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Math-29 · mathematics · Algebraic geometry

Deligne's Absolute Hodge Conjecture

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

A rational Hodge class $\alpha\in H^{2p}(X,{\mathbb{Q}}(p))$ becomes type $(0,0)$ in the twisted Hodge structure. Via algebraic de Rham comparison it has a conjugate on $X^\sigma$ for each $\sigma\in\operatorname{Aut}({\mathbb{C}})$. Absoluteness means \[ \forall\sigma\in\operatorname{Aut}({\mathbb{C}}),\qquad \alpha^\sigma\in H^{2p}(X^\sigma,{\mathbb{Q}}(p))\text{ is a Hodge class}. \] The conjecture requires this for every rational Hodge class on every smooth projective complex variety. Arbitrary field automorphisms of ${\mathbb{C}}$, not only continuous ones or complex conjugation, are quantified.

Formulation and concept references: [S1].

Short English statement

Does every rational Hodge class remain rational and of Hodge type after every automorphism of the complex numbers?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-29-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-29

Manuscript 1 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 2 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 3 (family 032, 2026-10-03): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 4 (family 032, 2026-09-18): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 5 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 6 (family 032, 2026-10-04): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 7 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Deligne's Absolute Hodge Conjecture

OpenAI's claim

Manuscript 1 (family 032, 2026-09-30): For tensor-Hodge-generic abelian-cover Jacobians and the specified very general diagonal complete intersections, OpenAI claims the rational Hodge assertion on every self-power; Manuscript 2 (family 032, 2026-09-30): On the specified quadratic-lattice locus of projective K3 surfaces, all rational Hodge classes on every power are claimed to be algebraic; an exact algebraic Kuga--Satake tensor also gives a conditional criterion beyond that locus; Manuscript 3 (family 032, 2026-10-03): For every projective complex K3 surface, the precisely normalized Kuga--Satake embedding into the full even-Clifford abelian target is claimed to come from an algebraic cycle. Manuscript 4 (family 032, 2026-09-18): Every rational Weil class of a split eight-dimensional abelian variety is claimed to have an algebraic representative, uniformly over imaginary-quadratic fields and compatible polarizations; Manuscript 5 (family 032, 2026-09-30): The manuscripts claim algebraicity of rational Hodge classes on CM abelian varieties, with consequences for generalized Hodge classes and finite-field Tate classes; Manuscript 6 (family 032, 2026-10-04): Rational Hodge classes on any finite product of projective complex K3 surfaces are claimed to be algebraic, even when the factors have unrelated periods or extra endomorphisms. Manuscript 7 (family 032, 2026-09-30): Algebraicity is claimed for Hodge classes on powers of split Weil sixfolds and of abelian varieties of dimension at most five carrying an imaginary-quadratic action, including special periods.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix $n\ge2$, $d\ge0$, $c\in\{0,1,2\}$ and $m=d+c+1$. For a rank-$c$ complex matrix $A=(a_{\alpha i})$ with every maximal coordinate minor nonzero, define the smooth diagonal complete intersection \[ X_A=\{[x_1:\cdots:x_m]\in\mathbb P^{m-1}:\ \sum_i a_{\alpha i}x_i^n=0\quad(1\le\alpha\le c)\}. \] OpenAI asserts surjectivity of $\operatorname{CH}^p(X_A^u)_{\mathbb Q}\to H^{2p}(X_A^u,\mathbb Q)\cap H^{p,p}(X_A^u)$ for every $u\ge1$ and every $p$, outside a union, indexed by a countable set, of proper algebraic closed subsets of this smooth parameter space. The exceptional set is unnecessary when $c\le1$ or $d=0$. A second assertion concerns a fixed connected finite abelian topological cover of a compact oriented surface with specified branch monodromy, allowing unbranched and trivial covers. Vary the full marked complex structure and branch points. At any parameter $s$ for which each rational tensor of type $(a,a)$ in $H^1(C_s,\mathbb Q)^{\otimes 2a}$ keeps its Hodge type at every point of the marked family, the same cycle-class surjectivity holds on \[ (\operatorname{Jac}(C_s)\times M)^N\quad(N\ge1),\qquad C_s^a\times M^b\quad(a,b\ge0), \] for every complex CM abelian variety $M$. CM means that its degree-one Hodge group is a torus. This is a Hodge-generic assertion, with the generic locus chosen independently of $M$.
Mathematical claim 2.
For a projective complex K3 surface $S$, write $T_S=\operatorname{NS}(S)_{\mathbb Q}^{\perp}$ with cup-product quadratic form $q$. Assume a rational isometric embedding \[ (T_S,q)\hookrightarrow\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4, \] where $\mathbb U$ is the hyperbolic plane with Gram matrix $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$. Choose any full even-Clifford realization $H^1(A,\mathbb Q)=C^+(T_S,q)$, any $w\in T_S$ with $q(w)\ne0$, and any rational weight-one polarization. The Kuga--Satake map is formed from $v\mapsto(a\mapsto vaw)$ and this polarization. OpenAI asserts existence of $\Gamma\in\operatorname{CH}^2(S\times A^2)_{\mathbb Q}$ inducing this exact prescribed map on $T_S$, also after transporting it to an isogenous realization. It further claims rational Hodge and generalized Hodge conjectures in all degrees of $S^m$ for every $m\ge1$. The quadratic embedding is a hypothesis, so this theorem alone is restricted to that locus.
Mathematical claim 3.
Let $S$ be any smooth projective complex K3 surface and let $T(S)=\operatorname{NS}(S)_{\mathbb Q}^{\perp}$ in $H^2(S,\mathbb Q)$. Fix the full even-Clifford Kuga--Satake construction, including the rational quadratic polarization, parity identification and rational weight-one polarization, with associated abelian variety $A_S$. These choices specify $\kappa_S:T(S)\to H^2(A_S^2,\mathbb Q)$. OpenAI claims \[ \exists\Gamma_S\in\operatorname{CH}^2(S\times A_S^2)_{\mathbb Q}\quad (p_{A_S^2})_*\left(\operatorname{cl}(\Gamma_S)\smile p_S^*x\right)=\kappa_S(x) \quad\text{for all }x\in T(S). \] Here $\operatorname{CH}^2$ is the group of rational codimension-two cycles modulo rational equivalence and $p_S,p_{A_S^2}$ are the projections. The assertion includes every isogenous realization with the prescribed map transported by its rational Hodge isomorphism.
Mathematical claim 4.
Let $K=\mathbb Q(\sqrt{-d})$ be imaginary quadratic, $A$ a complex abelian variety of dimension eight with a $K$-action, and $\lambda$ a compatible polarization of Weil type, with eigenspace multiplicities $(4,4)$. Put $V=H_1(A,\mathbb Q)$, let $\psi$ be the rational Riemann form and $J$ multiplication by $\sqrt{-d}$. Assume the associated Hermitian form \[ h(x,y)=\psi(Jx,y)+\sqrt{-d}\,\psi(x,y) \] vanishes on a four-dimensional $K$-subspace of $V$ (the split condition). Define the rational Weil space $W_K(A)=\bigwedge_K^8H^1(A,\mathbb Q)\subset H^8(A,\mathbb Q)$ using the two conjugate $K$-eigenspace exterior powers. OpenAI claims \[ W_K(A)\subset\operatorname{im}\left(\operatorname{CH}^4(A)_{\mathbb Q}\xrightarrow{\operatorname{cl}}H^8(A,\mathbb Q)\right). \] The field and compatible polarization type are arbitrary; extra endomorphisms are allowed. The statement concerns these Weil classes, rather than all Hodge classes on all varieties.
Mathematical claim 5.
A complex abelian variety $A$ is CM if $\operatorname{End}(A)\otimes\mathbb Q$ contains a commutative semisimple algebra of dimension $2\dim A$. For every such $A$ and every $p\ge0$, OpenAI claims \[ \operatorname{CH}^p(A)_{\mathbb Q}\longrightarrow H^{2p}(A,\mathbb Q)\cap H^{p,p}(A) \quad\text{is surjective}. \] This also applies to arbitrary finite products of CM abelian varieties, including their powers. If $W\subset H^k(A,\mathbb Q)$ is a rational Hodge substructure and $W(r)$ is effective (all Hodge bidegrees are nonnegative), it claims a closed algebraic $Z\subset A$ of codimension at least $r$ with $W\subset\ker[H^k(A,\mathbb Q)\to H^k(A\setminus Z,\mathbb Q)]$. The manuscript additionally asserts that, for every abelian variety $A/\mathbb F_q$, every $r\ge0$ and prime $\ell\ne\operatorname{char}\mathbb F_q$, \[ \operatorname{CH}^r(A)\otimes\mathbb Q_\ell\longrightarrow H^{2r}_{\mathrm{\acute et}}(A_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))^{\operatorname{Gal}(\overline{\mathbb F}_q/\mathbb F_q)} \quad\text{is surjective}. \] Numerical and $\ell$-adic homological equivalence are claimed to agree on rational cycles. Over any algebraically closed field, for an abelian $g$-fold and rational ample class $h$, the form $(-1)^r\deg(xyh^{g-2r})$ is claimed positive definite on the real extension of $\ker[h^{g-2r+1}:A^r_{\mathrm{num}}\to A^{g-r+1}_{\mathrm{num}}]$ for $0\le r\le g/2$.
Mathematical claim 6.
Choose any integer $m\ge1$ and any projective complex K3 surfaces $S_1,\ldots,S_m$, allowing repeated factors. For $X=\prod_{j=1}^m S_j$, OpenAI claims \[ \operatorname{CH}^p(X)_{\mathbb Q}\xrightarrow{\operatorname{cl}} H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) \quad\text{is surjective for every }0\le p\le2m. \] There are no Picard-number, period or endomorphism-field restrictions. The quantified varieties are products of K3 surfaces, so this statement has smaller scope than the Hodge conjecture for arbitrary smooth projective complex varieties.
Mathematical claim 7.
Let $E=\mathbb Q(u)$ be imaginary quadratic with $u^2=-d$, $d>0$, and let $A$ be a complex abelian variety with a unital $E$-action. OpenAI claims rational cycle-class surjectivity \[ \operatorname{CH}^p(A^m)_{\mathbb Q}\longrightarrow H^{2p}(A^m,\mathbb Q)\cap H^{p,p}(A^m) \quad(m\ge1,\ p\ge0) \] in either of these two cases: $\dim A\le5$; or $\dim A=6$ and the rational Hermitian form associated to a compatible polarization is hyperbolic with signature $(3,3)$. In the second case, with rational Riemann form $e$, this means $h_E(x,y)=e(x,uy)+u e(x,y)$ on $H_1(A,\mathbb Q)$ has a totally isotropic three-dimensional $E$-subspace. Compatibility means $e(ax,y)=e(x,\bar a y)$ for all $a\in E$.

Outline of the argument

Manuscript 1 (family 032, 2026-09-30): Deck-character projections and equivariant period calculations identify the necessary monodromy invariants of curve covers. Correspondences dominated by CM sources supply determinant tensors, while Gale duality transfers the character-level construction to diagonal complete intersections with the required supports and degree shifts. Manuscript 2 (family 032, 2026-09-30): Clifford multiplication carries exterior powers of the transcendental space into degree-two cohomology of an abelian target. Transposed cycles and a polynomial inverse on the finite-dimensional image recover these summands, so divisor algebraicity supplies the required classes without an inverse Lefschetz correspondence. Manuscript 3 (family 032, 2026-10-03): The construction engineers a Lagrangian whose second cohomology has a nineteen-dimensional deficit, yielding a corresponding bound on self-extensions after mirror realization. The resulting characteristic class is transported through K3 moduli by semiregularity and specialization, retaining the prescribed Clifford tensor.

Manuscript 4 (family 032, 2026-09-18): An embedded spin Lagrangian detects the exceptional middle-cohomology component and makes two homological obstruction maps injective. Finite-cutoff weighted Floer operations are assembled into algebraic module identities; cancellation in alternating specialization then transports the desired Chern character to an algebraic cycle. Manuscript 5 (family 032, 2026-09-30): The argument realizes four-factor balanced tensors using unitary theta constructions. Elementary switches between equal-sum lists, followed by algebraic polarization contractions and descent of coefficients, build arbitrary balanced CM tensors from these elementary cycle classes. Manuscript 6 (family 032, 2026-10-04): A Lagrangian immersion is used to produce prescribed Euler pairings while bounding the second self-extension space of the associated complex. Clifford-algebra calculations then identify its transcendental cohomology component; algebraic Kuga--Satake tensors and their contractions supply the classes needed on products.

Manuscript 7 (family 032, 2026-09-30): A coherent seed and its Fourier--Mukai transform produce the split-sixfold Weil plane through an obstruction calculation and deformation. A determinant description of the actual Hodge group, combined with the CM result, extends these algebraic generators to powers and to the stated low-dimensional cases.

Formal verification

Family 032 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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Math-29-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-29

Deligne's Absolute Hodge Conjecture

Definitions and mathematical statement

A rational Hodge class $\alpha\in H^{2p}(X,{\mathbb{Q}}(p))$ becomes type $(0,0)$ in the twisted Hodge structure. Via algebraic de Rham comparison it has a conjugate on $X^\sigma$ for each $\sigma\in\operatorname{Aut}({\mathbb{C}})$. Absoluteness means \[ \forall\sigma\in\operatorname{Aut}({\mathbb{C}}),\qquad \alpha^\sigma\in H^{2p}(X^\sigma,{\mathbb{Q}}(p))\text{ is a Hodge class}. \] The conjecture requires this for every rational Hodge class on every smooth projective complex variety. Arbitrary field automorphisms of ${\mathbb{C}}$, not only continuous ones or complex conjugation, are quantified.

Formulation and concept references: [S1].

Short English statement

Does every rational Hodge class remain rational and of Hodge type after every automorphism of the complex numbers?

Research attempt

Scope, process, and checked context.
This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026. The target is the assertion for every smooth projective complex variety, every rational Hodge class, and every field automorphism of $\mathbb C$. The attack is to build explicit certificates from already absolute classes and identify the information that such certificates cannot supply. The opposite track tests whether preservation of the algebraic Hodge filtration alone could force rationality.

Charles--Schnell [S1], Sections 2.2--2.4, supplies the comparison and functoriality conventions: algebraic cycle classes are absolute, and pullback, cup product and proper pushforward respect conjugation. Deligne's theorem in [E1] settles absoluteness of Hodge classes on abelian varieties. It does not assert the Hodge conjecture for those varieties or settle the present question for arbitrary $X$. These are external inputs, not results of the calculations below.

Separate the filtration from the rational lattice.
Temporarily use untwisted de Rham notation. Write $V_B=H^{2p}(X,\mathbb Q)$ and $V_{\rm dR}=H^{2p}_{\rm dR}(X/\mathbb C)$, and let $c_X:V_B\otimes\mathbb C\to V_{\rm dR}$ be comparison. For a field automorphism $\sigma$, algebraic base change gives a $\sigma$-semilinear map $s_\sigma$ between the de Rham spaces of $X$ and $X^\sigma$. It transports the Hodge filtration: \[ s_\sigma(F^pV_{\rm dR})=F^pH^{2p}_{\rm dR}(X^\sigma/\mathbb C). \] In the twisted notation of the statement, comparison must also include the Tate factor. It is the resulting conjugated class that must belong to the rational Betti space of $X^\sigma$. One cannot apply $\sigma$ entrywise to a rational Betti vector and declare that this is geometric conjugation: the two comparison maps lie between those operations.

Once a conjugated class is rational and belongs to $F^p$, its Hodge type is automatic. Indeed, after undoing the Tate twist, a rational class is real; if its components have first index at least $p$, complex conjugation forces second index at least $p$ too. In degree $2p$ only $(p,p)$ remains. Thus preservation of $F^p$ genuinely advances the argument, but leaves precisely the rationality of the conjugated comparison vector to prove.

Here is a linear-algebra stress test of that gap. Give the de Rham line $\mathbb Ce$ the full filtration and give its comparison rational structure the line $\mathbb Q\tau e$, with $\tau$ transcendental. The vector $\tau e$ is rational for this structure. An automorphism sending $\tau$ to $\tau+1$ exists: extend $\tau$ to a transcendence basis over $\overline{\mathbb Q}$, translate that variable, and extend the field isomorphism to an algebraic closure. Semilinear transport sends the vector to $(\tau+1)e$. Relative to the same comparison line its coefficient is $1+1/\tau$, which is not rational. The filtration was preserved throughout. This artificial line is not asserted to be the cohomology of a variety, and $\tau$ is not a substitute for a geometric Tate period. It isolates a missing compatibility condition, rather than giving a counterexample to Deligne's conjecture.

A complete elementary subcase.
Suppose the class is a rational linear combination of algebraic cycle classes. Write $\alpha=\sum_i a_i[Z_i]$, $a_i\in\mathbb Q$. Algebraic base change takes $Z_i$ to a codimension-$p$ cycle $Z_i^\sigma$ and commutes with its de Rham cycle class. The comparison convention therefore gives \[ \alpha^\sigma=\sum_i a_i[Z_i^\sigma]. \] Every term is rational and Hodge, for every $\sigma$, so $\alpha$ is absolute. In particular the Hodge conjecture would imply the target, but assuming that conjecture at this point would replace one open problem by a stronger input.

The codimension-one case can be completed using the classical Lefschetz $(1,1)$ theorem. Multiply a rational $(1,1)$ class by an integer to make it integral. The exponential sequence identifies such an integral class with the first Chern class of a holomorphic line bundle; on a projective variety GAGA makes the bundle algebraic. Conjugating that bundle proves the desired assertion, and division by the integer recovers the original class. At codimensions zero and $\dim X$, component and point classes give the same conclusion. No analogous general description of all intermediate Hodge classes is assumed.

Lemma 94.1: an explicit pairing certificate.
Let $d=\dim X$. Suppose $\beta_1,\ldots,\beta_r$ are absolute classes in $H^{2p}(X,\mathbb Q(p))$, and $\gamma_1,\ldots,\gamma_r$ are absolute classes in $H^{2d-2p}(X,\mathbb Q(d-p))$. Use the rationally normalized trace and set \[ G_{ji}=\operatorname{Tr}_X(\beta_i\smile\gamma_j),\qquad G\in\operatorname{GL}_r(\mathbb Q). \] For a rational Hodge class $\alpha$, put \[ b_j=\operatorname{Tr}_X(\alpha\smile\gamma_j),\quad c=G^{-1}b,\quad \alpha_0=\sum_i c_i\beta_i,\quad \delta=\alpha-\alpha_0. \] Then $\alpha_0$ is absolute, $\delta$ is rational Hodge and annihilates all the $\gamma_j$. If these pairings separate rational Hodge classes, then $\delta=0$ and $\alpha$ is absolute.

Proof. All entries of $G$ and $b$ are rational by the rational cup product and trace. Invertibility makes $c$ rational, so each $\alpha_0^\sigma=\sum_i c_i\beta_i^\sigma$ is rational Hodge. Subtracting gives a rational Hodge residual at the original embedding. The vector of its pairings is $b-Gc=0$. The additional separation hypothesis is exactly that this vanishing forces the residual to be zero. No information about the conjugate of the unknown residual was used.$\square$

For a checked instance take $X=\mathbb P^a\times\mathbb P^b$, with hyperplane classes $h,k$. The cohomology ring is $\mathbb Q[h,k]/(h^{a+1},k^{b+1})$. In codimension $p$ the nonzero monomials $h^ik^{p-i}$ form a basis. Their complementary classes are $h^{a-i}k^{b-p+i}$. The pairing matrix is the identity: an off-diagonal pair has an exponent exceeding $a$ or $b$, whereas a matching pair integrates to one. These algebraic classes separate the entire degree, so the certificate resolves all Hodge classes of this family. The check includes $p=0,a+b$ and the cases $a=0$ or $b=0$.

A route through abelian varieties and its precise limit.
Suppose there are algebraic correspondences inducing morphisms of rational Hodge structures \[ u:H^{2p}(X,\mathbb Q(p))\longrightarrow H^{2q}(A,\mathbb Q(q)), \qquad v:H^{2q}(A,\mathbb Q(q))\longrightarrow H^{2p}(X,\mathbb Q(p)), \] where $A$ is abelian, and suppose $v u(\alpha)=\alpha$. The class $u(\alpha)$ is Hodge and is absolute by Deligne's theorem. Functoriality of the conjugated correspondence then makes $v(u(\alpha))=\alpha$ absolute. This argument only needs a retraction on the class under consideration, not on the whole cohomology group. It supplies a concrete sufficient certificate which could be tested for particular geometric constructions.

The unsuccessful step is to assert that every Hodge class admits such a retraction. An arbitrary embedding of a rational vector space into another vector space is insufficient: the maps must be compatible with the Hodge structures and remain rational after geometric conjugation. Constructing $v$ merely by selecting a Betti basis assumes the compatibility that the attempt is trying to establish.

Likewise, spreading $X$ over a finitely generated field does not make a chosen Betti class a globally defined horizontal algebraic section. Monodromy can move it. Even when a section exists, an argument using transport from another fibre needs an absolute starting class and the hypotheses of a valid deformation theorem. The existence of a model over a small field alone supplies neither item.

Verification and next targets.
The certificate uses rational coefficients, so $\sigma$ fixes the coefficients but not the comparison matrices. Trace normalization includes the Tate twists in both complementary degrees. The toy example tests a logical implication only. The projective-product matrix can be verified symbolically from its ring relations; no numerical period approximation establishes rationality. All subcases above are classical consequences or explicit reorganizations of known facts; no novelty is claimed.

The first missing statement is that arbitrary rational Hodge classes are exhausted by certificates stable under every conjugation. Three specific next targets are to exhibit separating absolute complementary classes on a new family, construct an algebraic retraction to abelian cohomology for its primitive part, or control the residual $\delta$ under conjugated comparison maps. A putative counterexample must have nonzero residual beyond the available certificates and fail rationality for some field automorphism; the artificial line does not meet the geometric hypotheses.

Final status.
UNRESOLVED. The argument proves the pairing-certificate lemma, its projective-product application, and the conditional retraction criterion. It does not prove conjugate rationality for a general Hodge class and does not construct a geometric counterexample.

Sources

  • S1 F. Charles and C. Schnell, Notes on absolute Hodge classes $2011$, Section 2.5, Conjecture 25. https://www.math.ens.psl.eu/~charles/AH.pdf. Formulation source.
  • E1 P. Deligne (notes by J. S. Milne), Hodge cycles on abelian varieties, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900 $1982$, pp. 9--100; revised electronic version hosted by J. S. Milne. https://www.jmilne.org/math/Documents/Deligne82.pdf. Primary source for the abelian-variety theorem; consulted 27 September 2026.

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