Integral Hodge Conjecture

• Integral Hodge Conjecture is the problem of determining if every integral Hodge class on a smooth projective complex variety is algebraic, refining the rational version.
• Methodologies involve the cycle class map, spectral sequences, and unramified cohomology to reveal both supporting cases and counterexamples.
• Positive confirmations exist for one-cycles and specific varieties, while notable failures occur in higher codimension due to torsion and defect groups.

The integral Hodge conjecture is a central problem at the intersection of algebraic geometry, cohomology theory, and arithmetic geometry. It asks whether each integral Hodge class on a smooth projective complex variety is algebraic, that is, arises as the cohomology class of an algebraic cycle. This question probes the interplay between the topology of complex varieties and their algebraic cycle theory and has seen profound advances and notable failures through the introduction of cohomological, cycle-theoretic, and motivic techniques.

1. Formulation of the Integral Hodge Conjecture

Let $V$ be a smooth projective complex variety of dimension $d$, and let $CH^i(V)$ denote the Chow group of codimension $i$ algebraic cycles modulo rational equivalence. The singular cohomology group $H^{2i}(V, \mathbb{Z})$ admits the Hodge decomposition over $\mathbb{C}$, so that

$$H^{2i}(V,\mathbb{Z}) \otimes \mathbb{C} \simeq \bigoplus_{p+q=2i} H^{p,q}(V).$$

The cycle class map

$cl^i: CH^i(V) \rightarrow H^{2i}(V, \mathbb{Z})$

sends a cycle to its cohomology class. The integral Hodge conjecture asserts that

$$\operatorname{im}(cl^i) = Hdg^{2i}(V, \mathbb{Z}) := H^{i,i}(V) \cap H^{2i}(V, \mathbb{Z}),$$

i.e., every integral Hodge class (an integral cohomology class of type $(i, i)$) is the class of an algebraic cycle. This strengthens the rational Hodge conjecture, which states the analogous result after tensoring everything with $d$0.

Lefschetz’s $d$1 theorem establishes the integral conjecture for $d$2 (divisors), and it is trivial for $d$3 and $d$4 (the class of a point and the top cohomology). However, substantial evidence demonstrates that the conjecture fails for higher codimension $d$5.

2. Prototypical Counterexamples and Torsion Obstructions

Early counterexamples were provided by Atiyah–Hirzebruch, who constructed torsion Hodge classes not arising from algebraic cycles by analyzing the nontrivial differentials in the spectral sequence for topological $d$6-theory. Kollár gave non-torsion counterexamples using very general high-degree hypersurfaces, showing that there exist integral Hodge classes in $d$7 that cannot be algebraic, such as in hypersurfaces of degree $d$8 in $d$9 where all curve degrees are divisible by $CH^i(V)$0, but the Hodge group is generated by a class not so divisible.

A more recent systematic approach yields infinite families of counterexamples in various degrees and for all primes $CH^i(V)$1, particularly on approximations to classifying spaces $CH^i(V)$2 for type $CH^i(V)$3-groups of the form $CH^i(V)$4; here, for many $CH^i(V)$5, the image of the Chow group in the integral cohomology is strictly smaller than the lattice of integral Hodge classes (Tripathy, 2016).

Analogously, counterexamples appear in products involving Enriques surfaces and high-degree or odd-dimensional hypersurfaces, where the interplay of torsion classes from the surface and vanishing cycles from the hypersurface create non-algebraic 2-torsion classes in high-degree cohomology (Kok, 2023, Shen, 2019).

3. Defect Groups and Unramified Cohomology

The defect of the integral Hodge conjecture in degree $CH^i(V)$6 on $CH^i(V)$7 is measured by

$CH^i(V)$8

where $CH^i(V)$9 denotes the subgroup generated by algebraic cycles. When $i$0, this group can be related to the unramified cohomology via the Bloch–Kato conjecture and Gersten resolution:

$i$1

and in the limit,

$i$2

Thus, the torsion in $i$3 precisely corresponds to the failure of algebraicity for torsion Hodge classes of degree $i$4. This description applies to both the classical setting and to situations where the Chow group is minimal (so-called Chow-trivial varieties) (Colliot-Thélène et al., 2010, Diaz, 2022).

For uniruled threefolds, Voisin showed that $i$5 vanishes, so the integral Hodge conjecture holds. For certain unirational sixfolds and higher, examples exist with nontrivial torsion $i$6 arising from unramified cohomology, yielding failure of integral algebraicity.

4. The Role of Refined Unramified Cohomology and Specialization

Schreieder’s theory of refined unramified cohomology, building on Colliot-Thélène and Voisin, provides a framework for detecting non-algebraicity by considering the images of étale cohomology groups with torsion coefficients at successive coniveau stages. Specifically, for a smooth variety $i$7, a torsion class in $i$8 that is detected in $i$9 but not in algebraic cycles provides a witness to the failure of the integral Hodge conjecture modulo $H^{2i}(V, \mathbb{Z})$0:

$H^{2i}(V, \mathbb{Z})$1

Specialization arguments and vanishing-cycle techniques, such as in the product of an Enriques surface $H^{2i}(V, \mathbb{Z})$2 and a Lefschetz pencil of odd-dimensional hypersurfaces $H^{2i}(V, \mathbb{Z})$3 (with base field of characteristic $H^{2i}(V, \mathbb{Z})$4), demonstrate the explicit construction of nonalgebraic 2-torsion in $H^{2i}(V, \mathbb{Z})$5, arising from exterior products of torsion in $H^{2i}(V, \mathbb{Z})$6 and vanishing cycles in $H^{2i}(V, \mathbb{Z})$7 (Kok, 2023, Shen, 2019).

5. Cases of Confirmation: Positive Results and Geometric Criteria

Despite these failures, the integral Hodge conjecture holds in several important situations:

• For one-cycles on principally polarized abelian varieties whose minimal cohomology class is algebraic, a lift of the Fourier transform to the integral Chow group can be constructed, and surjectivity follows for the relevant degrees (Beckmann et al., 2022).
• Jacobians of curves and their products always satisfy the conjecture for one-cycles.
• For Fermat quartic and quintic fourfolds, explicit lattice-theoretic computations confirm the conjecture, showing the equality of the lattice of algebraic classes and the lattice of Hodge classes (Aljovin et al., 2017).
• For threefolds of Kodaira dimension zero with $H^{2i}(V, \mathbb{Z})$8, Totaro’s degeneration and Noether–Lefschetz arguments show that the conjecture holds in codimension two (Totaro, 2019).
• For quadric surface bundles and fourfolds fibred by quadric bundles, the dominance and rational connectedness of suitable Abel–Jacobi maps allows one to apply Voisin’s criterion to confirm the conjecture in degree four (Li et al., 2014).

6. Real and Non-Archimedean Versions of the Integral Hodge Conjecture

The conjecture admits real and arithmetic analogues involving equivariant cohomology and specialized cycle class maps. For real varieties, the “real integral Hodge conjecture” asks for surjectivity of the cycle class map

$H^{2i}(V, \mathbb{Z})$9

where the target incorporates both Hodge and topological constraints, such as compatibility with Steenrod squares (Benoist et al., 2018, Fortman, 2022). Notably, for real abelian threefolds, the conjecture holds modulo torsion, and in many cases integrally, with reduction to the question of algebraicity for certain minimal classes or Ceresa cycles. Over non-archimedean real closed fields, counterexamples emerge due to the failure of archimedean approximation and the breakdown of real-analytic techniques essential in the real case (Benoist et al., 2018).

7. Broader Implications and Open Problems

The dichotomy between the rational and integral Hodge conjectures is now categorical: classical and recent counterexamples demonstrate that rational algebraicity does not imply integral algebraicity, and subtle torsion phenomena proliferate in higher codimension and for special classes of varieties (e.g., classifying spaces, products with Enriques surfaces, abelian varieties in high dimension) (Engel et al., 21 Jul 2025). The landscape has also expanded to categorical (noncommutative) versions and to connections with stable rationality, for instance through the failure of the conjecture on generic intermediate Jacobians or ppav’s, ruling out stably rational very general cubic threefolds (Perry, 2020, Engel et al., 21 Jul 2025).

A nontrivial open direction concerns the finiteness and structure of the defect group $\mathbb{C}$0, the universality or failure mechanisms for the conjecture on higher-dimensional hyperkähler manifolds, real or arithmetic analogues, and the development of refined cycle-theoretic or motivic obstructions (e.g., via higher unramified cohomology or derived categorical techniques).

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This encyclopedic summary provides a technical consolidation of the integral Hodge conjecture, its mechanisms of failure and confirmation, and the cycle- and cohomology-theoretic frameworks that structure contemporary and ongoing research.