Hartshorne's conjecture on the Chow groups of hypersurfaces: is $A^i(X)=\mathbb{Q}[h^i]$ for $i<n/2$?

For a smooth complex hypersurface $X\subset\mathbb{P}^{n+1}$ with hyperplane class $h$, Hartshorne conjectured in 1975 that the Chow group $A^i(X)$ of codimension-$i$ algebraic cycles modulo rational equivalence is generated by $h^i$ whenever $i<n/2$. The case $i=1$ is the classical Lefschetz theorem on the Picard group, but for $i\geq2$ the conjecture is open in general, apart from small-degree (Fano) cases, as reconfirmed in Laterveer's 2025 paper "Questions on the Chow ring of complete intersections" (arXiv:2512.06430). The conjecture remains open in its original generality.

Background

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface and let $h\in A^1(X)$ denote its hyperplane class, where $A^i(X)$ is the Chow group: the $\mathbb{Q}$-vector space of algebraic cycles of codimension $i$ on $X$ modulo rational equivalence. By the Lefschetz hyperplane theorem (the statement that the cohomology of $X$ below the middle degree coincides with that of projective space), one expects the "obvious" codimension-$i$ cycles with $i<n/2$ to be exactly the multiples of $h^i$. The case $i=1$ is settled: the Picard group of a smooth hypersurface of dimension at least $3$ is generated by $h$. Motivated by his philosophy that subvarieties of small codimension in a smooth variety should behave like complete intersections, Hartshorne (Equivalence relations on algebraic cycles and subvarieties of small codimension, Proc. Sympos. Pure Math. 29 (1975)) conjectured that $A^i(X)=\mathbb{Q}[h^i]$ for all $i<n/2$. The conjecture fits the standard expectation (a consequence of the conjectural Bloch–Beilinson filtration on Chow groups, namely that the graded pieces $A^i_{(j)}$ vanish for $j>0$ below the middle dimension) that Chow groups of hypersurfaces are as small as cohomology allows below the middle. Known partial results concern only special regimes: Otwinowska (C.R. Acad. Sci. 1999) treated hypersurfaces of small degree (the Fano range), and a body of work initiated by Voisin (Geom. Topol. 2012) and Fu (Adv. Math. 2013) proved multiplicativity statements - that for Calabi–Yau hypersurfaces the complementary product $A^i(X)\cdot A^j(X)$ with $i+j=n$ is one-dimensional, and that the small diagonal of a Calabi–Yau hypersurface admits a Chow-theoretic decomposition. These multiplicativity results constrain products of Chow groups but do not bound the individual groups $A^i(X)$, and one-dimensionality of products does not imply one-dimensionality of the factors. Laterveer's paper "Questions on the Chow ring of complete intersections" (arXiv:2512.06430, to appear in J. Math. Soc. Japan) systematically records which of the surrounding conjectures - Hartshorne's Conjecture 1.1, the multiplicativity Conjectures 1.2 and 1.4, Voisin's Conjecture 2.16 on the cycle $\Gamma$ from lines, and the existence of multiplicative Chow–Künneth (MCK) decompositions, i.e. splittings of the diagonal of $X$ into mutually orthogonal idempotent correspondences indexed by cohomological degree that additionally behave multiplicatively for the composition product on self-correspondences of $X$ - are open, and proves partial cases (quintic threefolds admit MCK decompositions, and the tautological ring $R^*(X^m)$ - the subring of the Chow ring of $X^m$ generated by the exterior products of divisor classes and by the diagonals - injects into cohomology for $m\leq205$, injectivity for all $m$ being equivalent to Kimura finite-dimensionality of the motive of $X$, the conjectural property, expected of all motives, that the motive splits into an even part whose alternating powers eventually vanish and an odd part whose symmetric powers eventually vanish).

Problem Statement

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, let $h\in A^1(X)$ be its hyperplane class, and let $A^i(X)$ denote the Chow group of codimension-$i$ algebraic cycles on $X$ modulo rational equivalence with $\mathbb{Q}$-coefficients. Hartshorne's conjecture (Conjecture 1.1 of Laterveer, arXiv:2512.06430, following Hartshorne 1975) asserts that for every $i<n/2$ one has $A^i(X)=\mathbb{Q}[h^i]$: every codimension-$i$ cycle on $X$ is rationally equivalent to a rational multiple of the $i$-th power of the hyperplane class. The case $i=1$ is known (Lefschetz), as are small-degree/Fano cases. Prove the conjecture in general - in particular, that $A^i(X)$ is one-dimensional for all $2\leq i<n/2$ - or refute it by exhibiting an explicit smooth hypersurface $X$ and an explicit codimension-$i$ cycle with $2\leq i<n/2$ whose class in $A^i(X)$ is not a rational multiple of $h^i$.

References

1. R. Laterveer, "Questions on the Chow ring of complete intersections", arXiv:2512.06430 (2025), to appear in J. Math. Soc. Japan, DOI 10.48550/arXiv.2512.06430, https://arxiv.org/abs/2512.06430
2. R. Hartshorne, "Equivalence relations on algebraic cycles and subvarieties of small codimension", Proc. Sympos. Pure Math. 29 (1975), 129–164, DOI 10.1090/pspum/029/0369359, https://doi.org/10.1090/pspum/029/0369359
3. C. Voisin, "Chow rings and decomposition theorems for K3 surfaces and Calabi–Yau hypersurfaces", Geometry & Topology 16 (2012), 433–473, DOI 10.2140/gt.2012.16.433, arXiv:1102.1607, https://doi.org/10.2140/gt.2012.16.433
4. L. Fu, "Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi–Yau complete intersections", Advances in Mathematics 244 (2013), 894–924, DOI 10.1016/j.aim.2013.06.008, arXiv:1209.5616, https://doi.org/10.1016/j.aim.2013.06.008
5. A. Otwinowska, "Remarques sur les groupes de Chow des hypersurfaces de petit degré", C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 51–56, DOI 10.1016/S0764-4442(99)80460-1, https://doi.org/10.1016/S0764-4442(99)80460-1
6. S. Kimura, "Chow groups are finite dimensional, in some sense", Mathematische Annalen 331 (2005), 173–201, DOI 10.1007/s00208-004-0577-3, https://doi.org/10.1007/s00208-004-0577-3
7. Z. Xu, "Motivic multiplicativity of complete intersections", arXiv:2511.01362 (v2, 2026), https://arxiv.org/abs/2511.01362