Termination of the log minimal model program for polarized pairs and existence of minimal models for generalized pairs with an arbitrary nef real part

A polarized pair $(X, B+P)$, introduced by Birkar and Hu (Nagoya Math. J. 2014), is a pair $(X,B)$ of birational geometry together with a nef $\mathbb{R}$-Cartier divisor $P$ on which no rationality, effectivity, or support assumption is made. The two linked open problems at the core of Birkar-Hu's program for such pairs are: (1) their Conjecture 1.3, that an LMMP (log minimal model program) on $K_X+B+P$ can be run to termination for every $\mathbb{Q}$-factorial dlt polarized pair; and (2) the companion existence question, posed by Hu (2025) as Conjecture 1.1, whether a projective generalized log canonical pair whose adjoint divisor birationally has a Nakayama-Zariski decomposition with nef positive part has a minimal model - the direct descendant, without the standard NQC rational-decomposability hypothesis, of Birkar-Hu's Theorem 1.1 and their Section 6 program.

Background

Birational geometry over an algebraically closed field of characteristic zero is organized around the minimal model program (MMP, also called the log minimal model program, LMMP). Its input is a pair $(X,B)$: a normal projective variety $X$ together with a boundary, that is, an effective $\mathbb{R}$-divisor $B$ (a formal non-negative real combination of hypersurfaces) such that the adjoint divisor $K_X+B$ is $\mathbb{R}$-Cartier (a real linear combination of Cartier divisors), where $K_X$ is a canonical divisor, the divisor of a non-zero rational differential form. The program repeatedly finds curves $C$ with $(K_X+B)\cdot C<0$ and removes them, either by a divisorial contraction (a birational morphism contracting a divisor) or by a flip (a birational surgery replacing the negative-curve locus so that the strict transform of $K_X+B$ becomes relatively ample). The intended output is a minimal model, a birational model on which $K_X+B$ is nef (nef meaning $D\cdot C\ge 0$ for every curve $C$), or a Mori fibre space, a fibration with relatively ample adjoint divisor. Two structural facts govern whether the program can be run at all: the existence of flips, established for pairs of log general type by BCHM (2010) and for log canonical pairs by Birkar's flip existence theorem (2012); and termination, the requirement that some (equivalently, for standard purposes, every) sequence of flips be finite, which is open in general.

Singularities of pairs are measured by log discrepancies: numbers $a(E,X,B)$ attached to prime divisors $E$ over $X$ (prime divisors on birational models of $X$), with larger values meaning milder singularities. A pair is klt (Kawamata log terminal) if all log discrepancies are positive, log canonical (lc) if all are non-negative, and divisorially log terminal (dlt) if it is log canonical and, roughly, its boundary has simple normal crossings (snc, meaning the components meet transversally, like coordinate hyperplanes) at a general point of every codimension-one component of its non-klt locus; dlt is the standard running hypothesis of the LMMP. A variety is $\mathbb{Q}$-factorial if every Weil divisor has a positive integer multiple that is Cartier, the condition that keeps divisor arithmetic available at every step of the program.

On surfaces, the classical Zariski decomposition of an effective divisor into a nef part and a negative part drives the surface MMP. In higher dimensions no such decomposition exists on the nose, but birational substitutes do. A weak Zariski decomposition (WZD) of an $\mathbb{R}$-Cartier divisor $D$ is an expression $f^*D \equiv P' + N'$ on some birational model $f: X'\to X$ with $P'$ nef and $N'\ge 0$, where $\equiv$ denotes numerical equivalence ($D\equiv D'$ iff $D\cdot C=D'\cdot C$ for all curves $C$). The Nakayama-Zariski decomposition (also called the $\sigma$-decomposition; see the account in Birkar-Hu, Section 4) is a canonical divisorial decomposition $D=P_\sigma(D)+N_\sigma(D)$ defined for pseudoeffective divisors ($D$ pseudoeffective means its numerical class is a limit of effective classes); when the positive part $P_\sigma(D)$ is nef on some birational model one says that $D$ birationally has a Nakayama-Zariski decomposition with nef positive part. The Fujita-Zariski and CKM (Cutkosky-Kawamata-Moriwaki)-Zariski decompositions are intermediate notions requiring a positive part that is semi-ample (some positive multiple defines a morphism) or cut out by finitely many sections. The structural role of these decompositions was established by Birkar (2012): for lc pairs, assuming the LMMP in lower dimensions, the existence of a WZD of $K_X+B$ is equivalent to the existence of a log minimal model, so numerical positivity can be converted into birational geometry without first knowing termination.

Birkar and Hu (2014) introduced polarized pairs to handle adjoint divisors perturbed by a nef part carrying no rationality structure: a polarized pair is $(X, B+P)$ where $(X,B)$ is a pair as above and $P$ is a nef $\mathbb{R}$-Cartier divisor on $X$, with no assumption that $P$ is effective, has rational coefficients, or is a real combination of $\mathbb{Q}$-divisors. Such perturbations arise canonically as moduli parts of canonical bundle formulas (the decomposition of the canonical divisor of a fibration into a relative part and a nef moduli part) and in the study of Iitaka fibrations (birational models induced by spaces of pluricanonical sections), as developed by Birkar-Zhang (2016). Birkar-Hu extend the cone, contraction and flip theory to these pairs and prove: for an ordinary lc pair, a birational Nakayama-Zariski decomposition of $K_X+B$ with nef positive part implies a log minimal model (Theorem 1.1); in the big case, Fujita/CKM-Zariski decompositions are equivalent to log minimal models (Theorem 1.2); termination for $K_X+B+\alpha P$ with $\alpha\gg 0$ holds under rationality or support hypotheses on $P$ or on $N=K_X+B-P$ (Theorem 1.4; by Corollary 1.5 the additionally assumed termination of the LMMP on $K_X+B$ is automatic in dimension $\le 3$, the rationality or support hypotheses being retained); and a conditional finite-generation result for section rings $\bigoplus_{m\ge 0}H^0(X,\lfloor m(K_X+B+P)\rfloor)$ (Theorem 1.6). They leave open Conjecture 1.3, termination with no conditions on $P$, and Question 3.5, whether pseudoeffectivity of $K_X+B+P$ forces numerical equivalence with an effective divisor (numerical nonvanishing).

The same circle of ideas was formalized by Birkar-Zhang (2016) in the notion of a generalized pair $(X, B+M)$: a pair together with nef b-divisor data, meaning a nef divisor $M'$ on a higher birational model pushing forward to a class $M$ on $X$ (a b-divisor is a compatible system of divisors on all birational models). A polarized pair is exactly the generalized pair whose nef data descends to $X$ itself. Subsequent work split along an arithmetic line: an NQC generalized pair is one whose nef data is a non-negative real combination of nef $\mathbb{Q}$-Cartier divisors, a condition that restores the lattice and discreteness techniques of the classical theory, and nearly all progress beyond Birkar-Hu (Han-Li 2022; Tsakanikas-Xie 2024) required NQC. The Section 6 route from Zariski decompositions to minimal models is carried out for NQC generalized pairs (Han-Li 2022; Hacon-Moraga 2020) and largely for ordinary pairs (Lazić-Tsakanikas 2022; Tsakanikas-Xie 2024). In the non-NQC world genuinely new phenomena appear: numerical nonvanishing fails (the $E\times E$ example of Han and Liu (2020)), and even the expected truth value of the main existence question is unclear. Nevertheless Hu-Liu (2026) established that flips, and hence the MMP as a procedure, exist for arbitrary lc generalized pairs, with no NQC, klt or $\mathbb{Q}$-factoriality assumptions; what remains unknown is whether the procedure terminates and when it produces minimal models, and no counterexample to Conjecture 1.3 is known.

Problem Statement

The problem concerns the open core of Birkar-Hu's program for polarized pairs: extending termination and the Zariski-decomposition route to minimal models to nef real perturbations with no arithmetic hypothesis. It has two linked components.

Component 1 - termination (Birkar-Hu, Conjecture 1.3). Let $(X, B+P)$ be a $\mathbb{Q}$-factorial dlt polarized pair: $(X,B)$ a $\mathbb{Q}$-factorial dlt pair and $P$ a nef $\mathbb{R}$-Cartier divisor on $X$, with no rationality, effectivity or support assumption on $P$. Show that one can run an LMMP on $K_X+B+P$ which terminates: there is a finite sequence of flips and divisorial contractions for $K_X+B+P$ ending with a model on which the strict transform of $K_X+B+P$ is nef, or with a Mori fibre space. At $P=0$ this is the termination conjecture for the LMMP on $\mathbb{Q}$-factorial dlt pairs, so the component asks for that conjecture together with its extension to arbitrary nef real perturbations.

Component 2 - minimal models from Nakayama-Zariski decompositions without NQC (Hu 2025, Conjecture 1.1; the direct descendant of Birkar-Hu Theorem 1.1 and their Section 6 program). Let $(X, B+M)$ be a projective generalized log canonical pair with nef b-divisor data, and suppose $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part: for some birational model $\pi: X'\to X$, the positive part $P_\sigma(\pi^*(K_X+B+M))$ is nef. Show that $(X, B+M)$ has a minimal model: a birational generalized pair $(Y, B_Y+M_Y)$, carrying the same nef data, on which the adjoint divisor $K_Y+B_Y+M_Y$ is nef, $Y$ is $\mathbb{Q}$-factorial, and the log discrepancies satisfy $a(D,X,B+M)\le a(D,Y,B_Y+M_Y)$ for every prime divisor $D$ on $X$ exceptional over $Y$, with strict inequality in each case (Hu 2025, Section 2). Since a polarized pair is the special case where the nef data descends to $X$, this subsumes the polarized-pair question whether a birational Zariski decomposition with nef positive part yields a log minimal model.

Relationship between the components. They share one objective, minimal-model theory for pairs with an arbitrary nef real perturbation, and are linked: termination for $K_X+B+P$ together with the now-known existence of flips would give minimal models for all pseudoeffective polarized pairs, while Component 2 asks for the same conclusion from purely numerical, decomposition-theoretic input.

The problem is stated in full generality: no hypothesis may be added on the nef part being NQC, rational, or $\mathbb{Q}$-decomposable; on the effectivity of $P$ or $M$ or of $K_X+B+P$; on support conditions such as $\mathrm{Supp}N\subseteq\mathrm{Supp}B$ for $N=K_X+B-P$; or on the dimension. Component 2 is for log canonical pairs.

References

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