Negativity of Log Divisors on Q-Acyclic Surfaces

Definitions and mathematical statement

In the algebraic-geometric setting of the supplied record, a complex surface is a complex algebraic variety of dimension two. A nonempty surface $S$ is ${\mathbb{Q}}$-acyclic if its complex points, with their analytic topology, satisfy \[ H_0(S({\mathbb{C}});{\mathbb{Q}})\cong{\mathbb{Q}},\qquad H_i(S({\mathbb{C}});{\mathbb{Q}})=0\quad\text{for every integer }i>0, \] where $H_i$ denotes singular homology. Use the standard convention that a log-smooth completion $(X,D)$ consists of a smooth projective complex surface $X$ and a reduced simple normal crossings divisor $D$ with $S\cong X\setminus\operatorname{Supp}(D)$. Thus the components of $D$ are smooth and meet transversely, with at most two components through any point. Let $K_X$ denote a canonical divisor of $X$, and let $\kappa$ denote the Iitaka dimension of a rational divisor. In particular, \[ \kappa\!\left(K_X+\tfrac12D\right)=-\infty \quad\Longleftrightarrow\quad H^0\!\left(X,\mathcal O_X\left(m(2K_X+D)\right)\right)=0 \quad\text{for every integer }m\geq1. \] Negativity Conjecture, as supplied. For every complex ${\mathbb{Q}}$-acyclic surface $S$ and every log-smooth completion $(X,D)$ of $S$, is it true that \[ \kappa\!\left(K_X+\tfrac12D\right)=-\infty? \] Qualification concerning the domain. Under the completion convention above, $S$ is necessarily smooth because it is an open subset of the smooth surface $X$. The supplied statement does not separately specify smoothness or define its completion convention. If singular ${\mathbb{Q}}$-acyclic surfaces are intended through a resolution-based meaning of completion, the required resolution and boundary conventions are missing; that extension is not formulated here.

Short English statement

The supplied Negativity Conjecture asks whether every complex ${\mathbb{Q}}$-acyclic surface with a log-smooth completion $(X,D)$ has $\kappa(K_X+\tfrac12D)=-\infty$. Equivalently, every positive integral multiple of $2K_X+D$ has no nonzero global sections. The standard completion convention forces the surface to be smooth; any intended singular-surface formulation requires clarification.

Sources

• Local UnsolvedMath registry, supplied source record for permanent ID $30004856$, source problem code catalogue entry, titled "Negativity of Log Divisors on Q-Acyclic Surfaces." The record supplies status "open" and statement status corrected_verified, together with a claim of manual semantic, typographic, and LaTeX review. These are supplied registry assessments, not independent verification here.
• Supplied report citation: Singularities (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/47, pp. 2573--2639, https://doi.org/10.4171/owr/2021/47. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. It supplies proposed year 2021 and no named proposer. Both the cited 2022 title and the supplied 2021 year are retained without reconciliation.
• The report excerpt labels the conjecture "Negativity Conjecture, [Pal19, 4.7]." No full bibliographic entry for [Pal19] is supplied, so its author and publication details cannot be identified from this record alone.
• Supplied literature source: "Pełka, smooth Q-homology planes satisfying negativity," https://doi.org/10.1112/jlms.12779. The supplied assessment says that Pełka classifies the smooth ${\mathbb{Q}}$-homology planes satisfying the conjecture; this is a conditional classification, not a proof of the universal assertion. The excerpt attributes the corresponding thesis work to T. Pełka and mentions almost log exceptional curves as a tool for grouping such surfaces and rational cuspidal curves into series.
• Supplied literature source: "OWR statement that the conjecture remains open," https://doi.org/10.4171/owr/2021/47.
• Literature assessment date: 2026-08-22. The supplied dated curation describes the conjecture as still open and distinguishes Pełka's conditional classification from the universal conjecture. This assessment is supplied dated curation, not independently verified; the record explicitly requests independent verification of time-sensitive status and cited claims.