Algebraicity of Weil classes on abelian sixfolds of discriminant -1Research Paper
## Motivation The Hodge conjecture predicts that on a non-singular complex projective variety $X$ every rational cohomology class of type $(p,p)$ is a rational linear combination of the classes of algebraic subvarieties of $X$. Abelian varieties are the oldest testing ground for the conjecture, and the hardest known classes on them were isolated by A. Weil. A $2n$-dimensional complex abelian variety $A$ is **of Weil type** for an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$ if $K$ embeds into $\mathrm{End}_{\mathbb{Q}}(A)$ in such a way that both eigenspaces of $\eta(\sqrt{-d})$ meet $H^{1,0}(A)$ in an $n$-dimensional subspace. Such an $A$ carries a distinguished two-dimensional space of rational $(n,n)$-classes, the **Weil classes**, which for a generic $A$ of Weil type does not lie in the subring generated by divisor classes. Weil classes are therefore the standard obstruction to the Hodge conjecture in low dimension: for abelian fourfolds the conjecture reduces to their algebraicity. Timeline of the unconditional results on algebraicity of Weil classes: - A. Weil (1977) constructed the classes and showed that the generic abelian variety of Weil type has a three-dimensional space of rational $(n,n)$-classes, spanned by $h^n$ and the two-dimensional Weil plane. - C. Schoen (Compositio Math. 65 (1988) and its 1998 Addendum, Compositio Math. 114) proved algebraicity for fourfolds of Weil type with $K=\mathbb{Q}(\sqrt{-3})$ and arbitrary discriminant, for sixfolds with $K=\mathbb{Q}(\sqrt{-3})$ and trivial discriminant, and for fourfolds with $K=\mathbb{Q}(\sqrt{-1})$ and discriminant $-1$; a second proof of the last case is in B. van Geemen's survey (1994). - K. Koike (Canad. Math. Bull. 47 (2004)) proved algebraicity for sixfolds with $K=\mathbb{Q}(\sqrt{-1})$ and discriminant $-1$, which yields fourfolds with $K=\mathbb{Q}(\sqrt{-1})$ and arbitrary discriminant. - E. Markman (JEMS 25 (2023)) proved algebraicity for fourfolds, arbitrary $K$, and discriminant $1$. - E. Markman, [arXiv:2502.03415](https://arxiv.org/abs/2502.03415), proves algebraicity for **sixfolds of discriminant $-1$ and every imaginary quadratic $K$**, and deduces the Hodge conjecture for all abelian fourfolds. ## Setting Fix $g$ and present a $g$-dimensional complex torus by its first homology: a complex structure $J$ on $H_1(A,\mathbb{R})=\mathbb{R}^{2g}$ with $J^2=-1$, the lattice being $\mathbb{Z}^{2g}\subset\mathbb{R}^{2g}$. A **polarization** is a rational alternating form $E$ on $H_1(A,\mathbb{Q})=\mathbb{Q}^{2g}$ satisfying the Riemann relations $E(Jx,Jy)=E(x,y)$ and $E(x,Jx)>0$ for $x\neq 0$. Cohomology is $H^k(A,\mathbb{C})=\wedge^k H^1(A,\mathbb{C})$, realized as the alternating $\mathbb{C}$-multilinear forms on $H_1(A,\mathbb{C})=\mathbb{C}^{2g}$; a class is **rational** if its values on rational vectors are rational, and it has **type** $(p,q)$ if it is multiplied by $z^p\bar z^{q}$ under the scaling action $v\mapsto (\mathrm{Re}\,z)v+(\mathrm{Im}\,z)Jv$ of $z\in\mathbb{C}$. A **Hodge class** of degree $2p$ is a rational class of type $(p,p)$. Let $g=2n$ and $K=\mathbb{Q}(\sqrt{-d})$ with $d>0$. A **polarized abelian $2n$-fold of Weil type** is such an $(A,E)$ together with a rational endomorphism $M=\eta(\sqrt{-d})$ of $H_1(A,\mathbb{Q})$ with $$M^2=-d,\qquad MJ=JM,\qquad E(Mx,My)=d\,E(x,y),$$ the last condition saying that $\eta(k)$ multiplies the polarization class by the norm $\mathrm{Nm}(k)$, and such that each of the two eigenspaces $W,\overline{W}\subset H_1(A,\mathbb{C})$ of $M$ meets the $i$-eigenspace of $J$ in an $n$-dimensional subspace. The **Hodge–Weil classes** are the rational classes in $$\widehat{HW}\;=\;\big(\wedge^{2n}W\;\oplus\;\wedge^{2n}\overline{W}\big)\cap H^{2n}(A,\mathbb{Q}),$$ equivalently the rational degree-$2n$ classes that vanish on every tuple of vectors containing both a vector of $W$ and a vector of $\overline{W}$. The form $H(x,y)=E(Mx,y)+\sqrt{-d}\,E(x,y)$ is $K$-valued and hermitian on $H_1(A,\mathbb{Q})$, viewed as a $K$-vector space through $\eta$. The determinant of its Gram matrix in a $K$-basis lies in $\mathbb{Q}^{\times}$, and its class in $\mathbb{Q}^{\times}/\mathrm{Nm}(K^{\times})$ is the **discriminant** $\det H$ of $(A,\eta,h)$. Discriminant $-1$ means that this determinant equals $-(a^2+dc^2)$ for some rationals $a,c$ not both zero. **Algebraicity** is not modelled abstractly. An abelian variety is tied to a genuine non-singular projective variety $X\subseteq\mathbb{P}^N$ by a *projective realization*: a smooth, $\mathbb{Z}^{2g}$-periodic immersion $u:\mathbb{R}^{2g}\to X$, surjective onto $X$ and injective modulo the lattice, whose differential intertwines $J$ with the complex structure of $\mathbb{P}^N$. A class $w\in H^{2p}(A,\mathbb{C})$ is algebraic when some de Rham class of $X$ whose periods over the $2p$-cycles swept out by rational vectors $\lambda_1,\dots,\lambda_{2p}$ equal $w(\lambda_1,\dots,\lambda_{2p})$ is a rational combination of cycle classes $\mathrm{cl}(Z)$ of irreducible subvarieties $Z\subseteq X$ of dimension $g-p$. Cycle classes, de Rham cohomology of a projective variety, $(p,q)$-types and Hodge classes are taken from the platform's `HodgeConjecture` bundle, which formalizes §1 of Deligne's Clay problem description. ## Formalization targets ### Goal — Theorem 1.5.1 of arXiv:2502.03415 $$\text{For every } d>0:\ \text{the Hodge–Weil classes of a polarized abelian sixfold of Weil type with CM by }\mathbb{Q}(\sqrt{-d})\text{ and discriminant }-1\text{ are algebraic.}$$ The statement fixes neither the field $K$ nor the sixfold: it quantifies over every $d>0$, every polarized abelian sixfold of Weil type of discriminant $-1$, and every projective realization of it. ### Milestone — Weil's plane of Hodge–Weil classes (§1.1) $$\widehat{HW}\ \text{is a two-dimensional }\mathbb{Q}\text{-space, and each of its elements has type }(n,n).$$ ### Milestone — Schoen's degeneration step (§1.6) $$\text{Goal for all sixfolds of discriminant }-1\ \Longrightarrow\ \text{Hodge–Weil classes of every abelian fourfold of Weil type are algebraic,}$$ for every imaginary quadratic $K$ and every discriminant; this is the use made of Schoen's Proposition 10 (Compositio Math. 114 (1998)) in the paper. ### Milestone — Corollary 1.6.1 $$\text{The Hodge conjecture holds for abelian fourfolds.}$$ ### Milestone — Lefschetz $(1,1)$ Divisor classes: every Hodge class in $H^2$ of a non-singular projective variety is algebraic. This is an already published platform statement, imported here as a reference, since the reduction in Corollary 1.6.1 uses the algebraicity of divisor classes. ## Significance Weil classes are, by the results of Moonen–Zarhin (Duke Math. J. 77 (1995), Math. Ann. 315 (1999)), the only obstruction left in dimension four: for a simple abelian fourfold $H^{2,2}(A,\mathbb{Q})$ is spanned by quadratic expressions in divisor classes and by Weil classes, and the non-simple cases reduce to products treated by Ramón Marí (Collect. Math. 59 (2008)) and Moonen–Zarhin. The goal theorem therefore closes the Hodge conjecture for abelian fourfolds, the first dimension in which the conjecture for abelian varieties was open. Formalizing it produces, first, a reusable Lean model of polarized abelian varieties, of complex multiplication of Weil type, of the Hodge–Weil plane and of the discriminant, tied to an honest notion of algebraic cohomology class through projective realizations. None of these objects exists in Mathlib today. The result itself is proved in the source preprint and has no machine-checked proof; the milestones below are equally unformalized, including the classical statements of Weil and Schoen that the paper's Corollary depends on. ## Difficulty The naive attack — write down subvarieties whose classes span $\widehat{HW}$ — fails because Weil classes are not expressible through divisors: for a generic abelian variety of Weil type the Néron–Severi group is cyclic while $H^{n,n}(A,\mathbb{Q})$ is three-dimensional, so no product of divisor classes reaches the Weil plane. The source constructs instead a reflexive sheaf $\mathcal{E}$ on $X\times\hat X$, for $X$ the Jacobian of a genus-$3$ curve, whose characteristic class $\kappa(\mathcal{E})$ remains of Hodge type along all deformations of $(X\times\hat X,\eta,h)$ as a polarized abelian sixfold of Weil type, and deforms the pair over that moduli space using a semiregularity theorem for twisted sheaves. Each of these steps — Orlov's derived equivalence, spinor geometry of the Mukai lattice, semiregularity — is itself missing from Mathlib, which is why the milestone list stays on the Hodge-theoretic side of the argument rather than transcribing the sheaf-theoretic core. ## Formalization scope Conventions the Lean development commits to: 1. Complex tori are presented by $(J,E)$ on $\mathbb{R}^{2g}$ with the lattice $\mathbb{Z}^{2g}$; the polarization form is rational rather than integral, which is the isogeny-invariant form of the Riemann relations. 2. Cohomology is the space of alternating multilinear forms on $H_1(A,\mathbb{C})$, i.e. invariant forms on the torus; a period over a lattice cube is used to compare it with the de Rham cohomology of a projective realization. 3. The Weil condition is imposed symmetrically on both eigenspaces of $\eta(\sqrt{-d})$, so it does not depend on the choice of convention for $H^{1,0}$ versus $H^{0,1}$. 4. Discriminant $-1$ is stated as the existence of a $K$-basis in which the Gram determinant of $H$ is $-\mathrm{Nm}(k)$; changing the basis multiplies the determinant by a norm, so the condition is basis-independent. 5. Algebraicity always refers to cycle classes of subvarieties of an actual projective variety, in the sense of Deligne's formulation, never to an abstract subspace of "algebraic" classes; in particular the goal cannot be satisfied by exhibiting a formal object, and the hypotheses are satisfiable — abelian varieties of Weil type of discriminant $-1$ exist for every $K$, and abelian varieties admit projective realizations. A complete development needs, beyond what is drafted here: the spin representation of the Mukai lattice of an abelian $n$-fold, pure spinors and $K$-secant lines, Orlov's equivalence, Atiyah classes and semiregularity for twisted sheaves. Contributions establishing any of these, or proving the Hodge-theoretic milestones, are welcome. ## Selected references - E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. https://arxiv.org/abs/2502.03415 - A. Weil, *Abelian varieties and the Hodge ring*, Collected Papers III, Springer 1980, 421–429. - C. Schoen, *Hodge classes on self-products of a variety with an automorphism*, Compositio Math. 65 (1988), 3–32; *Addendum*, Compositio Math. 114 (1998), 329–336. https://eudml.org/doc/89880 - B. van Geemen, *An introduction to the Hodge conjecture for abelian varieties*, Lecture Notes in Math. 1594, Springer 1994, 233–252. https://doi.org/10.1007/BFb0094425 - K. Koike, *Algebraicity of some Weil Hodge classes*, Canad. Math. Bull. 47 (2004), 566–572. https://doi.org/10.4153/CMB-2004-055-3 - B. Moonen, Y. Zarhin, *Hodge classes and Tate classes on simple abelian fourfolds*, Duke Math. J. 77 (1995), 553–581. https://doi.org/10.1215/S0012-7094-95-07717-5 - B. Moonen, Y. Zarhin, *Hodge classes on abelian varieties of low dimension*, Math. Ann. 315 (1999), 711–733. https://doi.org/10.1007/s002080050333 - J. Ramón Marí, *On the Hodge conjecture for products of certain surfaces*, Collect. Math. 59 (2008), 1–26. https://doi.org/10.1007/BF03191179 - E. Markman, *The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians*, J. Eur. Math. Soc. 25 (2023), 231–321. https://doi.org/10.4171/JEMS/1199 - P. Deligne, *The Hodge conjecture*, Clay Mathematics Institute Millennium Prize Problem description, 2000. https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf