Existence of weak Zariski decompositions for pseudo-effective R-Cartier divisors (Birkar's Question 1.4, from Nakayama's problem)

The source paper (Caucher Birkar, Math. Ann. 2012) introduced the weak Zariski decomposition — a purely birational, numerical analogue of the classical surface Zariski decomposition — and proved that, assuming the log minimal model program (LMMP) for Q-factorial dlt pairs one dimension lower, a log canonical pair (the broadest singularity class of pairs for which the program is conjectured to work) has a log minimal model exactly when its log canonical divisor has such a decomposition. The paper explicitly leaves open its Question 1.4, going back to Nakayama's problem from 2004: does every pseudo-effective R-Cartier divisor D on a normal variety X/Z admit a weak Zariski decomposition/Z, i.e. a projective birational morphism f: W→X and a numerical equality $f^{*}D \equiv P+N/Z$ with $P$ nef/Z and $N$ effective? The question is open: a positive answer in all dimensions would imply the minimal model conjecture for log canonical pairs — itself open from dimension 5 upward — while a counterexample would refute Nakayama's expectation. For adjoint divisors (those of the form $K_X+B$) the answer is known unconditionally only up to dimension 4, and for arbitrary divisors even dimension 3 is open, with positive results confined to surfaces, big divisors (divisors of positive volume, with ample-like asymptotic growth of sections), and special families such as projectivized vector bundles. Later work (Hacon–Moraga, Han–Li, Tsakanikas–Xie, and others) extends the decomposition–minimal-model equivalence to generalized pairs (pairs equipped with an additional nef part defined on birational models above $X$) but uniformly treats the existence of weak Zariski decompositions as an open hypothesis.

Background

Work over an algebraically closed field $k$ of characteristic zero, and fix a projective morphism $X \to Z$ of normal quasi-projective varieties, written $X/Z$ (the relative setting; the absolute case is $Z = \mathrm{pt}$, a point). An $\mathbb{R}$-Cartier divisor $D$ on $X$ is a finite real linear combination of Cartier divisors. Two such divisors are numerically equivalent$/Z$, written $D \equiv D'/Z$, if $D \cdot C = D' \cdot C$ for every curve $C$ on $X$ mapped to a point in $Z$. $D$ is nef$/Z$ if $D \cdot C \geq 0$ for all such curves, and pseudo-effective$/Z$ if its class is a limit of classes of effective $\mathbb{R}$-divisors in the Néron–Severi space $N^1(X/Z)$ of numerical classes. Nef divisors are pseudo-effective, but not conversely, and whether a pseudo-effective divisor can be written birationally and numerically as a nef plus an effective part is exactly the phenomenon at issue.

On a smooth projective surface, every pseudo-effective $\mathbb{R}$-Cartier divisor admits a classical Zariski decomposition $D = P + N$ with $P$ nef, $N \geq 0$, $P$ orthogonal to the components of $N$, and the intersection matrix of those components negative definite — the classical theory recalled in the introduction of the source paper and treated in detail in Nakayama's book. Such a decomposition reduces positivity questions about $D$ to the nef part $P$. In dimension three and higher the classical extensions — the Fujita–Zariski decomposition, where $P$ is required to dominate every competing decomposition $f^{*}D = P' + N'$ via $P' \leq f^{*}P$, and the Cutkosky–Kawamata–Moriwaki (CKM) decomposition, where additionally the section rings (the graded algebras of sections of multiples) of $\lfloor mP \rfloor$ and $\lfloor mD \rfloor$ agree — often fail to exist; Nakayama's book constructs big divisors with no decomposition of either kind, and Prokhorov's survey compares the various notions.

The source paper introduces the weakest meaningful version. A weak Zariski decomposition$/Z$ of $D$ (Definition 1.3 of the source) is a projective birational morphism $f : W \to X$ from a normal variety together with an expression $f^{*}D \equiv P + N/Z$ where $P$ and $N$ are $\mathbb{R}$-Cartier, $P$ is nef$/Z$, and $N \geq 0$. If $D \geq 0$ one takes $f = \mathrm{id}$ and $P = 0$; if $D$ is big (of positive volume, that is, its multiples have asymptotically as many sections as those of an ample divisor), Kodaira's lemma — a standard consequence of bigness — gives $D \sim_{\mathbb{R}} A + E$ with $A$ ample and $E \geq 0$, so a weak Zariski decomposition again exists on $X$ itself. The substantive content of the question therefore lies in the pseudo-effective, non-big case. The question originates in Nakayama's divisorial Zariski decomposition: for pseudo-effective $D$, Nakayama defines non-negative asymptotic coefficients $\sigma_E(D)$ on prime divisors $E$ via limits of volumes of $D - \varepsilon E$, sets $N_\sigma(D) = \sum_E \sigma_E(D) E$ and $P_\sigma(D) = D - N_\sigma(D)$, and asks (Problem, p. 4 of his book) whether some birational model makes $P_\sigma(f^{*}D)$ nef; a positive answer gives $f^{*}D = P_\sigma(f^{*}D) + N_\sigma(f^{*}D)$ with $N_\sigma \geq 0$, hence a weak Zariski decomposition.

The problem is inseparable from the minimal model program (MMP, also called LMMP). A pair $(X/Z, B)$ is a normal variety $X/Z$ together with an $\mathbb{R}$-divisor $B$ with coefficients in $[0,1]$ such that $K_X + B$ is $\mathbb{R}$-Cartier, where $K_X$ is the canonical class. The pair is log canonical (lc) if every birational valuation has non-negative log discrepancy $a(D,X,B)$ (no worse than the standard lower bound); dlt (divisorially log terminal: log canonical with at worst simple-normal-crossing singularity on a dense open subset) and klt (Kawamata log terminal: all log discrepancies strictly positive) are the standard stronger variants. The LMMP modifies an lc pair by contracting $K_X + B$-negative extremal rays — by divisorial contractions, or by flips, which replace the contracted curves by a small birational model — until $K_X + B$ becomes nef; a resulting model $(Y/Z, B_Y)$ with $K_Y + B_Y$ nef$/Z$ is a log minimal model, and if an extremal contraction instead has positive-dimensional fibres one obtains a Mori fibre space (a fibration whose general fibre has ample anti-canonical class). The minimal model conjecture predicts that every lc pair with $K_X + B$ pseudo-effective$/Z$ has a log minimal model. Existence of log canonical flips is known unconditionally (Birkar), so the missing ingredient in high dimensions is termination of sequences of flips; special termination means termination of flips that no longer meet the reduced boundary $\lfloor B \rfloor$, a statement reducible to lower dimensions.

Birkar's main theorem (Theorem 1.5 of the source) connects the two threads: assuming the LMMP for $\mathbb{Q}$-factorial dlt pairs in dimension $d-1$ ($\mathbb{Q}$-factorial: every Weil divisor has a positive integral multiple that is Cartier), an lc pair $(X/Z, B)$ of dimension $d$ has a log minimal model if and only if $K_X + B$ has a weak Zariski decomposition$/Z$ (equivalently, a birational CKM- or Fujita–Zariski decomposition). Corollary 1.7 draws the unconditional conclusion that every lc pair of dimension 4 has a log minimal model or a Mori fibre space — a result first proved by Shokurov, for which the corollary gives another proof. For adjoint divisors $K_X + B$ the picture is thus: surfaces by the classical Zariski decomposition theory; klt pairs with big boundary by Birkar–Cascini–Hacon–McKernan; dimension up to 4 unconditionally (dimension 4 first by Shokurov, another proof in Corollary 1.7 of the source); and dimension at least 5 open, with the conjecture reduced to nonvanishing on smooth zero-boundary pairs (Hashizume) and to minimal models of smooth varieties (Lazić–Tsakanikas). For divisors not of adjoint type, no general results beyond surfaces are known.

Problem Statement

Question (Birkar, 'On existence of log minimal models and weak Zariski decompositions', Question 1.4; going back to Nakayama's problem). Let $X/Z$ be a projective morphism of normal quasi-projective varieties over an algebraically closed field of characteristic zero. An $\mathbb{R}$-Cartier divisor on $X$ is a finite real linear combination of Cartier divisors; two such divisors are numerically equivalent$/Z$ if they take the same value on every curve on $X$ mapped to a point in $Z$; a divisor is nef$/Z$ if it is non-negative on every such curve; and a divisor is pseudo-effective$/Z$ if, in the Néron–Severi space $N^1(X/Z)$ of divisor classes modulo numerical equivalence$/Z$, its class is a limit of classes of effective $\mathbb{R}$-divisors. Does every pseudo-effective$/Z$ $\mathbb{R}$-Cartier divisor $D$ on $X$ admit a weak Zariski decomposition$/Z$ — that is, a projective birational morphism $f : W \to X$ from a normal variety and $\mathbb{R}$-Cartier divisors $P, N$ on $W$ such that $P$ is nef$/Z$, $N \geq 0$, and

$f^{*}D \equiv P + N/Z \ ?$

The question is asked at full generality: every dimension, every normal variety, and every pseudo-effective $\mathbb{R}$-Cartier divisor, with no restriction to adjoint divisors (those of the form $K_X+B$, the canonical class of $X$ plus an effective boundary) and no singularity hypothesis on $X$. The absolute case $Z = \mathrm{pt}$ (a point) contains Nakayama's original problem — is there a birational model on which the positive part $P_\sigma$ of the divisorial Zariski decomposition (Nakayama's canonical candidate for the nef summand, constructed from asymptotic coefficients on prime divisors) is nef? — a positive answer to which yields a weak Zariski decomposition. A negative answer means that there is an explicit normal variety $X/Z$ and pseudo-effective$/Z$ $\mathbb{R}$-Cartier divisor $D$ for which no projective birational model $W \to X$ admits such a decomposition. The logical stakes are explicit in the source: by Theorem 1.5 and induction on dimension, a positive answer in all dimensions implies the minimal model conjecture for log canonical pairs (the prediction that every log canonical pair with pseudo-effective log canonical divisor has a log minimal model, i.e. a birational model on which that divisor becomes nef), while a negative answer would refute Nakayama's expectation and show that the minimal model conjecture cannot be settled through weak Zariski decompositions.

References

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