Completeness Tests for Integral Subvarieties

Definitions and mathematical statement

The supplied problem concerns a smooth projective variety $X$ over ${\mathbb{C}}$, with $\dim X>2$, and a section \[ \omega\in H^0\!\left(X,L\otimes S^r\Omega^1_{X/{\mathbb{C}}}\right), \] where $\Omega^1_{X/{\mathbb{C}}}$ is the cotangent bundle and $S^r$ denotes the $r$-th symmetric power. The source does not explicitly specify the nature of $L$, the range of $r$, or the meaning of an $\omega$-integral hypersurface.

For a provisional precise formulation, interpret $L$ as a line bundle and $r$ as a positive integer. For a reduced irreducible closed subvariety $Y\subset X$, let $j:Y_{\mathrm{sm}}\hookrightarrow X$ denote the inclusion of its smooth locus. Under the restriction-vanishing convention, call $Y$ $\omega$-integral if the image of $j^*\omega$ under the natural map \[ j^*L\otimes S^r(j^*\Omega^1_{X/{\mathbb{C}}}) \longrightarrow j^*L\otimes S^r\Omega^1_{Y_{\mathrm{sm}}/{\mathbb{C}}} \] is identically zero. This convention includes singular subvarieties by testing on their smooth loci; its agreement with the intended source terminology, especially for hypersurfaces, requires confirmation.

For $d\in\{1,\dim X-1\}$, write \[ \mathcal I_d(X,\omega)= \{Y\subset X:Y\text{ is reduced, irreducible, closed, }\dim Y=d, \text{ and }Y\text{ is }\omega\text{-integral}\}. \] Problem, conditional on these conventions. Given $X,L,r,\omega$ and a listed collection $\mathcal L_d\subseteq\mathcal I_d(X,\omega)$, find a method or criterion for checking completeness: \[ \mathcal I_d(X,\omega)=\mathcal L_d, \] equivalently, \[ \forall Y\in\mathcal I_d(X,\omega),\qquad Y\in\mathcal L_d. \] The cases $d=1$ and $d=\dim X-1$ concern curves and hypersurfaces, respectively. The source does not specify whether the list is finite, how it is represented, or whether the requested check must be an algorithm with a termination guarantee. No nonvanishing assumption on $\omega$ is supplied. These qualifications remain unresolved.

Short English statement

Given a smooth projective complex variety of dimension greater than two and a twisted symmetric differential $\omega$, seek a way to certify that a supplied list contains every $\omega$-integral curve or hypersurface. The precise integrality convention and the computational meaning of a completeness test require clarification; the formulation above makes its provisional conventions explicit.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003254$, title Completeness Tests for Integral Subvarieties, and exact source problem code catalogue entry. No individual proposer is identified. The record gives proposed year 2016.
• Supplied report citation: Definability and Decidability Problems in Number Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/49, pp. 2793--2866. https://doi.org/10.4171/owr/2016/49. This citation is reproduced as supplied; the problem's precise location within the report is not given.
• Supplied literature source: García-Fritz, Curves on surfaces. https://qspace.library.queensu.ca/bitstreams/007eb1de-ecf4-4747-a756-f7cac789ba56/download.
• Supplied literature source: Oberwolfach Reports 2017. https://ems.press/content/serial-article-files/46652.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that completeness criteria are available in dimension two, while no general method for checking completeness of the higher-dimensional list is known. This is supplied dated curation, not an independently verified literature assessment. The record also labels its statement corrected_verified and reports a manual semantic, typographic, and LaTeX review; these are supplied editorial claims, not independent mathematical verification.