The Weak Nonvanishing Conjecture: is every pseudo-effective log canonical divisor numerically equivalent to an effective divisor?

The Weak Nonvanishing Conjecture, recorded as Conjecture 1.3 in Birkar's 'On existence of log minimal models II' (Journal für die reine und angewandte Mathematik 658 (2011), 99–113), predicts that for every $\mathbb{Q}$-factorial dlt pair $(X/Z,B)$ in characteristic zero whose log canonical divisor $K_X+B$ is pseudo-effective over $Z$ — a purely numerical condition, namely that its class is a limit of effective classes — the divisor is effective over $Z$: $K_X+B\equiv M/Z$ for some effective $\mathbb{R}$-divisor $M\ge 0$. The source paper calls this 'the main conceptual obstacle' to the Abundance Conjecture and proves (Theorem 1.4) that a positive answer in dimension $d$ implies the full Minimal Model Conjecture in dimension $d$. The conjecture is known in dimension at most 3 and in special classes — uniruled pairs (varieties covered by rational curves) one dimension above the settled range, pairs of Calabi–Yau type (pairs admitting an effective boundary for which the log canonical divisor is numerically trivial), generalized pairs (pairs augmented by an auxiliary nef divisor) with a polarization, and irregular varieties (those with non-zero irregularity $q(X)=h^1(X,\mathcal{O}_X)>0$) under lower-dimensional assumptions — and Hashizume (2018) reduced it to the case of smooth projective varieties with zero boundary. Despite continuous activity through 2025 — Liu–Xu (2025) prove abundance in numerical dimension at most 1 from nonvanishing and still state it as an open conjecture — it remains open in dimension at least 4, with the surviving core being smooth non-uniruled varieties of Kodaira dimension $-\infty$ (no pluricanonical sections at all) whose canonical class is nonetheless pseudo-effective.

Background

The minimal model program (MMP) is birational geometry's central classification scheme for projective varieties of dimension at least 3. Its input is a pair $(X/Z,B)$: a normal quasi-projective variety $X$ over an algebraically closed field of characteristic zero, a projective morphism $X\to Z$, and a boundary — an $\mathbb{R}$-divisor $B\ge 0$ (a finite $\mathbb{R}$-linear combination of irreducible codimension-one subvarieties) with coefficients in $[0,1]$ such that the log canonical divisor $K_X+B$ is $\mathbb{R}$-Cartier ($K_X$ denoting a canonical divisor). Singularities of pairs are measured by log discrepancies $a(D,X,B)$, one real number attached to each divisor $D$ on birational models of $X$: the pair is log canonical (lc) if $a(D,X,B)\ge 0$ for every such $D$, Kawamata log terminal (klt) if $a(D,X,B)>0$ for every $D$, and divisorial log terminal (dlt) if it is lc and already well behaved (simple normal crossing) at the generic point of every divisor with discrepancy $0$. The variety $X$ is $\mathbb{Q}$-factorial if every Weil divisor on it has some integer multiple that is Cartier. The MMP prescribes repeatedly contracting $(K_X+B)$-negative extremal rays — the edges of the closed cone spanned by curve classes on $X$ — via divisorial contractions and log flips (a birational surgery replacing a flipping contraction) — aiming to reach a log minimal model (a birational model $Y/Z$ on which $K_Y+B_Y$ is nef over $Z$, i.e. non-negative on every curve contracted over $Z$) or a Mori fibre space (a Fano-type fibration). The Minimal Model Conjecture asserts that this program always succeeds for lc pairs; the Abundance Conjecture asserts the stronger property that a nef log canonical divisor is semi-ample over $Z$ (some multiple defines a morphism over $Z$), which implies the equality of the Kodaira dimension $\kappa$ (the growth rate of spaces of pluricanonical sections) with the numerical dimension $\nu$ (the number of 'positive directions' of a divisor; $\nu\ge 0$ holds exactly for pseudo-effective classes).

The conjecture under consideration sits at the numerical heart of this program. Let $N^1(X/Z)$ denote the finite-dimensional real vector space of numerical equivalence classes of $\mathbb{R}$-Cartier divisors over $Z$, where $D\equiv D'/Z$ means $D-D'$ intersects trivially every curve contracted over $Z$. The pseudo-effective cone is the closure, inside $N^1(X/Z)$, of the cone spanned by effective divisors: $K_X+B$ is pseudo-effective over $Z$ if its class is a limit of effective classes, and the pair is effective over $Z$ if its class is actually represented by an effective $\mathbb{R}$-divisor $M\ge 0$ on $X$. For big classes — the interior of the pseudo-effective cone — the distinction is vacuous, and the foundational theorem of Birkar–Cascini–Hacon–McKernan (2010) on finite generation of the canonical ring (the ring $\bigoplus_{m\ge 0} H^0(X, mK_X)$ of pluricanonical sections) settles the existence of minimal models in the big case. The open difficulty is the boundary of the pseudo-effective cone: a limit of effective classes need not itself be an effective class. Birkar's paper 'On existence of log minimal models II' isolates exactly this gap as 'the main conceptual obstacle' to abundance, stating the Weak Nonvanishing Conjecture (its Conjecture 1.3): every $\mathbb{Q}$-factorial dlt pair with pseudo-effective log canonical divisor is effective. The same paper proves (Theorem 1.4) that this statement in dimension $d$ implies the Minimal Model Conjecture in dimension $d$, so weak nonvanishing is not a side branch of abundance but a sufficient foundation for the whole program; the paper also proves termination of the LMMP with scaling — a guided version of the MMP, due to Shokurov, in which the coefficient of an ample divisor is successively scaled down — but only under additional hypotheses.

The conjecture holds in dimension at most 3, where it follows from the threefold abundance theorem completed in the 1990s (as surveyed by Lazić–Meng, 2021). Hashizume (2018) reduced the general conjecture — in its standard, slightly stronger form with $\mathbb{R}$-linear equivalence $K_X+\Delta\sim_{\mathbb{R}}D$ in place of numerical equivalence — to the single most special case, that of smooth projective varieties with zero boundary. The surviving core is therefore sharp: a smooth projective variety of dimension at least 4 which is not covered by rational curves, has Kodaira dimension $\kappa=-\infty$ (no pluricanonical sections at all), yet has pseudo-effective canonical class. No known technique — algebraic, analytic, or Hodge-theoretic (from Hodge theory, the study of the extra structures on the cohomology of projective varieties) — reaches this case.

Problem Statement

Source-faithful statement (Conjecture 1.3, 'Weak nonvanishing', in Birkar, On existence of log minimal models II). Let $k$ be an algebraically closed field of characteristic zero, and let $(X/Z,B)$ be a $\mathbb{Q}$-factorial dlt pair of arbitrary dimension: $X$ normal and quasi-projective over $k$, $X\to Z$ a projective morphism, $B\ge 0$ an $\mathbb{R}$-divisor with coefficients in $[0,1]$ such that $K_X+B$ is $\mathbb{R}$-Cartier, and the pair divisorial log terminal. Assume that $K_X+B$ is pseudo-effective over $Z$, that is, that the class of $K_X+B$ in $N^1(X/Z)$ is a limit of classes of effective $\mathbb{R}$-divisors:

$$K_X+B \equiv \lim_{i\to\infty} M_i/Z$$

for some sequence of $\mathbb{R}$-divisors $M_i\ge 0$ on $X$. Must $K_X+B$ be effective over $Z$ — must there exist an $\mathbb{R}$-divisor $M\ge 0$ on $X$ with

$$K_X+B \equiv M/Z?$$

The conjecture is stated in full generality: all dimensions, arbitrary $Z$, and no restrictions on the pair beyond those listed above. The stakes are as recorded in the source: by Theorem 1.4 of the same paper, a positive answer in dimension at most $d$ implies the Minimal Model Conjecture in dimension at most $d$.

References

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