Smoothability of algebraic cycles below the middle dimension in positive characteristic: the surviving core of Hironaka's smoothing problem

Hironaka (Amer. J. Math. 90, 1968) proved that, over an algebraically closed field of characteristic zero, every algebraic $d$-cycle on a smooth variety of dimension $n$ with $d\le\min(3,(n-1)/2)$ is rationally equivalent to a $\mathbb{Z}$-linear combination of smooth subvarieties, and asked whether the theorem extends to every $d$ with $2d<n$ and whether it holds over fields of positive characteristic, noting that his proof depends on resolution of singularities, then available only in characteristic zero. The characteristic-zero question is now resolved: Kollár and Voisin (Annals of Mathematics 200, 2024) proved that for every smooth projective variety over a field of characteristic zero and every $d$ with $2d<n$, the Chow group $CH_d(X)$ is generated by classes of smooth subvarieties. The open core that survives is the positive-characteristic case: the statement is known for $d\le3$ over infinite perfect fields, for $d=1$ over finite fields, and, over infinite perfect fields, in characteristics $p\ge n-d$, but it remains open for $d\ge4$ when $p<n-d$ — where current methods only show that suitable $p$-power multiples of a cycle are smoothable — and over finite fields for $d\ge2$. The problem asks for a complete answer in these remaining cases; the source's auxiliary questions (a classification of the generic linkage singularities for $d\ge4$, and whether the embedded component of its Proposition 3.9 can occur) also remain open.

Background

Let $k$ be a field and $X$ a smooth projective variety of dimension $n$ over $k$. An algebraic $d$-cycle on $X$ is a finite integer linear combination $Z=\sum_i a_i Z_i$ of closed irreducible $d$-dimensional subvarieties. Two $d$-cycles are rationally equivalent if their difference is generated by differences of fibers of algebraic families of cycles parametrized by the projective line $\mathbb{P}^1$ (that is, by cycles on $X\times\mathbb{P}^1$ flat over $\mathbb{P}^1$); this is the algebraic analogue of homotopy, and $d$-cycles modulo rational equivalence form the Chow group $CH_d(X)$, the algebro-geometric counterpart of singular homology (over $k=\mathbb{C}$, a cycle determines a homology class, and rational equivalence implies homological equivalence, that is, equality of homology classes). A $d$-cycle is called smoothable if it is rationally equivalent to a $\mathbb{Z}$-linear combination of classes of smooth $d$-dimensional subvarieties of $X$.

Borel and Haefliger asked in 1961, for homological equivalence over $\mathbb{C}$, whether the classes of smooth subvarieties generate all algebraic homology classes; the rational-equivalence version over an arbitrary field asks whether $CH_d(X)$ is generated by classes of smooth subvarieties. The answer depends sharply on where $d$ sits relative to $n/2$. Above the middle dimension ($2d\ge n$) the answer is negative in general: Hartshorne, Rees and Thomas proved in 1974 that the second Chern class $c_2$ of the tautological rank-3 subbundle on the Grassmannian $G(3,6)$ (the smooth projective variety parametrizing 3-dimensional subspaces of $\mathbb{C}^6$; its tautological subbundle has as fiber over a point the corresponding 3-space, and $c_2$ is a characteristic class of it, defining a codimension-2 algebraic cycle) is not a $\mathbb{Z}$-linear combination of classes of smooth subvarieties. Subsequent work surveyed in Benoist's Bourbaki exposé 1228 produced many further counterexamples above the middle dimension, including at the boundary $2d=n$ (Benoist, 2024, for every $d$ whose binary digit sum $\alpha(d+1)$ is at least 3, the smallest case being $d=6$). Below the middle dimension ($2d<n$) — the Whitney range, by analogy with Whitney's weak embedding theorem in differential topology, where a generic map from a $d$-manifold to an $n$-manifold is an embedding when $2d<n$ — no counterexample is known in any characteristic, and the topological analogue is a theorem.

The first general positive result below the middle dimension is the source of this problem. Hironaka (1968) proved that, over an algebraically closed field of characteristic zero, every $d$-cycle with $d\le\min(3,(n-1)/2)$ is smoothable. His method: resolve the singularities of the subvarieties $Z_i$ (using his resolution-of-singularities theorem, which desingularizes any variety in characteristic zero by a sequence of blow-ups — modifications replacing a subvariety by the projectivization of its normal directions), embed a resolution in $X\times\mathbb{P}^N$, and replace the cycle by a rationally equivalent one in general position by linkage: writing a general complete intersection of hypersurfaces (that is, a subvariety cut out by the right number of hypersurfaces) containing the resolved subvariety $\tilde Z$ as $\tilde Z\cup W$, so that $[\tilde Z]$ is rationally equivalent to the difference of the complete intersection and $[W]$, and iterating. By a Bertini-type theorem (general members of a sufficiently ample linear system are smooth away from the base locus), the residual $W$ of a general linkage is smooth outside a locus of dimension at most $d-4$; this is exactly why the method stops at $d=3$. For $d=4$ the generic singularity is isolated and formally isomorphic to the rank-$\le1$ determinantal singularity $k[x_1,x_2,x_3,y_1,y_2,y_3]/(x_i y_j-x_j y_i)$ (the equations say that the $2\times3$ matrix with rows $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ has rank at most 1), which is the cone over the Segre embedding $\mathbb{P}^1\times\mathbb{P}^2\subset\mathbb{P}^5$ (the Segre map sends a pair of points to the six coordinatewise products of their homogeneous coordinates); and Hironaka already observed — with the computation made explicit in Example 9 of Kollár's lecture notes — that a general linkage residual is always singular once $d\ge4$, so the method cannot be iterated to full smoothness. Independently, Kleiman (1969) proved, via the Grothendieck–Riemann–Roch formula without denominators — the formula $c_c(\mathcal{O}_Z)=(-1)^{c-1}(c-1)![Z]$ (due to Jouanolou) for the $c$-th Chern class of the structure sheaf of a codimension-$c$ subvariety $Z$ — together with a study of Schubert varieties (the subvarieties of a Grassmannian defined by incidence conditions with a fixed flag of subspaces) in Grassmannians, that in the Whitney range $2d<n$ the multiple $(c-1)!\,z$ of any cycle $z$ of dimension $d$ and codimension $c=n-d$ is smoothable; and Hironaka's paper itself proves that in arbitrary characteristic a fixed multiple $\varepsilon Z$, with $\varepsilon$ depending only on $(n,d)$, is smoothable. The substantive question is whether the cycle itself — with no multiple — is smoothable.

In characteristic zero this is now a theorem. Kollár and Voisin (Annals of Mathematics, 2024) proved that for every smooth projective variety $X$ over a field of characteristic zero and every $d$ with $2d<n$, the group $CH_d(X)$ is generated by classes of smooth subvarieties. Their proof does not attempt to control linkage singularities; instead they prove the structural theorem that every cycle on $X$ is a flat pushforward of an intersection of divisor classes (flat meaning equidimensional fibers, for morphisms between smooth projective varieties), and then smooth such pushforwards by a general-position projection argument in the Whitney range.

What remains open is the positive-characteristic analogue, and the obstruction is concrete. The known proofs consume resolution of singularities of the subvarieties involved, and resolution is known in positive characteristic only up to dimension three (Cossart–Piltant). The best substitute — de Jong's alterations (proper, surjective, generically finite maps from a smooth variety onto the given variety) and their refinements — only lets one combine a family of such covers whose degrees have greatest common divisor a power of $p$, which is exactly why it produces smoothability of $p$-power multiples but not of the cycle itself. Over finite fields an additional obstacle appears: general members of a linear system cannot be chosen by a naive genericity argument, and one needs sieve-theoretic Bertini theorems (Bertini-type smoothness statements proved over finite fields by counting closed points of bounded degree instead of genericity arguments), so far adapted to this problem only for $d=1$.

Problem Statement

The problem is the smoothing problem for algebraic cycles of small dimension posed by Hironaka in 1968, restricted to the cases still open after the later literature.

Setup. Let $k$ be a field, $X$ a smooth projective variety of dimension $n$ over $k$, and $d\le(n-1)/2$ an integer (equivalently, the Whitney range $2d<n$). An algebraic $d$-cycle on $X$ is a finite sum $Z=\sum_i a_i Z_i$ with $a_i\in\mathbb{Z}$ and $Z_i$ closed irreducible $d$-dimensional subvarieties of $X$. Two $d$-cycles are rationally equivalent if their difference is generated by differences of fibers, over pairs of $k$-rational points of the projective line $\mathbb{P}^1$, of cycles on $X\times\mathbb{P}^1$ whose fibers over $\mathbb{P}^1$ all have dimension exactly $d$; $d$-cycles modulo rational equivalence form the Chow group $CH_d(X)$. A $d$-cycle $Z$ is smoothable if its class in $CH_d(X)$ is a $\mathbb{Z}$-linear combination of classes of smooth closed $d$-dimensional subvarieties of $X$.

What the source asked. Hironaka proved that, over an algebraically closed field of characteristic zero, every $d$-cycle with $d\le\min(3,(n-1)/2)$ is smoothable, and asked: (i) does the same hold for every $d$ with $2d<n$? (ii) is the theorem valid over fields of positive characteristic? He recorded that his proof depends on birational desingularization of subvarieties, known only in characteristic zero at the time, and that his technique already gives, for every $d$ in the range, rational equivalence to a cycle with singularities of dimension at most $d-4$. The paper also proves that in arbitrary characteristic a fixed multiple $\varepsilon Z$ (with $\varepsilon$ depending only on $(n,d)$) is smoothable.

Status established by later work. Question (i) is resolved for smooth projective varieties in characteristic zero: Kollár and Voisin proved that for every $d$ with $2d<n$, the Chow group $CH_d(X)$ is generated by classes of smooth subvarieties. Question (ii) is partially resolved: the statement holds over infinite perfect fields (a perfect field is one all of whose algebraic extensions are separable; finite fields and algebraically closed fields are perfect) when $d\le3$ (any characteristic, via resolution of singularities in dimension at most 3) or when $\operatorname{char}k\ge n-d$ (via alterations), and over finite fields when $d=1$; in all these cases the cycle itself (no multiple) is smoothable.

The open core to be answered. Let $X$ be a smooth projective variety of dimension $n$ over a field $k$ of characteristic $p>0$, with $2d<n$:

(a) If $k$ is infinite and perfect and $p<n-d$ with $d\ge4$: is every $d$-cycle on $X$ smoothable? (Known at present: only that $p^m Z$ is smoothable, $p^m$ the largest power of $p$ dividing $(n-d-1)!$, and that a fixed multiple of $Z$ — with the multiple depending only on $(n,d)$ — is smoothable in arbitrary characteristic.)

(b) If $k$ is finite and $d\ge2$: is every $d$-cycle on $X$ smoothable?

Auxiliary open questions from the same source, not part of the completion boundary: whether the embedded component $W$ of the linkage construction of the source paper (its moving technique, described in the background above) in Proposition 3.9 (case $t>n-d$) can be nonzero, and the classification of the generic singularities of a general linkage for $d\ge4$ (for $d=4$, an isolated rank-$\le1$ determinantal singularity); both remain open in general and are bypassed by the new methods.

References

1. Heisuke Hironaka, Smoothing of Algebraic Cycles of Small Dimensions, American Journal of Mathematics 90, No. 1 (1968), pp. 1-54. DOI: 10.2307/2373425, https://doi.org/10.2307/2373425
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