gpt 6 astra pro attempt on Math-25

Abundance Conjecture for the minimal model program

Definitions and mathematical statement

For a projective klt pair in the source's minimal-model setting, put $D=K_X+\Delta$. A divisor is nef if $D\cdot C\ge0$ for every curve $C$. In the usual rational-divisor formulation it is semiample if for some integer $m>0$, $mD$ is Cartier and its line bundle is generated by global sections. The conjecture is \[ D\text{ nef}\quad\Longrightarrow\quad \exists m>0:\mathcal O_X(mD)\text{ globally generated}. \] The rational-boundary convention is the standard one for this displayed statement; a real-boundary version requires the source's corresponding notion of semiampleness. No bigness hypothesis is added.

Formulation and concept references: [S1].

Short English statement

Must a nef canonical-plus-boundary divisor on a minimal klt variety eventually have enough sections to define a morphism?

Research attempt

Current frontier.

Lazić studies criteria involving metrics with minimal singularities and multiplier ideals, including an equivalence under a nonzero Euler characteristic hypothesis; these additional assumptions are not universal abundance [E1]. Results about anti-canonical nonvanishing in [S4] concern a different divisor. Here the attempt uses the rational-boundary convention of the displayed target and makes no bigness assumption. The aim is to isolate the difference between a nef numerical class, nonzero actual sections, and a section ring whose base ideals ultimately disappear.

Attack route.

A metric approach tries to remove singularities, but even a smooth flat metric can coexist with no sections. A finite-generation approach could turn asymptotic vanishing of actual base orders into semiampleness, provided nonvanishing has first been secured. The latter route is pursued by an explicit base-ideal calculation. The missing geometric steps are then separated from an invalid replacement of actual base orders by numerical orders after an ample perturbation.

Attempt: a base-ideal criterion with all hypotheses exposed.

Let $L$ be a Cartier divisor on a normal projective variety $X$. Assume there is a positive integer $m_0$ such that $H^0(X,m_0L)\ne0$ and the Veronese section ring \[ R^{(m_0)}(L)=\bigoplus_{k\geq0}H^0(X,km_0L) \] is generated by its degree-one piece. Finite generation of the section ring, together with positive-degree nonvanishing, permits such a sufficiently divisible Veronese; one can alternatively retain generation in degree one as the explicit sufficient hypothesis. Let $\mathfrak b$ be the base ideal of $|m_0L|$, defined by the image of its evaluation map after tensoring by $\mathcal O_X(-m_0L)$.

Surjectivity of the multiplication maps from the symmetric powers of $H^0(X,m_0L)$ gives the exact equality of ideal sheaves \begin{equation}\label{wsref-0-65-71-37-38-3a-70-6f-77-65-72-73} \mathfrak b_{|km_0L|}=\mathfrak b^k\qquad(k\geq1). \end{equation} Locally trivialize $m_0L$. Products of $k$ local generators of $\mathfrak b$ are evaluations of products of global sections, giving one inclusion; generation of all degree-$k$ sections by sums of such products gives the other. Thus (eq78:powers) concerns actual ideals, not just set-theoretic base loci.

For a divisorial valuation $v$ of the function field with centre on $X$, write $v(\mathfrak b)$ for the minimum valuation of local generators at its centre. Equation (eq78:powers) gives \begin{equation}\label{wsref-0-65-71-37-38-3a-6f-72-64-65-72} \frac{1}{km_0}v(\mathfrak b_{|km_0L|}) =\frac1{m_0}v(\mathfrak b). \end{equation} If $\mathfrak b\ne\mathcal O_X$, some such $v$ has $v(\mathfrak b)>0$. To see this geometrically, blow up the nonzero ideal $\mathfrak b$ and normalize. Its pullback is an invertible ideal $\mathcal O(-E)$ with $E$ effective. It is not everywhere a unit: over a point of the base locus all local generators lie in the maximal ideal at points above it. Hence $E$ is nonzero. A prime component of $E$ defines the required valuation. If the original ideal is already invertible, its nonzero effective Cartier divisor gives the same conclusion directly.

It follows that the simultaneous hypotheses \begin{equation}\label{wsref-0-65-71-37-38-3a-63-72-69-74-65-72-69-6f-6e} \begin{gathered} H^0(X,m_0L)\ne0,\qquad R^{(m_0)}(L)\text{ generated in degree one},\\ \lim_{k\to\infty}\frac{v(\mathfrak b_{|km_0L|})}{km_0}=0 \quad\text{for every divisorial }v. \end{gathered} \end{equation} force $\mathfrak b=\mathcal O_X$. Thus $m_0L$ is globally generated and $L$ is semiample. The proof uses neither a numerical approximation to $v$ nor an exchange of a limit with a minimum over all valuations: one fixed positive valuation would contradict the last condition.

Attempt: application to the adjoint target.

For the original rational pair choose $q>0$ with $L=q(K_X+\Delta)$ Cartier. If the nonvanishing, generation and actual base-order assertions in (eq78:criterion) can be established for this $L$, the target follows, since a globally generated multiple of $L$ is a globally generated multiple of $K_X+\Delta$. This packages three distinct arithmetic-geometric obligations. It does not assert that nefness alone proves any of them. In particular, a section ring with no positive-degree sections is trivially finitely generated but useless for semiampleness; the first condition excludes this pitfall.

An analytic continuation of the route might compare the actual base ideals with multiplier ideals of minimal metrics, then use vanishing or Euler characteristic information to produce sections. That is exactly where extra hypotheses in the metric literature matter [E1]. No equality between those metric ideals and the actual base ideals is proved here. Without it, substituting metric Lelong numbers for the valuations in (eq78:criterion) is an unproved bridge.

Stress test: a nef flat line bundle with no sections.

Let $C$ be a complex elliptic curve and choose a non-torsion degree-zero line bundle $P\in\operatorname{Pic}^0(C)$. Such bundles exist because the complex torus $\operatorname{Pic}^0(C)$ is uncountable and its torsion subgroup is countable. For every $m>0$, a nonzero section of $P^m$ would have an effective divisor of degree zero, hence no zeros, and would trivialize $P^m$. This contradicts non-torsion. Therefore \[ P\text{ is nef},\qquad H^0(C,P^m)=0\quad(m>0). \] It admits a smooth unitary flat metric; its curvature and Lelong numbers are zero. Also $P+\varepsilon A$ has positive degree for an ample $A$ and $\varepsilon>0$, so numerical vanishing orders after ample perturbation carry no obstruction of this sort. These facts show why numerical positivity and zero metric singularities alone do not create sections.

This is not an abundance counterexample. On an elliptic curve an effective boundary $\Delta$ with $\deg(K_C+\Delta)=0$ must be zero, and $K_C$ is trivial. The non-torsion $P$ cannot be the stipulated adjoint divisor in that degree-zero situation. The missing adjoint input is therefore genuine, not a reason to discard the conjecture.

There is another weak argument to avoid. Along factorial multiples, base loci form a descending family of closed sets whenever the relevant sections are present. Noetherianity can make that family stabilize at a nonempty closed set; it does not make the set disappear. Likewise finding more sections at each step does not imply a section avoiding every remaining base point. The valuation in (eq78:order) detects such a persistent base ideal quantitatively only after the generation hypothesis has been established.

Outcome.

Outcome: Conditional reduction.

A direct base-ideal proof reduces semiampleness to nonvanishing, degree-one generation after a Veronese, and zero actual asymptotic base orders. It isolates an invalid metric-to-section shortcut by an explicit nef flat example outside the adjoint class. The argument supplies no new unconditional abundance case and makes no priority claim for its elementary base-ideal lemma.

The unresolved task is to prove the three properties for arbitrary nef klt adjoint divisors without adding bigness, Euler characteristic, or a lower-dimensional MMP assumption. A real-boundary formulation would also need its own notion of semiampleness; it is not proved by this rational-multiple calculation.

Sources

• S1 Vladimir Lazic, Metrics with minimal singularities and the Abundance conjecture $2024$. https://arxiv.org/abs/2406.18233. Catalogue formulation source.
• S2 Inequalities of Miyaoka-Yau type and Uniformisation of varieties of intermediate Kodaira Dimension. https://arxiv.org/abs/2601.15138. Catalogue research/status reference; not a proof endorsement.
• S3 An approach to the abundance conjecture for Kaehler varieties via algebraic reduction. https://arxiv.org/abs/2604.03879. Catalogue research/status reference; not a proof endorsement.
• S4 Two non-vanishing results concerning the anti-canonical bundle. https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/two-nonvanishing-results-concerning-the-anticanonical-bundle/1E68F2B88B818A7CBD47B96B7D700548. Catalogue research/status reference; not a proof endorsement.
• E1 Vladimir Lazić, Metrics with minimal singularities and the Abundance conjecture, arXiv:2406.18233, introduction and hypotheses of the metric/nonvanishing criteria, particularly the nonzero Euler characteristic setting. https://arxiv.org/abs/2406.18233. Inspected 27 September 2026; the notebook does not assume that metric singularity control automatically identifies all actual base ideals.