Termination of log flips in dimension four for non-pseudo-effective log canonical pairs

Birkar (Math. Ann. 2010) proved that any sequence of log flips over the base $Z$ terminates for a 4-dimensional klt pair $(X/Z, B)$ of log canonical dimension $\lambda(X/Z, B) \ge 2$, and recorded as open the removal of the effectivity hypothesis $\lambda \ge 2$, the extension to log canonical pairs, the weak finiteness of log minimal models up to isomorphism, and the stronger versions of his theorem implied by the ACC conjectures for log canonical thresholds and for minimal log discrepancies. Later literature resolved most of this program. ACC for log canonical thresholds was proved by Hacon–McKernan–Xu (2014), which via Birkar's earlier Duke (2007) reduction extends 4-dimensional termination to klt pairs with $K_X + B$ numerically equivalent to an effective divisor. Chen–Tsakanikas (2023) and independently Moraga (2025) then proved termination of every sequence of flips for 4-dimensional log canonical pairs — indeed generalized pairs — with $K_X + B$ pseudo-effective over $Z$, subsuming both the log canonical extension and the weakening of $\lambda \ge 2$ to pseudo-effectivity. The surviving open case is exactly the non-pseudo-effective case (every such pair has $\lambda(X/Z, B) = -\infty$; the class includes 4-dimensional log Fano pairs): Chen–Tsakanikas state in print that it “remains unresolved, even in dimension 4” and is “essentially the only obstacle” to deriving termination of flips from the existence of minimal models and Mori fiber spaces, and Han–Liu (2025) confirm that termination of flips remains one of the major open problems of birational geometry, reducing the 4-dimensional log canonical case to the ACC for global minimal log discrepancies of exceptionally non-canonical pairs. The problem: prove — or refute by an explicit non-terminating example — that for every log canonical pair $(X/Z, B)$ of dimension 4 with $K_X + B$ not pseudo-effective over $Z$, every sequence of $(K_X + B)$-flips over $Z$ terminates; combined with the published pseudo-effective theorems this would settle the termination of flips conjecture in dimension 4 completely.

Background

The minimal model program (MMP) is the central organizing framework of birational geometry over algebraically closed fields of characteristic zero, such as the complex numbers. Its goal is, given a variety — more generally a pair — to construct a birational model on which the canonical class is as positive as possible, reducing the classification of varieties to two terminal shapes: minimal models, on which the canonical class is nef (numerically effective, i.e. non-negative on every curve), and Mori fiber spaces, which are fibered over a lower-dimensional base by Fano-type fibers (fibers with anti-ample canonical class). One works with pairs $(X/Z, B)$: $X$ is a normal quasi-projective variety equipped with a projective surjective morphism $X \to Z$ to a normal quasi-projective base (the relative, or “over $Z$”, setting; the absolute case is $Z$ a point), and $B$ is a boundary, an effective $\mathbb{R}$-divisor with coefficients in $[0, 1]$ such that $K_X + B$ is $\mathbb{R}$-Cartier, where $K_X$ is a canonical divisor. The singularities of a pair are measured by log discrepancies $a(E, X, B)$, rational numbers indexed by divisors $E$ on birational models of $X$; the pair is klt (Kawamata log terminal) when every log discrepancy over $X$ is positive, and lc (log canonical) when every log discrepancy is non-negative — the largest class in which the program is expected to run. (A dlt pair — divisorially log terminal — is an lc pair whose log discrepancies are positive away from a distinguished divisorial locus; dlt is the technical class in which the program's operations are best behaved.) The program proceeds by elementary surgeries: when $K_X + B$ is not nef over $Z$, the cone theorem describes the cone of curves and produces an extremal contraction of a $(K_X + B)$-negative curve; if that contraction is divisorial the Picard rank drops (so only finitely many can occur in a program), but if it is small — its exceptional locus has codimension at least 2 — one must perform a flip: a non-trivial surgery $X \to W \leftarrow X^+$ replacing the $(K_X + B)$-negative locus by a $(K_X + B)$-positive one, which preserves the dimension and the Picard rank. A sequence of flips $X = X_1 \dashrightarrow X_2 \dashrightarrow X_3 \dashrightarrow \cdots$ over $Z$, where each $X_{i+1}$ is the flip of an extremal flipping contraction of $(X_i/Z, B_i)$ and $B_i$ is the birational transform of $B$, could a priori continue forever; the termination of flips conjecture asserts that it never does. Existence of the needed flips is known in wide generality (for klt pairs in all dimensions by Birkar–Cascini–Hacon–McKernan 2010, and for log canonical pairs; see the introduction of Moraga 2025 for references), so termination is the remaining pillar of the conjecture. In dimension at most 3 it is a theorem (Kawamata 1992; see the accounts in Chen–Tsakanikas 2023 and Han–Liu 2025 for the complete attribution), and for programs with scaling — a restricted, practically sufficient way of ordering the flips in which each step is selected using an auxiliary ample divisor — termination is known for 4-dimensional $\mathbb{Q}$-factorial dlt pairs (see the account in Chen–Tsakanikas 2023). For arbitrary sequences, termination in dimension 4 is open exactly in the non-pseudo-effective case described below, and in dimensions at least 5 even the pseudo-effective case is known only conditionally on termination in lower dimensions (Birkar 2007; Chen–Tsakanikas, Theorem 1.3); Han–Liu describe the conjecture as one of the major remaining open problems in birational geometry. In dimension 4 the theorem was approached through progressively larger singularity classes of $X$ itself: terminal (all log discrepancies over $X$ strictly greater than 1; smooth points are terminal), canonical (at least 1), then klt and lc. Terminal 4-fold flips terminate by Kawamata–Matsuda–Matsuki (1987), canonical 4-fold flips by Fujino (2004), and a large class of klt pairs by Alexeev–Hacon–Kawamata (2007). The source paper, Birkar (Math. Ann. 2010), proved termination for klt pairs of dimension 4 under a positivity hypothesis: the log canonical dimension $\lambda(X/Z, B) \ge 2$. Here $\kappa(X/Z, B)$ denotes the relative Kodaira dimension (the Iitaka dimension of $K_X + B$ over $Z$, i.e. the growth rate of the spaces of sections of multiples of the pushforward divisor), while $\lambda(X/Z, B)$ is Birkar's log canonical dimension: the dimension of the log canonical model of a log minimal model of the pair — a log minimal model being a birational model on which the pushforward of $K_X + B$ is nef, and its log canonical model the projective model over $Z$ defined by that semi-ample (nef, with enough sections to define a morphism) pushforward — with $\lambda = -\infty$ if no such model exists; if the minimal model conjecture and abundance (the expectation that nef canonical classes on minimal models are semi-ample) hold, then $\lambda = \kappa$. A divisor on $X$ is pseudo-effective over $Z$ if it is pseudo-effective on a very general fiber of $X \to Z$, i.e. a limit of effective divisors there. The chain of implications $\lambda \ge 2$ ⇒ $K_X + B$ is $\mathbb{R}$-linearly equivalent over $Z$ to an effective divisor ⇒ $K_X + B$ is pseudo-effective over $Z$ (the first step is Step 2 of the source paper's proof) is what connects the source theorem to the later results below. The paper's closing remarks record the open directions: (i) the lc generalisation (“It should not be difficult to generalise these results to the lc case but for simplicity we only treat the klt case”); (ii) the removal of $\lambda \ge 2$, for which it would suffice that the log minimal models of the pair be finite in number, or even only finite up to isomorphism — a weak finiteness predicted by the Morrison–Kawamata–Totaro cone conjecture for Calabi–Yau fiber spaces, which asserts that on a fiber space with trivial canonical class the pseudo-automorphism group (birational self-maps isomorphic in codimension one) acts on the relevant cone of effective divisors with a rational polyhedral fundamental domain, implying finitely many minimal models up to isomorphism (Kawamata 1997; Totaro 2008); and (iii) the observation that the ACC conjecture for log canonical thresholds and the ACC conjecture for minimal log discrepancies in dimension 4 each imply stronger versions of the theorem. In (iii), for an auxiliary effective divisor $M$, the log canonical threshold $\mathrm{lct}(X, B; M)$ is the supremum of $t \ge 0$ such that $(X, B + tM)$ is lc; the minimal log discrepancy $\mathrm{mld}(P, X, B)$ at a point $P$ is the infimum of the log discrepancies of divisors over $X$ whose center contains $P$; the ACC conjectures assert that these quantities admit no infinite strictly ascending chains within naturally restricted classes of pairs. A generalized pair (g-pair) is a pair $(X/Z, B)$ together with the data of a b-nef divisor $M$ on a higher biratorial model, the pair being written $(X/Z, B + M)$; NQC means that on that higher model the moduli part is a non-negative $\mathbb{Q}$-linear combination of $\mathbb{Q}$-Cartier divisors that are nef over $Z$; every statement below for g-pairs specializes to usual pairs by taking $M = 0$. Later work transformed the landscape. Hacon–McKernan–Xu (Ann. of Math. 2014) proved the ACC for log canonical thresholds, and their Corollary 1.2, combined with Birkar's earlier reduction (Duke Math. J. 2007), yields termination for $\mathbb{Q}$-factorial klt projective pairs of dimension $n$ with $K_X + B$ numerically equivalent to an effective divisor, assuming termination of flips in dimension $n - 1$ (for $n = 4$ the 3-dimensional input is known). Chen–Tsakanikas (2023) and independently Moraga (2025, in the projective case) then settled the pseudo-effective range in dimension 4: for a 4-dimensional lc pair — indeed an NQC lc g-pair — with $K_X + B$ (respectively $K_X + B + M$) pseudo-effective over $Z$, every sequence of flips over $Z$ terminates (Chen–Tsakanikas, Theorem 1.1; Moraga's Theorem 1 covers projective lc 4-folds). These theorems subsume Birkar's theorem and its conjectural lc extension, and weaken $\lambda \ge 2$ all the way to pseudo-effectivity. What remains is precisely the non-pseudo-effective case: a pair with $K_X + B$ not pseudo-effective over $Z$ has $\lambda(X/Z, B) = -\infty$, and the class includes 4-dimensional log Fano pairs (pairs with $-(K_X + B)$ ample over $Z$). Chen–Tsakanikas state that this case remains unresolved even in dimension 4 and is essentially the only obstacle between the current state of knowledge and the termination of flips conjecture. The sharpest conditional reduction is due to Han–Liu (2025) via exceptionally non-canonical (enc) singularities — pairs that are not canonical and for which all but one exceptional prime divisor over $X$ have log discrepancy at least 1, a class automatically contained in klt — proving that the ACC for global minimal log discrepancies of 4-dimensional enc pairs whose boundary coefficients belong to a fixed finite set implies termination of all lc flips in dimension 4.

Problem Statement

Prove, or disprove by an explicit example, the following statement.

(†) Let $k$ be an algebraically closed field of characteristic zero. Let $Z$ be a normal quasi-projective variety over $k$, let $X$ be a normal quasi-projective 4-dimensional variety over $k$ equipped with a projective surjective morphism $\pi \colon X \to Z$, and let $B \ge 0$ be a boundary, i.e. an $\mathbb{R}$-divisor (a finite $\mathbb{R}$-linear combination of prime divisors) with coefficients in $[0, 1]$, such that $K_X + B$ is $\mathbb{R}$-Cartier (locally a $\mathbb{Q}$-linear combination of Cartier divisors), where $K_X$ is a canonical divisor. Assume that the pair $(X/Z, B)$ is log canonical: for every birational morphism $f \colon Y \to X$ from a normal variety $Y$ and every prime divisor $E$ on $Y$, the log discrepancy $a(E, X, B) = 1 + \mathrm{mult}_E(K_Y - f^*(K_X + B))$ is non-negative. Assume moreover that $K_X + B$ is not pseudo-effective over $Z$: its restriction to a very general fiber of $\pi$ is not pseudo-effective, i.e. not a limit of effective $\mathbb{R}$-divisors. In Birkar's terminology every such pair has log canonical dimension $\lambda(X/Z, B) = -\infty$, and the class includes all 4-dimensional log Fano pairs, e.g. $Z$ a point with $-(K_X + B)$ ample.

Then there is no infinite sequence of $(K_X + B)$-flips over $Z$ starting from $(X/Z, B)$: one cannot have $X = X_1 \dashrightarrow X_2 \dashrightarrow X_3 \dashrightarrow \cdots$ over $Z$ indefinitely, where each step is a flip — that is, $X_{i+1}$ is the flip of an extremal flipping contraction of $(X_i/Z, B_i)$, a small (exceptional locus of codimension at least 2) projective contraction $X_i \to W_i$ over $Z$ of an extremal ray of the cone of curves on $X_i$ over $Z$ that is negative for $K_{X_i} + B_i$, and $B_i$ denotes the birational transform of $B$ on $X_i$. Equivalently: every sequence of $(K_X + B)$-flips over $Z$ starting from $(X/Z, B)$ is finite.

A proof of (†), combined with the published termination theorems for the pseudo-effective case (Chen–Tsakanikas 2023, Theorem 1.1, DOI: 10.1007/s10114-023-0116-3; Moraga 2025, Theorem 1, DOI: 10.1007/s00209-025-03766-y), establishes the termination of flips conjecture in dimension 4 in full: for every log canonical pair $(X/Z, B)$ of dimension 4, with no restriction on $\kappa$, $\lambda$, or pseudo-effectivity, every sequence of flips over $Z$ terminates. Without loss of generality $X$ may be assumed $\mathbb{Q}$-factorial (every Weil divisor has a positive multiple that is Cartier), since a sequence of flips lifts through a $\mathbb{Q}$-factorialisation (Step 1 of the source paper); by the standard reductions used in the literature (dlt modifications together with special termination — the technique that handles flips whose centers lie near the log canonical centers, the subvarieties of $X$ along which all log discrepancies vanish — as in Chen–Tsakanikas 2023 and Moraga 2025), it suffices to treat $\mathbb{Q}$-factorial dlt pairs, and the klt case is the essential difficulty.

A disproof of (†) consists of a single pair $(X/Z, B)$ as above together with one explicit infinite sequence of $(K_X + B)$-flips over $Z$ starting from it.

References

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