Voisin's conjecture: is the class of a very moving subvariety big?

A smooth subvariety $Y$ of a smooth complex projective variety $X$ is called very moving (Voisin, 2010) if through a general point of $X$ and for a general choice of $k$-plane in the tangent space there passes a deformation of $Y$ with exactly that tangent direction. Voisin conjectured that the cycle class of such a subvariety is big, that is, it lies in the interior of the cone of effective cycles. The conjecture is known for divisors, for curves with ample normal bundle (Peternell, Ottem), for very moving subvarieties of Grassmannians of lines, and for certain codimension-two subvarieties with split conormal bundle under strong positivity and movability hypotheses (Halic); Voisin showed that subvarieties which are merely moving with ample normal bundle need not have big class, so the tangent-preserving condition is essential. Whether the conjecture holds for all smooth very moving subvarieties of intermediate dimension remains open; a particular case would imply the generalized Hodge conjecture for coniveau-2 complete intersections.

Background

Let $X$ be a smooth complex projective variety. For each $0\le k\le\dim X$, the finite-dimensional real vector space $N_k(X)$ of $k$-cycles (formal real combinations of $k$-dimensional subvarieties) modulo numerical equivalence carries the cone of effective cycles $\mathrm{Eff}_k(X)$, spanned by the classes of actual subvarieties; a class is called big if it lies in the interior of this cone. This generalizes big divisors, and for curves the statement that a moving curve with ample normal bundle has big class is classical in spirit. In sharp contrast to divisors and curves, almost nothing is known about effective cones in intermediate dimensions.

Motivated by her strategy toward the generalized Hodge conjecture for coniveau-2 complete intersections (a refinement of the Hodge conjecture predicting which cohomology classes are supported on subvarieties of a given coniveau, i.e. supported away from a subset of prescribed codimension), Voisin introduced a purely geometric mobility condition: a smooth $k$-dimensional subvariety $Y\subset X$ is very moving if, through a general point $x\in X$ and for a general $k$-dimensional subspace $W\subset T_{X,x}$ of the tangent space, there is a deformation of $Y$ which is smooth at $x$, passes through $x$, and has tangent space exactly $W$ at $x$. She conjectured (Conjecture 0.7 of her 2010 paper) that the class $[Y]\in N_k(X)$ of a very moving subvariety is big. Voisin proved the conjecture for very moving subvarieties of the Grassmannian $G(1,n)$ of lines in $\mathbb{P}^n$ — including the varieties of lines on general complete intersections of multidegree $(d_1,\dots,d_r)$ in the numerical range $n\ge\sum_i d_i+d_r$ — and noted that the curve case follows from results of Peternell on curves with ample normal bundle. She also constructed, in every dimension $\ge 4$ and codimension 2, subvarieties that are moving in a weaker sense (a deformation can be prescribed only to pass through a general point, with no condition on its tangent direction) and have ample normal bundle, yet whose class is not big; hence any proof of the conjecture must use the full tangent-preserving strength of the very moving condition.

Halic's paper on G3-criteria approaches this circle of ideas from formal geometry. The G3-property of Hironaka–Matsumura asks that the formal completion of $X$ along $Y$ (the ringed space defined by powers of the ideal of $Y$) determine $X$ birationally via its field of rational formal functions; it is a strong, hard-to-check positivity property of the embedding. Halic proves that Voisin-type mobility hypotheses (movable, strongly movable, very moving) imply a partial positivity property, denoted $1^{>0}$, which is slightly stronger than G3, and, as a partial answer to Voisin's question, that a codimension-two subvariety $Y$ whose conormal bundle (the dual of the normal bundle) splits as $(\mathcal{L}_1^{-1}\oplus\mathcal{L}_2^{-1})\otimes\mathcal{O}_Y$ for semi-ample line bundles (line bundles some positive power of which is globally generated) $\mathcal{L}_1,\mathcal{L}_2$ pulled back from $X$, with $X$ simply connected and $Y$ movable in a family whose general fiber over the incidence map is irreducible and meeting every effective divisor non-trivially, is big once the normal bundle is ample (for instance when $\mathcal{L}_2$ is sandwiched between $\mathcal{L}_1$ and a positive power of $\mathcal{L}_1$). This line of attack on codimension-two subvarieties builds on Faltings' theorem that a codimension-two subvariety of $\mathbb{P}^n$ with split normal bundle and $\dim Y\ge n/2$ is a complete intersection, and is motivated by Hartshorne's 1974 conjecture that for $n\ge 6$ every smooth codimension-two subvariety of $\mathbb{P}^n$ is a complete intersection — a famous open problem that remains unsolved.

Problem Statement

Let $X$ be a smooth complex projective variety and let $Y\subset X$ be a smooth subvariety of dimension $k$ which is very moving in the sense of Voisin: through a general point $x\in X$ and for a general $k$-dimensional linear subspace $W\subset T_{X,x}$, there exists a deformation $Y'$ of $Y$ which is smooth at $x$, contains $x$, and has tangent space $T_{Y',x}=W$.

Voisin's conjecture (2010, Conjecture 0.7): the class $[Y]\in N_k(X)$ is big, that is, it lies in the interior of the cone of effective $k$-cycles of $X$.

Is this true in full generality - in particular in the intermediate range $2\le k\le\dim X-2$, beyond the settled cases (divisors; curves, which by Peternell and Ottem have big class whenever the normal bundle is ample; very moving subvarieties of Grassmannians of lines; and Halic's codimension-two split-conormal case with semi-ampleness, movability, and ampleness-of-normal-bundle hypotheses)? A complete answer is either a proof for all smooth pairs $(X,Y)$ as above, or an explicit counterexample consisting of a smooth very moving subvariety whose class is not in the interior of the effective cone.

References

1. Mihai Halic, G3-Criteria and Applications, arXiv:2401.16196 (2024; v3 May 2026), DOI 10.48550/arXiv.2401.16196, https://arxiv.org/abs/2401.16196
2. Claire Voisin, Coniveau 2 complete intersections and effective cones, Geometric and Functional Analysis 19 (2010), 1494-1513, DOI 10.1007/s00039-009-0036-0, arXiv:0809.0870, https://arxiv.org/abs/0809.0870
3. Thomas Peternell, Submanifolds with ample normal bundles and a conjecture of Hartshorne, arXiv:0804.1023 (2008), DOI 10.48550/arXiv.0804.1023, https://arxiv.org/abs/0804.1023
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6. Mihai Halic, Subvarieties with partially ample normal bundle, Mathematische Zeitschrift 293 (2019), 371-382, DOI 10.1007/s00209-018-2175-1, arXiv:1805.06418, https://arxiv.org/abs/1805.06418
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