Posed by W. V. D. Hodge, ICM 1950 · Clay Millennium Prize problem (2000), $1,000,000 · field: algebraic & complex geometry
How to read this page. The Hodge conjecture lives at the border between two ways of studying a geometric shape: topology (how it is connected, its holes, measured by cohomology) and algebraic geometry (the polynomial subshapes that actually sit inside it). Throughout, X is a smooth projective variety over ℂ: the common zero locus of homogeneous polynomials in complex projective space, with no singular points. Hk(X, ℚ) and Hk(X, ℂ) are its singular cohomology groups with rational / complex coefficients; a superscript like Hp,q denotes a piece of the Hodge decomposition (defined in §2). A cycle is a formal ℤ- or ℚ-combination of irreducible closed subvarieties; codim means complex codimension. The symbol ⊗ ℚ means "allow rational coefficients." When a claim is proven I say so with a name and date; when it is open or merely expected, I say that too. Nothing on this page proves the conjecture: it is open, and the hard walls are laid out explicitly in §5.
0 · The claim, in one line
On a smooth projective complex variety, every rational cohomology class that "looks algebraic" to the Hodge decomposition (a Hodge class) really is algebraic: a ℚ-linear combination of the fundamental classes of actual subvarieties.
Hdgp(X) := H2p(X, ℚ) ∩ Hp,p(X) =? image of cl : Zp(X) ⊗ ℚ → H2p(X, ℚ)Every rational (p,p)-class should be a rational combination of classes of algebraic cycles of codimension p. The conjecture asserts the "=?" is an equality.
1 · The object: what it is and why anyone cares
Start with the most innocent question in geometry: which subshapes live inside a given shape? Inside a smooth projective variety X sit its algebraic subvarieties (curves, surfaces, divisors, points) each itself cut out by polynomials. These are the "visible," constructible objects of algebraic geometry. We want to understand the collection of all of them up to the natural equivalences.
Topology hands us a coarser but far more computable shadow of the same information. Every closed subvariety Z ⊆ X of complex codimension p (real codimension 2p) carries a fundamental class, and integrating forms over Z turns it into a cohomology class
cl(Z) ∈ H2p(X, ℤ) → H2p(X, ℚ).The cycle class map: it forgets the geometry of Z and remembers only "how Z sits homologically" inside X.
So we have a one-way street: algebra (subvarieties) → topology (cohomology classes). The why, chased to its root, is this: we want to reverse the arrow. Cohomology is computable, functorial, and finite-dimensional. Subvarieties are the actual geometric content we care about but are hard to get our hands on. If we knew exactly which cohomology classes come from subvarieties, we could convert topological computations into existence theorems for genuine algebraic geometry: and conversely import the rigidity of algebraic cycles into topology.
The deep reason to believe the arrow can be reversed is Hodge theory. Not every cohomology class can come from an algebraic cycle: a subvariety is a complex-analytic object, and that forces its class to be of a special type under the Hodge decomposition. The conjecture is the bold claim that this one necessary condition (being a Hodge class) is also sufficient. In other words: the only obstruction to a rational class being algebraic is the obvious Hodge-theoretic one, and there is no hidden, deeper obstruction.
Why care beyond elegance? Algebraic cycles are the raw material of motives: Grothendieck's conjectural "universal cohomology" organizing all cohomology theories (Betti, de Rham, ℓ-adic, crystalline) of all varieties. The Hodge conjecture is one of the twin pillars (with the Tate conjecture, its arithmetic cousin) on which that entire edifice rests. If it fails, cohomology classes exist that no geometry can explain; if it holds, topology and algebraic geometry agree to a degree that would reshape the field.
2 · The precise statement: the mathematical setup
The Hodge decomposition
Because X is a compact Kähler manifold (every smooth projective variety is one), Hodge theory gives a canonical direct-sum decomposition of each complex cohomology group into pieces indexed by a bidegree(p, q):
Hk(X, ℂ) = ⊕p+q=k Hp,q(X), with Hp,q = Hq,p.A degree-k class splits into "holomorphic-degree p, antiholomorphic-degree q" parts; complex conjugation swaps H^{p,q} and H^{q,p}. Here H^{p,q} is realized by harmonic forms with p holomorphic and q antiholomorphic differentials (dz's and dz̄'s).
The classes that can possibly be algebraic live in the middle, where p = q. A cycle of codimension p lands in degree 2p, and complex-analytic geometry forces its image into the single summand Hp,p. That is the necessary type condition.
Hodge classes
A Hodge class of codimension p is a rational class whose complexification is purely of type (p,p):
Hdgp(X) = H2p(X, ℚ) ∩ Hp,p(X) ⊆ H2p(X, ℂ).Rational classes sitting entirely in the (p,p) slot. The rationality is essential: see §5, the integral version is false.
The conjecture
Hodge Conjecture. For every smooth projective variety X over ℂ and every p ≥ 0, the cycle class map
clℚ : CHp(X) ⊗ℤ ℚ ⟶ Hdgp(X)from the Chow group of codimension-p cycles modulo rational equivalence, tensored with ℚ, onto the Hodge classes.
is surjective. Equivalently: every Hodge class is a ℚ-linear combination ∑ ai cl(Zi) of classes of irreducible algebraic subvarieties Zi ⊆ X of codimension p.
The cycle class map goes left-to-right unconditionally. The Hodge conjecture asserts the dashed arrow exists after tensoring with ℚ: every Hodge class is hit.
3 · Every connection: what it is equivalent to and what hangs on it
The Lefschetz (1,1)-theorem (the one proven case). For p = 1, Hodge classes are exactly the divisor classes. This is a theorem, due to Lefschetz (1924), provable via the exponential sheaf sequence 0 → ℤ → OX → OX* → 0 and the fact that line bundles are classified by H1(X, OX*). It is the foundation stone, and crucially it uses machinery (the exponential sequence) with no analogue in higher codimension.
The Tate conjecture (arithmetic sibling). Over a finitely generated field, Tate predicts that Galois-invariant ℓ-adic classes are algebraic. Hodge (over ℂ, Hodge symmetry) and Tate (over number fields, Galois action) are two realizations of the same philosophy; a full theory of motives wants both.
Grothendieck's standard conjectures. The Lefschetz-type standard conjecture B (algebraicity of the inverse of the hard Lefschetz operator) and the Künneth standard conjecture would follow from Hodge, and would themselves give a workable theory of pure motives with a well-defined category and the expected semisimplicity. Hodge is strictly stronger than these in general.
The Hodge-theoretic criterion for algebraic cycles underwrites the construction of the category of motives for homological equivalence; if Hodge fails, "Hodge classes" become genuine ghost cohomology that no motive can account for.
Absolute Hodge classes (Deligne). Deligne isolated a class of Hodge classes (the absolute Hodge ones) that behave as algebraic cycles ought to across all comparison isomorphisms, and proved every Hodge class is absolute Hodge for abelian varieties (1982). This is strictly weaker than algebraicity but is the best unconditional structural result in its direction, and it rescues the functoriality that the Hodge conjecture would otherwise be needed for.
Variational / Cattani–Deligne–Kaplan (1995). In a family of varieties, the locus where a given class stays a Hodge class is a countable union of algebraic subvarieties of the base. This is overwhelming circumstantial evidence (Hodge loci behave exactly as they would if they were cut out by the conjecturally-existing algebraic cycles) yet it stops short of producing a single cycle.
4 · Every serious attack: and how far each got
Codimension 1, and its dual
Lefschetz (1924): the (1,1) case, proven in full, as above. By the Hard Lefschetz theorem (stated by Lefschetz, first given a complete proof via Hodge theory; iterating cup product with a hyperplane class gives isomorphisms Hn−k ≅ Hn+k on an n-fold), the divisor result transports to give the conjecture for classes of dimension 1 as well: i.e. codimension p = dim X − 1. So the top and bottom of the codimension range are secured; everything in the middle is open.
Abelian varieties
The richest testing ground. Mattuck, Mumford and others handled special families; Tankeev and Ribet (1980s) proved the conjecture for abelian varieties of prime dimension and various CM cases using Mumford–Tate groups: the idea that Hodge classes are the invariants of a reductive group MT(X), so computing that group bounds the Hodge classes. When the group is as large as possible, all Hodge classes are forced to be the obvious (Lefschetz) ones, which are algebraic.
Deligne's absolute Hodge theorem (1982)
For abelian varieties, every Hodge class is absolute Hodge. This does not prove algebraicity, but it proves the classes transform under field automorphisms exactly as algebraic cycles must, closing off one entire family of potential counterexamples to functoriality and letting the theory of motives proceed for abelian varieties on a conditional-but-controlled footing.
Specific "exotic" classes verified by hand
Weil (1977) constructed Hodge classes on abelian varieties with complex multiplication ("Weil classes") that are not generated by divisors, i.e. not explained by the Lefschetz construction. These are the sharpest known test cases. In special instances they have been shown algebraic: Schoen (1988) and van Geemen verified families of Weil classes; later work by Abdulali and André added more. Each is a hand-built existence proof for one family: none generalizes.
Low-dimensional and special varieties
The conjecture is known for: curves and surfaces (trivially / by Lefschetz), uniruled and rationally connected threefolds and fourfolds in degrees worked out by Conte–Murre and others, many complete intersections and hypersurfaces with large Hodge groups, products of curves and abelian varieties in verified cases, and varieties whose cohomology is generated in degree 2 (where everything reduces to divisors). Shioda settled many Fermat varieties by explicit computation.
5 · Why it breaks down: the honest crux
Wall 1: There is no construction. The whole proof strategy is missing. The Lefschetz (1,1) proof works because a codimension-1 Hodge class is a line bundle (via H1(X,OX*)) and a line bundle has a divisor of zeros: a geometric object literally falls out of the cohomology. For p ≥ 2 there is no analogue of the exponential sequence, no object classifying codimension-p cycles that a Hodge class maps into. We can verify a class is Hodge (a linear-algebra condition on harmonic forms) but we have no mechanism whatsoever to manufacture a subvariety from that data. Every known positive case either reduces to divisors via Hard Lefschetz, or diagonalizes a Mumford–Tate group to show the "exotic" Hodge classes simply do not exist, or constructs cycles by hand. None of these is a general method, and the general case has resisted precisely because the problem is existential: assert something exists with no recipe to build it.
Wall 2: The integral version is false, so ℚ is not a cosmetic choice. Replace ℚ by ℤ (ask that every integral Hodge class be an integral combination of cycles) and the statement is provably wrong. Atiyah–Hirzebruch (1962) used topological K-theory and Steenrod operations to exhibit torsion cohomology classes that are Hodge yet not algebraic. Kollár (1990s) went further with non-torsion counterexamples: on very general hypersurfaces of high degree, an integral Hodge class exists whose smallest algebraic multiple is forced to be a large integer > 1, so the class itself is not a ℤ-combination of cycles. This is brutal: it means any proof of the rational conjecture must crucially exploit the ability to divide by integers, and cannot be a direct "build the cycle" argument: because the honestly-built cycle may only exist after multiplying the class by some integer nobody can control. The failure of the integral statement is a standing warning that our intuition about "a class should bound an algebraic cycle" is simply false on the nose.
Wall 3: Weil classes on CM abelian varieties: the designated assassins. Weil's Hodge classes of CM type are the leading candidate counterexamples. They are genuinely of type (p,p), genuinely rational, and demonstrably not in the subring generated by divisors: so they cannot be reached by the one general tool (Lefschetz + Hard Lefschetz) we possess. For the conjecture to hold, each must nonetheless be algebraic. Where this has been confirmed it required bespoke constructions tied to the specific CM field and the arithmetic of the abelian variety; the arguments do not transport to a neighboring CM type. We are in the worst position: we neither have a proof covering all Weil classes, nor a single Weil class demonstrably non-algebraic. The conjecture is balanced on a knife-edge exactly here, and most experts who suspect it might be false point to this family.
Wall 4: Circularity with the standard conjectures. A natural strategy is to prove the Künneth and Lefschetz standard conjectures first and bootstrap to Hodge. But the standard conjectures are themselves open, and the tools that would prove them (algebraicity of certain correspondences, the existence of a well-behaved category of motives) are entangled with the very algebraicity claims Hodge is supposed to supply. Attempts to decouple them run in circles: to construct the needed algebraic self-correspondence on X × X (e.g. the Künneth projectors) one wants cycles whose existence is a special case of the problem. No one has broken this loop in general; it has only been cut in cases where enough cycles are already known for independent reasons (abelian varieties, surfaces).
Wall 5: FRINGE / speculative: motivic and p-adic end-runs, and why they stall.Clearly labelled speculation. Several unconventional routes have been floated. (a) Absolute-Hodge / motivic Galois: push Deligne's theorem from "absolute Hodge" to "algebraic" by proving the Hodge = absolute-Hodge = motivated classes all coincide with algebraic ones (André's "motivated cycles" give a formal category where the standard conjectures hold by fiat, but inserting the extra motivated classes is precisely admitting we cannot produce them as honest cycles: it relabels the gap rather than closing it). (b) p-adic / crystalline transfer: prove the Tate conjecture in positive characteristic and lift, hoping arithmetic rigidity forces algebraicity; this founders on the same cycle-construction wall after reduction, plus lifting obstructions. (c) Arithmetic-statistics / "Hodge loci are suspiciously algebraic" (Cattani–Deligne–Kaplan and successors): the Hodge locus is algebraic, so one dreams of reading an algebraic cycle off its defining equations: but CDK produce the locus, never the cycle, and the locus being algebraic does not by itself yield a cycle. (d) Occasional claims via physics-flavored period relations or "numerical experiments finding the cycle" are not rigorous and have not survived scrutiny. Every fringe route so far either smuggles in the conclusion (motivated cycles), transfers the gap to another open problem (Tate, standard conjectures), or produces a shadow (locus, period, numerical hit) that stops one logical step short of an actual subvariety. The recurring failure mode is the same as Wall 1: no route yet turns topological data into geometry.
6 · Status: confirmed
OPEN. The Hodge conjecture is one of the seven Clay Mathematics Institute Millennium Prize Problems (stated 2000, carrying a US $1,000,000 prize), and it remains unsolved as of this writing. What is settled: codimension 1 (Lefschetz (1,1), 1924) and its Hard-Lefschetz dual; many abelian varieties and special varieties; the algebraicity of Hodge loci (CDK, 1995); and that every Hodge class on an abelian variety is absolute Hodge (Deligne, 1982). What is settled against the naive form: the integral Hodge conjecture is false (Atiyah–Hirzebruch 1962; Kollár 1990s), which is why the statement must be over ℚ. The general rational conjecture is neither proven nor disproven for any single codimension p ≥ 2 across all smooth projective varieties.
A genuine proof would have to do what no argument currently does: produce, from the bare fact that a rational class is of type (p,p), an actual algebraic subvariety (after clearing some denominator) whose class realizes it: uniformly, including for the Weil classes on CM abelian varieties, and without assuming the standard or Tate conjectures as unproven inputs. Equivalently, a disproof would exhibit one explicit smooth projective X and one rational Hodge class demonstrably not a ℚ-combination of cycle classes. Both remain out of reach. Anyone claiming a short or elementary proof is almost certainly tripping over Wall 1 or Wall 4.
Credit where it is due. The conjecture is W. V. D. Hodge's, announced at the 1950 International Congress of Mathematicians in Cambridge, Massachusetts. The proven codimension-1 case is Solomon Lefschetz's (1924). The framework of standard conjectures and motives is Alexander Grothendieck's; the absolute-Hodge theorem for abelian varieties is Pierre Deligne's (1982). The counterexamples to the integral form are due to Michael Atiyah and Friedrich Hirzebruch (1962) and to János Kollár (1990s). The exotic test classes are André Weil's (1977), with special cases proven algebraic by Chad Schoen, Bert van Geemen, Salman Abdulali, Yves André and others; the algebraicity of Hodge loci is Eduardo Cattani, Pierre Deligne and Aroldo Kaplan's (1995). Mumford–Tate group methods trace to David Mumford, John Tate, Sergei Tankeev and Kenneth Ribet. The unsolved problem is claimed by no one here: if it is ever solved, that proof is offered freely to the world.