Genus Bounds from Positive Closed Currents

Definitions and mathematical statement

Question. If a positive closed current $T$ with potential $\varphi$ satisfies\[T[\varphi]\ge\lambda[C]\qquad\text{for some }\lambda>0\text{ and a curve }C,\] must the geometric genus of $C$ be less than $2$? For a reduced irreducible compact complex curve, the geometric genus means $g(\widetilde C)=\dim_{{\mathbb{C}}}H^0(\widetilde C,\Omega^1_{\widetilde C})$, where $\widetilde C$ is its smooth normalization; the desired conclusion is then $g(\widetilde C)\in\{0,1\}$. The integration current $[C]$ acts on compatible test forms by integration along the regular part of $C$. An inequality of real currents means that the difference is a positive current of the same degree.

The source does not define the operation $T[\varphi]$, the ambient complex space, the degree of $T$, the sense of potential, or the hypotheses on $C$. Accordingly, $T[\varphi]$ is retained as an undefined source-specific notation; it is not identified with $T$, a wedge product, a restriction, or a regularization. The normalization definition above is conditional on $C$ being reduced, irreducible, and compact, properties not stated in the extracted question. A precise quantified conjecture requires the missing construction and geometric hypotheses; positivity and closedness alone do not specify them. In particular, no globally defined potential or ambient compact surface is imposed silently.

The supplied literature triage dated 2026-08-21 labels this partially solved. It reports that the genus bound is proved in important foliated-surface cases, whereas the general positive-current formulation was described as only partially advanced and no general proof or counterexample was located in that triage. No precise foliated-surface theorem or its hypotheses are supplied here, so that report does not fill the formulation gap or certify the general question's status.

Short English statement

The source asks whether a curve detected with positive coefficient by the current construction $T[\varphi]$ must have normalization of genus zero or one. The construction and ambient hypotheses are missing, so the inequality is preserved without assigning it an unsupported interpretation.

Sources

• Local UnsolvedMath registry, permanent ID 30000011, code catalogue entry; Genus Bounds from Positive Closed Currents, Oberwolfach Reports Open Problems.
• Multiplier Ideal Sheaves in Algebraic and Complex Geometry (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/21, pp. 1108--1166. https://doi.org/10.4171/owr/2004/21. Proposed year and year tag: 2004; no proposer supplied.
• Supplied literature label original OWR report: https://doi.org/10.4171/owr/2004/21.
• Supplied literature label McQuillan, curves on surfaces: https://www.mat.uniroma2.it/~mcquilla/files/charp.pdf. No fuller citation or theorem statement is supplied.
• Literature triage checked 2026-08-21, status partially_solved; record updated 2026-09-16. The registry's corrected_verified label records restoration of notation from supplied prose, not a supplied definition of $T[\varphi]$. Verify the dated assessment independently.