Termination of flips in dimension at least 5 and non-vanishing for minimal models in dimension at least 4: the open core of the higher-dimensional minimal model program raised in Mori's flip theorem paper (1988)

Mori's 1988 proof of the flip theorem completed the minimal model program (MMP) for projective 3-folds and explicitly raised three issues for higher dimensions: existence of flips, termination of flip sequences, and the higher-dimensional generalization of Miyaoka's non-vanishing theorem (a 3-fold admitting a minimal model has non-negative Kodaira dimension). The first issue has since been resolved in full: Birkar-Cascini-Hacon-McKernan (2010) proved that flips exist for all kawamata log terminal pairs in every dimension. The open problem is the surviving open core, consisting of the other two issues, as of the 2026 literature. (1) Termination: prove that no infinite sequence of $(K_X+\Delta)$-flips exists for projective log terminal pairs in dimension at least 5 - open even for terminal pairs with empty boundary, where termination is known only up to dimension 4 - and in dimension 4 beyond the classes where it is known (terminal, canonical, and pseudo-effective log canonical pairs, and Mori-with-scaling sequences under additional hypotheses). (2) Non-vanishing: prove that a projective kawamata log terminal pair of dimension at least 4 with a minimal model has non-negative Kodaira dimension, that is, that some pluricanonical system is non-empty; this is open in dimension 4 and above, where Miyaoka's characteristic-$p$ argument does not reach. A 2025 Geometry & Topology paper (Han-Liu) describes termination of flips as one of the major remaining open problems in birational geometry and reduces dimension-4 termination to an ascending chain condition for minimal log discrepancies. Together the two components are the remaining obstacles to extending Mori's 3-fold program - minimal models or Mori fiber spaces, and the Kodaira-dimension/uniruledness dichotomy - to all dimensions.

Background

From surfaces to the minimal model program. Let $X$ be a smooth complex projective variety and let $K_X$ be its canonical class, the divisor class of any holomorphic volume form. The minimal model program (MMP), also called Mori's program, classifies projective varieties up to birational equivalence (two varieties are birational when they are isomorphic on dense open subsets) by producing, for each $X$, either a minimal model - a birational copy $X'$ with mild singularities whose canonical class is nef, that is, has non-negative degree on every curve - or a Mori fiber space, a morphism $X' \to Y$ whose anti-canonical class is relatively ample (a family of Fano-type fibers, for example a family of projective spaces). The invariant governing which outcome occurs is the Kodaira dimension $\kappa(X)$: the growth rate of the spaces of pluricanonical sections $H^0(X, \mathcal{O}_X(mK_X))$ for $m = 1, 2, \ldots$, normalized so that the vanishing of all these spaces means $\kappa = -\infty$ and maximal growth means $\kappa = \dim X$ (general type). A variety is uniruled if it is covered by rational curves; uniruled varieties have $\kappa = -\infty$, and the expectation - a theorem in dimension 3 - is that the converse holds. Since pluricanonical sections are birational invariants for varieties with mild singularities, a minimal model, when it exists, should have $\kappa \ge 0$; this is the non-vanishing statement at the center of the problem below.

How the program runs. The MMP is an algorithm driven by the cone theorem, which describes the closed cone of curves (the closure of the cone spanned by classes of effective curves): its $K_X$-negative part is generated by rational curves, and one repeatedly contracts an extremal ray - a one-dimensional face of the cone on which $K_X$ is negative. A contraction is divisorial if it contracts a divisor (the variety becomes simpler), or small if it contracts only curves and no divisors. A small contraction $f \colon X \to Z$ cannot be a step towards a minimal model, because $K_X$ does not descend as a well-behaved divisor to the target; the flip of $f$ is the repair: another birational modification $X^+ \to Z$, an isomorphism outside the exceptional sets, such that $K_{X^+}$ is relatively ample. The singularities are measured by discrepancies: on a resolution $g \colon Y \to X$ one compares $K_Y$ with $g^*K_X$, and the resulting rational coefficients $a(E, X)$ of the exceptional divisors measure how singular $X$ is; $X$ is terminal (respectively canonical) when $a(E, X) > 0$ (respectively $\ge 0$) for all $E$. Terminal is the mildest class and the one carried by minimal models. For pairs one works with a boundary, an effective $\mathbb{Q}$-divisor $\Delta = \sum d_i \Delta_i$ with $0 \le d_i \le 1$, and the pair $(X, \Delta)$ is kawamata log terminal (klt) when $a(E, X, \Delta) > -1$ for every divisor $E$ over $X$, exceptional or a boundary component (this forces all $d_i < 1$); klt is the widest class in which the program is expected to run. $\mathbb{Q}$-factorial (every Weil divisor has a Cartier multiple) is a standing technical assumption, and for a pair the program is run relative to the divisor $K_X + \Delta$.

Dimension 3: the source of the problem. For surfaces the program is classical. In dimension 3 it was completed in the 1980s. Mori proved the flip theorem - flips exist for 3-folds with terminal $\mathbb{Q}$-factorial singularities - and deduced that every smooth projective 3-fold is birational to a minimal model or to a Mori fiber space (Mori 1988). Two companion results enter. Termination: Kawamata proved that klt 3-fold pairs admit no infinite sequence of flips (Kawamata 1992), and Shokurov proved the analogue for real coefficients (Shokurov 1996), after his log-flip existence theorem (Shokurov 1993). Non-vanishing: Miyaoka proved that a 3-fold which admits a minimal model has $\kappa \ge 0$ (Miyaoka 1988), by a characteristic-$p$ argument based on his semipositivity theorem for minimal varieties (Miyaoka 1987); an algebraic non-vanishing theorem for 3-fold pairs had been proved earlier by Shokurov (1986). Combining these with the numerical criterion for uniruledness of Miyaoka-Mori (1986) yields the 3-fold dichotomy: a projective 3-fold has $\kappa = -\infty$ if and only if it is uniruled. Mori's 1988 paper explicitly formulates the higher-dimensional issues left open: existence of flips, termination of flips, and non-vanishing in dimension at least 4.

Higher dimensions: flips exist, but do they stop? Kawamata-Matsuda-Matsuki (1987) extended termination to 4-folds with terminal singularities, and Matsuki (1991) to 4-dimensional flops (flips for which the canonical class is relatively trivial). Fujino (2004) handled canonical 4-fold flips. Alexeev-Hacon-Kawamata (2007) proved termination for many klt 4-dimensional flips, namely the sequences appearing in the MMP with scaling (where an ample scaling parameter in the boundary is progressively reduced) when $-(K_X+\Delta)$ has non-negative Kodaira dimension. On existence, Shokurov proved 4-dimensional log flips (Shokurov 2003), and Hacon-McKernan (2005) proved the inductive principle that flips exist in dimension $n$ assuming termination of flips in dimension $n-1$, recovering 4-dimensional flips; see also their expository account (Hacon-McKernan 2007). The existence problem was then completely solved by Birkar-Cascini-Hacon-McKernan (2010): proving finite generation of adjoint canonical rings, they showed that the flip of any small $(K_X+\Delta)$-extremal contraction of a klt pair exists in every dimension (their Corollary 1.4.1), and they established the MMP with scaling, including termination, for klt pairs of log general type. Existence of flips is thus no longer open in the klt category - but termination and non-vanishing are.

Termination after BCHM. The state of the art: Han-Liu (2025) reduce termination of flips in dimension 4 to the ascending chain condition (ACC: the absence of infinite strictly increasing sequences) for the minimal log discrepancies - the quantities $\operatorname{mld}(x, X, \Delta)$ measuring how far a pair is from being log canonical, the boundary case $a \ge -1$ - of exceptionally non-canonical pairs (a class of singularities detected by discrepancies failing to be canonical along a unique exceptional divisor), noting that termination of terminal flips is known only up to dimension 4. Moraga (2025) proves termination for pseudo-effective (a limit of effective divisors) log canonical 4-fold pairs; Chen-Tsakanikas (2023) prove it for pseudo-effective log canonical generalized pairs (pairs endowed with an auxiliary nef part) in dimension 4, with higher dimensions only conditional on weak Zariski decompositions and lower-dimensional termination; Lazić-Moraga-Tsakanikas (2023) develop special termination (termination of sequences whose centers avoid a prescribed divisor) for log canonical pairs. Conditional results: Shokurov (2003; 2004) reduced termination to the ACC plus lower-semicontinuity conjectures for log discrepancies; Birkar (2007) reduced it, assuming $\kappa \ge 0$, to the ACC for log canonical thresholds and the MMP in one dimension less; Hacon-Moraga (2020) connect termination to the existence of weak Zariski decompositions (writing a divisor as the sum of a nef part and a limit of effective parts); Lazić-Xie (2025) pursue the approach through Nakayama-Zariski decompositions, which write a pseudo-effective divisor on any variety as the sum of a positive and a negative part, generalizing the classical Zariski decomposition of surfaces. No published result proves termination of flip sequences in dimension 5 or higher in any nontrivial singularity class.

Non-vanishing after BCHM. The non-vanishing conjecture - a klt pair with $K_X + \Delta$ pseudo-effective has $\kappa(K_X+\Delta) \ge 0$ - is one of the three standard pillars of the MMP (existence of flips, termination, abundance), as listed in Hacon-McKernan (2005); abundance itself is the stronger statement that a nef canonical class is semiample, that is, some multiple defines a morphism. Beyond the big case (covered by the canonical models of Birkar-Cascini-Hacon-McKernan 2010) and Miyaoka's dimension-3 theorem, partial cases in dimension at least 4 include: Hashizume (2018), reducing the non-vanishing and log minimal model conjectures for log canonical pairs to non-vanishing for smooth varieties with empty boundary; Meng (2019), a non-vanishing criterion for log canonical pairs with nef canonical class on uniruled varieties (assuming the MMP conjectures in lower dimensions) and semi-ampleness for irregular varieties (those with a non-constant morphism to an abelian variety), with consequences for 4-folds; and Liu-Xu (2025), proving abundance in dimension at most 5 for pairs with $\kappa \ge 0$ and numerical dimension (the number of positive intersection directions of a nef class) at most 1. Adjacent to the source problem, Hacon-Liu (2023) prove the existence of flips for generalized log canonical pairs under additional hypotheses. The general non-vanishing statement for minimal models in dimension 4 and above remains open.

Problem Statement

The two open issues left by Mori's flip-theorem paper (1988) after the resolution of flip existence by Birkar-Cascini-Hacon-McKernan (2010). Throughout, $(X, \Delta)$ is a projective pair over the complex numbers with $\Delta$ an effective $\mathbb{Q}$-divisor with coefficients in $[0, 1]$, and a $(K_X+\Delta)$-flip sequence is a chain

$$X = X_0 \dashrightarrow X_1 \dashrightarrow X_2 \dashrightarrow \cdots$$

of successive flips of $(K_{X_i}+\Delta_i)$-negative small contractions, each $X_i$ being $\mathbb{Q}$-factorial with $(X_i, \Delta_i)$ kawamata log terminal.

(1) Termination of flips in high dimension. Prove, or disprove by an explicit example, that for every integer $n \ge 5$ and every $\mathbb{Q}$-factorial klt pair $(X, \Delta)$ of dimension $n$ - in particular for terminal varieties with $\Delta = 0$ - every $(K_X+\Delta)$-flip sequence terminates. The terminal case alone is open for every $n \ge 5$; termination of terminal flip sequences is known only up to dimension 4. Component (1) also comprises the dimension-4 case: the same question remains open for klt pairs outside the known classes (terminal; canonical; pseudo-effective log canonical; scaling sequences under the hypotheses of Alexeev-Hacon-Kawamata 2007).

(2) Non-vanishing for minimal models in dimension at least 4 (Miyaoka's theorem in higher dimension). Prove, or disprove by an explicit example, that if $(X, \Delta)$ is a projective klt pair of dimension $n \ge 4$ admitting a minimal model - a birational copy $(X', \Delta')$ with log terminal singularities whose canonical class $K_{X'} + \Delta'$ is nef - then $\kappa(K_X+\Delta) \ge 0$, that is, $H^0(X, \mathcal{O}_X(m(K_X+\Delta))) \ne 0$ for some integer $m \ge 1$. Because pluricanonical sections are birational invariants of klt pairs, it is equivalent to treat the case where $(X, \Delta)$ is itself the minimal model, that is, where $K_X + \Delta$ is nef. (The general non-vanishing conjecture assumes only that $K_X+\Delta$ is pseudo-effective, a limit of effective divisors; it follows from (1) and (2) together, since (1) plus the existence of flips produces the minimal model on which (2) applies.)

Common objective. These two statements are the remaining pillars of the higher-dimensional minimal model program identified in Mori's 1988 paper. Since flips exist in all dimensions for klt pairs (Birkar-Cascini-Hacon-McKernan 2010, Corollary 1.4.1), an affirmative answer to (1) yields a minimal model for every klt pair with pseudo-effective canonical class, and an affirmative answer to (2) shows that minimal models in dimension at least 4 have non-negative Kodaira dimension. Together they would complete the higher-dimensional analogue of Mori's theorem - every projective variety is birational to a minimal model or to a Mori fiber space - and would give, via the Miyaoka-Mori numerical criterion, the dichotomy that a projective variety has $\kappa = -\infty$ if and only if it is uniruled, in all dimensions. The two components are interlocking rather than independent: non-vanishing is the standard route to minimal models in the non-big pseudo-effective case, whose completion requires termination.

References

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