The study of solution sets of polynomial equations and of the topology they carry; Hodge theory splits that topology into pieces that tell algebraic cycles apart from merely topological ones.
Algebraic geometry supplies the language for arithmetic geometry (elliptic curves, the Weil conjectures, Fermat's Last Theorem) and for much of mathematical physics. Hodge theory is the main tool linking the complex-analytic and the algebraic structure of a variety.
The study of shapes cut out by polynomial equations, and the hidden structure those shapes carry.
1Shapes defined by polynomial equations
2One framework for many number systems
3Topology of the shape splits into types
4Hodge conjecture: which holes are algebraic
An algebraic variety is the common zero set of a collection of polynomials: a plane curve y2=x3+ax+b, or a surface in projective space . It is often studied over the complex numbers, where it becomes a geometric object with a topology. (1950s–60s) generalize varieties so that polynomial equations over the integers, over finite fields and over sit in one framework. This allows cohomological tools from topology to be applied to arithmetic questions. A landmark example is , which are the analogue of the Riemann Hypothesis for varieties over finite fields: the number of points on a variety over is a signed sum of -th powers of numbers whose sizes are exactly half-integer powers of , the finite-field version of zeros lying on a single line.
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The full technical text.
Varieties and what they remember
An algebraic variety is the common zero set of polynomials, such as a curve y2=x3+ax+b or a surface in . Grothendieck's schemes (1950s-60s) put polynomial equations over , finite fields and in one framework, so that a statement about integer solutions can be studied with the same cohomological tools as one about complex manifolds. This is the setting in which Deligne proved the Weil conjectures (1974): the analogue of the Riemann Hypothesis for varieties over finite fields.
OpenAI claims the Hodge conjecture for CM abelian varieties (unrefereed)
An OpenAI manuscript claims the rational Hodge conjecture for every complex CM abelian variety. It has no Lean formalization and had not been independently checked by mid-October 2026. Three related OpenAI manuscripts were withdrawn on 7 October after a sign error was found.
Markman: Weil classes on abelian fourfolds are algebraic
Markman proves that Weil classes on abelian sixfolds of Weil type with discriminant are algebraic. Via Schoen's degeneration argument this gives algebraicity of Weil classes on all abelian fourfolds of Weil type, from which the Hodge conjecture for abelian fourfolds follows (arXiv preprint).
Discussion
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Grothendieck's schemes
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Deligne's proof (1974) of the Weil conjectures
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Cohomology assigns to a space a sequence of vector spaces Hk that detect holes of dimension k. For a smooth complex projective variety X (the case with the nicest structure), the singular cohomology with complex coefficients has a canonical splitting
Hk(X,C)=⨁p+q=kHp,q(X),
In words: a k-dimensional hole is written as a sum of pieces, where Hp,q is represented by differential forms with p holomorphic directions and q antiholomorphic ones. This is the Hodge decomposition. It exists because X is Kähler, and it brings constraints: for example, odd-degree Betti numbers are even.
A Hodge class is a rational cohomology class in Hp,p (the 'balanced' piece). Any algebraic subvariety of codimension p produces such a class. The Hodge conjecture says the converse: every Hodge class is a rational linear combination of classes of algebraic subvarieties. It is known for p=1 (the Lefschetz (1,1)-theorem, 1924), hence for all varieties of dimension at most three, and in some further families; Markman (preprint, 2025) proved it for abelian fourfolds. It is open in general and is a Clay Millennium Prize Problem. The version with integer instead of rational coefficients is false (Atiyah–Hirzebruch). An October 2026 AI-generated manuscript claims the case of abelian varieties with complex multiplication; it is unrefereed and unchecked.
The conjecture matters because it would say that a purely topological and analytic condition, living in the cohomology, detects whether a class comes from solving polynomial equations. It also fits into the conjectural framework of motives, which seeks a universal cohomology theory.
The subject has practical offshoots. Gröbner bases make polynomial systems computable; algebraic curves over finite fields give error-correcting codes; and elliptic curves support much of modern cryptography.
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Hodge theory
Let X be a smooth complex projective variety. Its singular cohomology with complex coefficients splits canonically as
Hk(X,C)=⨁p+q=kHp,q(X),Hp,qHq,p,
where Hp,q is represented by differential forms with p holomorphic and q antiholomorphic directions. The decomposition exists because X is Kähler, and it is a source of strong constraints: Betti numbers of odd degree are even, and the Lefschetz theorems relate cohomology in different degrees.
A Hodge class of codimension p is an element of H2p(X,Q)∩Hp,p(X). Every algebraic subvariety of codimension p defines such a class (its cycle class), so the rational span of algebraic cycles sits inside the Hodge classes. The Hodge conjecture says the inclusion is an equality. It is known for p=1 (Lefschetz's theorem on (1,1)-classes, 1924), and hence by duality for all varieties of dimension at most three, and in a few further families. It is open in general. Counterexamples are known to the natural variants: with integral coefficients (Atiyah and Hirzebruch, 1962) and for compact Kähler manifolds that are not projective (Voisin, 2002), so any proof must use algebraicity in an essential way.
Related structures
Mixed Hodge structures (Deligne) extend the theory to singular and non-compact varieties and underlie modern work on degenerations and period maps.
Hodge loci, the places in a family where extra Hodge classes appear, are algebraic (Cattani, Deligne and Kaplan, 1995). Whether they are defined over a number field is a deeper open question.
Standard conjectures and motives. Grothendieck's standard conjectures and the Tate conjecture (the ℓ-adic analogue of the Hodge conjecture) aim to show that cohomology of varieties is governed by algebraic cycles. Their truth would give a universal "motivic" cohomology theory.
Birational geometry. The minimal model program, completed in key cases by Birkar, Cascini, Hacon and McKernan (2006-2010), classifies varieties up to birational equivalence, showing for example that the canonical ring of a smooth projective variety is finitely generated.
Current state
The Hodge conjecture is one of the Clay Millennium Problems and has no accepted resolution. The most notable recent result in the human literature is Markman's (arXiv preprint, February 2025). He proves that Weil classes on abelian sixfolds of Weil type with discriminant −1 are algebraic, deduces by a degeneration argument of Schoen that Weil classes on all abelian fourfolds of Weil type are algebraic, and hence (by known reductions) the Hodge conjecture for abelian fourfolds. In September 2026 Claire Voisin surveyed the problem in a guest post on Terence Tao's blog, calling Markman's theorem "a big recent progress" and discussing the variational Hodge conjecture and the approaches that have not worked.
On 6 October 2026 OpenAI released a repository of manuscripts produced by an internal model. Among them is a claimed proof of the rational Hodge conjecture for all complex abelian varieties with complex multiplication (CM). Through theorems of Milne, this would also give the Tate conjecture for abelian varieties over finite fields. The manuscript has no Lean formalization and is unrefereed, and no independent check had been reported by 15 October 2026. On 7 October OpenAI withdrew three related manuscripts (Weil classes on split abelian eightfolds, Kuga–Satake correspondences for K3 surfaces, and the Hodge conjecture for products of K3 surfaces) after a sign error was found. The general conjecture remains open, and the Clay Institute lists it as unsolved.
Why it matters outside geometry
Algebraic geometry feeds number theory (the modularity theorem for elliptic curves and Fermat's Last Theorem), coding theory (algebraic-geometry codes from curves over finite fields), cryptography (elliptic-curve groups), and mathematical physics (mirror symmetry, moduli spaces of gauge-theory solutions). The machinery of schemes and sheaf cohomology is also far from fully formalized in proof assistants, which makes the subject an informative stress test for tools such as Lean and Mathlib.
Ordinary space with extra points at infinity added, so parallel lines meet.
Minimal models exist for varieties of general type
Birkar, Cascini, Hacon and McKernan prove finite generation of the canonical ring and existence of minimal models in the general-type case, a landmark for the classification of higher-dimensional varieties (arXiv v1; journal version 2010).
What to read before Algebraic geometry and Hodge theory (left), and what builds on it: entries that depend on it and the open problems it hosts (right). Every box links to its page.