Twisted symmetric differential vanishing for smooth subvarieties of projective space in the Hartshorne complete-intersection regime

For a smooth complex subvariety $X\subset\mathbb{P}^N$, Etesse (2022; published 2025) proved that when $X$ is a complete intersection of codimension $c<N/2$, the graded space of twisted symmetric differentials $\bigoplus_{m\geq\max(d-1,1)}H^0(X,S^m\Omega_X(m+\max(d-2,0)))$ vanishes if and only if $X$ is contained in no hypersurface of degree $2\leq i\leq d$, for every $d\geq 2$ (the grading is stated in its corrected form; see the Background). He posed, as a natural open problem and as potential evidence toward Hartshorne's complete-intersection conjecture, the extension of this equivalence to arbitrary smooth subvarieties of codimension $c\leq N/3$ - the range $c<N/3$ in which Hartshorne's conjecture predicts that $X$ should be a complete intersection, together with its boundary $c=N/3$, where the conjecture makes no prediction. Only the codimension $\leq 2$ situation with $d=2$ (the quadric case) is known for general subvarieties, by Bogomolov–De Oliveira and De Oliveira–Langdon; the problem, including its $d=2$ case in codimension $\geq 3$ (to which the open De Oliveira–Langdon conjecture is closely related), remains open.

Background

Let $X$ be a smooth complex projective variety embedded in projective space $\mathbb{P}^N$. Its cotangent bundle $\Omega_X=\Omega_X^1$ is the bundle of holomorphic 1-forms; $S^m\Omega_X$ denotes its $m$-th symmetric power (the bundle of holomorphic symmetric differential $m$-forms), and $S^m\Omega_X(k)$ is its twist by $k$ times the hyperplane line bundle $\mathcal{O}_X(1)$. Sections of these twisted bundles - twisted symmetric differentials - encode positivity properties of $\Omega_X$: their presence or absence controls embedding obstructions (Schneider used them to obstruct re-embeddings), the algebra $Q(X,1)=\bigoplus_{m\geq 0}H^0(X,S^m\Omega_X(m))$ reflects first-order infinitesimal information about the embedding $X\subset\mathbb{P}^N$, and the existence of many symmetric differentials is a route to algebraic hyperbolicity, i.e. to showing that every nonconstant holomorphic map from $\mathbb{C}$ into $X$ has image contained in a proper subvariety. A classical vanishing theory for these groups goes back to Brückmann–Rackwitz (1990), who proved strong vanishing for smooth complete intersections (subvarieties cut out by exactly codimension many homogeneous equations of degrees $d_1,\dots,d_c$); Schneider (1992) and later Brotbek (2011) showed that the higher-cohomology parts of these theorems extend to arbitrary smooth subvarieties in the same codimension range, but the global-section statements remained available only for complete intersections. Bogomolov–De Oliveira (2008) initiated the study for arbitrary smooth subvarieties: they proved that for a smooth subvariety of codimension $c\in\{1,2\}$ with $c<N/2$, the algebra $\bigoplus_{m\geq 1}H^0(X,S^m\Omega_X(m))$ vanishes exactly when $X$ is not contained in a quadric hypersurface, using polynomials that are tangentially homogeneous (vanishing on the cone over the embedded tangent space at each point). De Oliveira–Langdon (2019) conjectured (their Conjecture 1.1) that for smooth nondegenerate $X\subset\mathbb{P}^N$ of low codimension, $Q(X,1)$ is isomorphic as a graded algebra to the quadric algebra $QA(X)=\mathrm{Sym}(H^0(\mathbb{P}^N,\mathcal{I}_X(2)))$, where $\mathcal{I}_X(2)$ is the vector space of quadrics vanishing on $X$, so that $QA(X)$ is the graded algebra generated by the quadrics containing $X$; they proved this for codimension $2$ with $\dim X\geq 3$. Etesse (arXiv:2211.08955; European Journal of Mathematics, 2025) gave completely explicit resolutions of the structural sheaves of the projectivized tangent bundles of smooth complete intersections and used them to compute all the cohomology of $S^m\Omega_X(m+n)$. Among his applications, his Theorem 0.0.10 (= Theorem 4.1.3) states: if $X\subset\mathbb{P}^N$ is a nondegenerate (i.e. not contained in any hyperplane) smooth complete intersection of codimension $c<N/2$, then for every integer $d\geq 2$ (as printed in arXiv:2211.08955 the summation range and twist are one step lower, $m\geq\max(d-2,1)$ with twist $m+\max(d-3,0)$; the grading displayed here and throughout this task is the corrected one),

$$\bigoplus_{m\geq\max(d-1,1)}H^0\left(X,S^m\Omega_X\left(m+\max(d-2,0)\right)\right)=0$$

if and only if $X$ is contained in no hypersurface of degree $2\leq i\leq d$. The interest in codimension at most $N/3$ comes from Hartshorne's 1970 conjecture: a smooth subvariety $X\subset\mathbb{P}^N$ of dimension greater than $\tfrac{2}{3}N$ (equivalently codimension less than $N/3$) should be a complete intersection. In that range one therefore expects the complete-intersection theorem to hold for all smooth subvarieties, and Etesse explicitly poses proving such an extension as a natural open problem, noting that it would constitute evidence toward Hartshorne's conjecture; he also records that even the underlying Brückmann–Rackwitz global-section vanishing is not known to extend to arbitrary smooth subvarieties of codimension less than $N/2$.

Problem Statement

Let $X\subset\mathbb{P}^N$ (over $\mathbb{C}$) be a nondegenerate smooth subvariety (i.e. not contained in any hyperplane of $\mathbb{P}^N$) of codimension $c\leq N/3$. Prove, or disprove, that for every integer $d\geq 2$ one has the equivalence

$$\bigoplus_{m\geq\max(d-1,1)}H^0\left(X,S^m\Omega_X\left(m+\max(d-2,0)\right)\right)=0$$

if and only if $X$ is contained in no hypersurface of degree $i$ with $2\leq i\leq d$. This is Etesse's Theorem 0.0.10, in the corrected grading described in the Background, with the complete-intersection hypothesis removed (there proved for smooth complete intersections of any codimension $c<N/2$); the codimension restriction $c\leq N/3$ is the range $c<N/3$ in which Hartshorne's conjecture predicts that such $X$ is a complete intersection, together with its boundary $c=N/3$, where the conjecture makes no prediction; Etesse poses the problem as a test of that expectation. The central open direction is the vanishing statement: a smooth $X$ of codimension $\leq N/3$ that lies on no hypersurface of degree $2\leq i\leq d$ should have no twisted symmetric differentials in the stated grading and twist. Known cases: arbitrary smooth $X$ of codimension $c\leq 2$ with $d=2$ (Bogomolov–De Oliveira; De Oliveira–Langdon), and all $d\geq 2$ for complete intersections of codimension $<N/2$ (Etesse).

References

1. Antoine Etesse, Cohomology of twisted symmetric powers of cotangent bundles of smooth complete intersections, arXiv:2211.08955 (2022), https://arxiv.org/abs/2211.08955
2. Antoine Etesse, Cohomology of twisted symmetric powers of cotangent bundles of smooth complete intersections, European Journal of Mathematics 11(3), Article 51 (2025), doi:10.1007/s40879-025-00834-9, https://doi.org/10.1007/s40879-025-00834-9
3. Fedor Bogomolov and Bruno De Oliveira, Symmetric tensors and geometry of $\mathbb{P}^N$ subvarieties, Geometric and Functional Analysis 18(3), 637–656 (2008), doi:10.1007/s00039-008-0666-7, https://doi.org/10.1007/s00039-008-0666-7
4. Bruno De Oliveira and Christopher Langdon, Twisted symmetric differentials and the quadric algebra of subvarieties of $\mathbb{P}^N$ of low codimension, European Journal of Mathematics 5(2), 454–475 (2019), doi:10.1007/s40879-018-0265-6, https://doi.org/10.1007/s40879-018-0265-6
5. Michael Schneider, Symmetric differential forms as embedding obstructions and vanishing theorems, Journal of Algebraic Geometry 1(2), 175–181 (1992), Zbl 0790.14009, https://zbmath.org/232512
6. P. Brückmann and H.-G. Rackwitz, $T$-symmetrical tensor forms on complete intersections, Mathematische Annalen 288, 627–635 (1990), doi:10.1007/BF01444555, https://doi.org/10.1007/BF01444555
7. Damian Brotbek, Differential equations as embedding obstructions and vanishing theorems, arXiv:1111.5324 (2011), https://arxiv.org/abs/1111.5324
8. Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics 156, Springer-Verlag, Berlin (1970), doi:10.1007/BFb0067839, https://doi.org/10.1007/BFb0067839