Numerical and Nef Dimensions of Line Bundles

Definitions and mathematical statement

The source asks for a geometric explanation of strict inequality between numerical and nef dimensions of a nef line bundle. To make the standard invariants explicit, use a smooth projective irreducible complex variety $X$ of dimension $d$; the original does not specify its base field, and the nef-reduction formulation over other fields requires separate hypotheses.

A line bundle $L$ is nef if $L\cdot C\geq0$ for every integral curve $C\subset X$. For an ample line bundle $A$, its numerical dimension is \[ \nu(L)=\max\{0\leq k\leq d:c_1(L)^k\cdot c_1(A)^{d-k}>0\}, \] independent of the ample choice in this nef setting. Its nef dimension $n(L)$ is the dimension of the base of a nef reduction $f:X\dashrightarrow Y$: a dominant almost-holomorphic map with connected general fibers, numerically trivial restriction of $L$ to a general compact fiber, and $L\cdot C>0$ for curves through a very general point which are not contracted. The map is considered up to the usual birational equivalence.

The question is to describe geometric conditions and structures accounting for \[ \nu(L)<n(L), \] rather than merely computing the two numbers. In this standard setting $\nu(L)\leq n(L)$; the gap compares intersection-theoretic positivity with the size of the base needed to account for numerical triviality along nef-reduction fibers. The source gives no proposed necessary-and-sufficient criterion or specified meaning of "mechanism." Numerically trivial foliations and nef-reduction geometry are reported as partial structural tools, not asserted to give a complete explanation of every strict gap.

The supplied literature assessment also mentions the Kodaira dimension $\kappa(X,L)$, defined by the growth/images of the powers of $L$, with value $-\infty$ if no positive power has a nonzero section. The usual comparison $\kappa(X,L)\leq\nu(L)\leq n(L)$ provides that broader context; the original question itself specifically concerns the numerical-versus-nef gap. The curation dated 21 August 2026 calls the general geometric explanation unsettled and partially solved. That is a supplied dated assessment, not a new classification or literature judgment.

Short English statement

The source asks what geometry accounts for $\nu(L)<n(L)$ for a nef line bundle. Numerical dimension measures intersection positivity, while nef dimension comes from the nef-reduction base. The supplied curation cites foliations and nef reduction as partial tools without a complete criterion.

Sources

• Local UnsolvedMath registry, permanent ID $30000009$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; verification says only variable notation was restored.
• Supplied report: "Multiplier Ideal Sheaves in Algebraic and Complex Geometry (2005)," Oberwolfach Reports, pp. 1108--1166, DOI 10.4171/owr/2004/21: https://doi.org/10.4171/owr/2004/21. Proposed-year metadata and the DOI refer to 2004; the citation label says 2005.
• Supplied literature label "Demailly, positive line bundles and adjunction": https://arxiv.org/abs/alg-geom/9410022. The other supplied literature source is the original report DOI above; no fuller bibliographic details are given.
• Literature checked 21 August 2026: supplied status partially_solved, with numerically trivial foliations and nef reduction as structural progress. Registry created 25 August 2026 and updated 16 September 2026. This is dated curation triage requiring independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report/literature retain their supplied provenance; this statement-only draft requires human review.