For a smooth projective complex variety XX, ask whether every algebraic cohomology class with rational coefficients is a rational linear combination of classes of smooth algebraic subvarieties - the rational version of the Borel–Haefliger smoothing question, posed as Question 0.1 by Benoist and Voisin (2024/2026). With integral coefficients the answer is negative (Hartshorne–Rees–Thomas 1974), but no rational counterexample is known, and classical topological obstructions vanish rationally. Benoist and Voisin proved that a positive answer would refute Hartshorne's 1974 conjecture that smooth subvarieties of small codimension in projective space are complete intersections, so the two central 'tameness' expectations cannot both hold. As of August 2026 the question is open in general, including for the original Hartshorne–Rees–Thomas class c2c_2 of the tautological bundle on the Grassmannian G(3,6)G(3,6).

Background

By an algebraic cycle on a smooth projective complex variety XX we mean a finite integer combination of irreducible algebraic subvarieties; a cycle has a cohomology class, and two cycles are homologically equivalent when their classes coincide. For c≥0c\geq 0, H2c(X,Q)algH^{2c}(X,\mathbb{Q})_{\mathrm{alg}} denotes the Q\mathbb{Q}-vector subspace of cohomology spanned by classes of codimension-cc algebraic subvarieties. Borel and Haefliger asked in 1961 whether, with integral coefficients, this space is generated by classes of smooth subvarieties - i.e. whether arbitrary algebraic cycles can be 'smoothed' up to homological equivalence. The answer with Z\mathbb{Z}-coefficients is negative in general: Hartshorne, Rees and Thomas showed that on the Grassmannian G(3,6)G(3,6) (the variety of 33-planes in C6\mathbb{C}^6) the second Chern class of the tautological rank-33 vector bundle (Chern classes are the standard characteristic cohomology classes of a vector bundle; the tautological bundle on a Grassmannian is the bundle whose fiber over a point [W][W] is the subspace WW itself) is not an integral combination of classes of smooth codimension-22 subvarieties, and further integral counterexamples were later given by Benoist–Debarre (2023) and Benoist (2024). Two positive results bound the phenomenon: Kleiman (1969) proved that with rational coefficients the answer is positive in the high-codimension range c>n/2−1c>n/2-1 (where n=dim⁡Xn=\dim X), and Kollár and Voisin (2024), extending Hironaka, proved smoothability below the middle dimension even with integral coefficients (c>n/2c>n/2). Rational coefficients also kill the classical topological obstructions, so no counterexample with Q\mathbb{Q}-coefficients is known. In a different direction, Hartshorne's famous 1974 conjecture on complete intersections predicts that a smooth subvariety of codimension cc in projective space Pn\mathbb{P}^n is a complete intersection (a variety defined by exactly cc equations) whenever nn is sufficiently large compared to cc; its codimension-22 case is equivalent to Conjecture 1.1 in Benoist–Voisin, asserting that there is n0≥5n_0\geq 5 such that every rank-22 algebraic vector bundle on Pn\mathbb{P}^n, n≥n0n\geq n_0, is decomposable, i.e. a direct sum of two line bundles (Hartshorne's suggested value is n0=7n_0=7). The apparently weaker 'unstable' variant - restricted to bundles that are not slope-stable (slope stability is the standard stability condition for vector bundles, requiring every proper subsheaf to have smaller slope, i.e. a smaller ratio of first Chern class to rank) - was attacked by Grauert–Schneider (1977) and in a 2015 preprint of Siu, but Benoist–Voisin report that both proofs have serious gaps. Benoist and Voisin (2024, published 2026) proved a striking incompatibility (their Theorem 0.3): the rational smoothing question and Hartshorne's conjecture cannot both have positive answers, even in codimension 22 - if smooth codimension-22 subvarieties of projective space are eventually complete intersections, then codimension-22 cycles on Grassmannians G(k,n)G(k,n) with k,n−k≫0k,n-k\gg 0 fail rational smoothability.

Problem Statement

Let XX be a smooth projective variety over C\mathbb{C} and let c≥0c\geq 0 be an integer. Is the Q\mathbb{Q}-vector space H2c(X,Q)algH^{2c}(X,\mathbb{Q})_{\mathrm{alg}} - the subspace of cohomology spanned by classes of codimension-cc algebraic subvarieties of XX - generated by classes of smooth subvarieties of codimension cc in XX? This is Question 0.1 of Benoist–Voisin (the rational-coefficient form of the Borel–Haefliger smoothing question). A complete answer is either (i) a proof that the answer is yes for every smooth projective XX and every cc, or (ii) an explicit counterexample: a specific smooth projective variety XX, a codimension cc, and a class in H2c(X,Q)algH^{2c}(X,\mathbb{Q})_{\mathrm{alg}} together with a proof that it is not a Q\mathbb{Q}-linear combination of classes of smooth codimension-cc subvarieties. Note that by Theorem 0.3 of Benoist–Voisin a positive answer in general would refute Hartshorne's conjecture on complete intersections, so the two possible resolutions have directly identifiable consequences for that famous conjecture; the paper's own first test case is the class c2c_2 (the second Chern class) of the tautological rank-33 bundle on the Grassmannian G(3,6)G(3,6) (the variety of 33-planes in C6\mathbb{C}^6), for which rational smoothability is currently unknown.

References

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