomorphism induces the indicated Hodge pullback. This proves the first assertion; equivalently it is the dual of the rational form of Riemann’s theorem [?], Proposition 6.5, Theorems 6.7–6.8, and Corollary 6.9.

The Albanese map of $S$ induces an isomorphism $H^1(\operatorname{Alb}S,\mathbb{Q})\simeq H^1(S,\mathbb{Q})$. Apply the first assertion to the map with target $H^1(\operatorname{Alb}S,\mathbb{Q})$, clear a denominator, and compose the resulting homomorphism $\operatorname{Alb}S\to B$ with an Albanese map.

The Lefschetz $(1,1)$ theorem says that $\operatorname{Hdg}^{2}(X)=Z^{2}(X)$. In conjunction with Lemma 2.1, it has the following useful consequence: if $t\in H^2(X,\overline{\mathbb{Q}})$ is of type $(1,1)$ in every scalar conjugate, then $t$ is algebraic. This observation will supply all two-factor tensors used below.

Rank-one CM structures

For a polarizable rational Hodge structure $H$ of weight one, let $J(H)$ be the smallest rational algebraic subgroup of $\operatorname{GL}(H)$ containing the Hodge circle. The circle acts by $z$ on $H^{1,0}$ and by $\bar{z}$ on $H^{0,1}$, for $|z| = 1$. It preserves an alternating polarization. Its connectedness shows that it lies in the rational identity component of $J(H)$; minimality therefore makes $J(H)$ connected.

Lemma 2.3. For a complex abelian variety $A$, the following conditions are equivalent: