Infinite Transverse Actions on Codimension-One Foliations

Definitions and mathematical statement

In the usual complex algebraic setting, a possibly singular codimension-one foliation on a smooth projective variety $X$ of dimension $n$ is described by a saturated coherent subsheaf $T_{\mathcal F}\subseteq T_X$ of rank $n-1$ that is closed under the Lie bracket. The supplied record does not explicitly specify the ground field or the permitted singularities; this conventional description requires confirmation against the source.

The supplied conjecture is the universally quantified implication \[ \left. \begin{gathered} X\text{ is a projective manifold},\\ \mathcal F\text{ is a codimension-one foliation on }X,\\ f\in\operatorname{End}(X,\mathcal F),\\ f\text{ has infinite transverse action} \end{gathered} \right\} \quad\Longrightarrow\quad \mathcal F\text{ is virtually transversely additive}. \] This is recorded as a conjecture, not as a proved implication.

The record does not define $\operatorname{End}(X,\mathcal F)$, "infinite transverse action," or "virtually transversely additive." Consequently, these expressions retain their source terminology in the displayed statement. In particular, it remains to determine whether the allowed maps are regular or dominant rational self-maps, what equivalence or induced action defines transverse infinitude, and what kind of covering and additive structure are required by the conclusion. These unresolved definitions prevent a fully specified formulation from the supplied record alone.

Short English statement

The conjecture asks whether a self-map preserving a codimension-one foliation on a projective manifold must force the foliation to be virtually transversely additive whenever its transverse action is infinite. The precise map class, transverse action, and meaning of the virtual additive structure require source confirmation.

Sources

• Local UnsolvedMath registry, supplied record: permanent ID $30004491$, title "Infinite Transverse Actions on Codimension-One Foliations," exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: Komplexe Analysis -- Algebraicity and Transcendence (2021), Oberwolfach Reports, pp. 1263--1313, DOI 10.4171/owr/2020/24. https://doi.org/10.4171/owr/2020/24. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Lo Bianco--Pereira--Rousseau--Touzet, Rational endomorphisms of codimension one foliations. https://doi.org/10.1515/crelle-2022-0023.
• Supplied literature source: "OWR Complex Analysis report." https://ems.press/content/serial-article-files/46857.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general conjecture open and reports proofs in restricted settings, including low-dimensional orbit closures and Picard-rank-one endomorphisms. It also reports that the unrestricted statement is explicitly presented as a conjecture in the source. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming preservation of the mathematical target and self-containedness. These are supplied review claims and do not resolve the missing definitions identified here.