Existence of log minimal models or Mori fibre spaces for log canonical pairs in arbitrary dimension (the minimal model conjecture)

The log minimal model program (LMMP) predicts that every log canonical pair $(X/Z,B)$ over an algebraically closed field of characteristic zero admits either a log minimal model, on which $K_X+B$ becomes nef, or a Mori fibre space, a Fano-type fibration negative for $K_X+B$. Birkar's 2010 paper 'On existence of log minimal models' proved that the LMMP with scaling in dimension $d-1$ implies this conjecture for effective log canonical pairs in dimension $d$ (unconditionally in dimension five), posed the weak non-vanishing conjecture (every pseudo-effective $\mathbb{Q}$-factorial dlt pair is effective) as the key route, and framed termination of log flips as the remaining structural obstacle. Subsequent literature: existence of log canonical flips is now unconditional (Birkar 2012, extended to generalized pairs by Hacon-Liu), the ascending chain condition for log canonical thresholds is a theorem (Hacon-McKernan-Xu 2014), and the conjecture is reduced to non-vanishing for smooth projective varieties (Hashizume 2018; Lazic-Tsakanikas 2022). The general conjecture nevertheless remains open beyond dimension four for arbitrary log canonical pairs, beyond dimension five for effective ones, and beyond the big-boundary klt case; weak non-vanishing is open from dimension four upward, and termination of log flips is open in general dimension despite substantial 2023-2025 progress in dimension four. The problem asks for a proof or refutation of the conjecture in full generality.

Background

The birational classification of algebraic varieties over an algebraically closed field $k$ of characteristic zero is organized by the log minimal model program (LMMP). Its input is a pair $(X/Z,B)$: $X$ and $Z$ are normal quasi-projective varieties over $k$ with a projective morphism $X \to Z$, and $B$ is an $\mathbb{R}$-divisor with coefficients in $[0,1]$ (a boundary) such that the log canonical class $K_X+B$ is $\mathbb{R}$-Cartier, where $K_X$ is a canonical divisor. Singularities are measured by log discrepancies $a(D,X,B)$, defined for each prime divisor $D$ over $X$ by writing $K_Y = f^*(K_X+B) + \sum a(D,X,B)D$ on a log resolution $f: Y \to X$: the pair is log canonical (lc) if $a(D,X,B) \ge 0$ for all $D$, Kawamata log terminal (klt) if all $a(D,X,B) > 0$, and divisorially log terminal (dlt) if some log resolution is simple normal crossings at every divisorial lc centre; the working category usually adds $\mathbb{Q}$-factoriality, meaning every Weil divisor has a Cartier multiple. A pair is pseudo-effective if $K_X+B$ is, over $Z$, a numerical limit of effective divisors, and effective if $K_X+B \equiv M/Z$ (numerical equivalence over $Z$) for some effective $\mathbb{R}$-divisor $M \ge 0$; for a projective variety, non-negative Kodaira dimension implies effectivity.

The LMMP improves a pair birationally: an extremal contraction that is negative on $K_X+B$ and small (contracts no divisor) is repaired by a log flip, a birational surgery replacing the contracted curves by curves on which $K_X+B$ is relatively ample; a sequence of flips may a priori be infinite, and its termination is a major open problem. The desired outputs are: a log minimal model, meaning a birational model $(Y/Z, B+E)$ that is $\mathbb{Q}$-factorial dlt, with $E$ the sum of divisors exceptional over $X$, such that $K_Y+B+E$ is nef over $Z$ (non-negative on every curve over $Z$) and such that every divisor $D$ on $X$ exceptional over $Y$ satisfies $a(D,X,B) < a(D,Y,B+E)$; or a Mori fibre space, meaning a birational model $(Y/Z, B+E)$ that is $\mathbb{Q}$-factorial dlt, with $E$ again the sum of divisors exceptional over $X$, carrying a $K_Y+B+E$-negative extremal contraction $Y \to T/Z$ of fibre type with $\dim Y > \dim T$ — no nefness of $K_Y+B+E$ over $Z$ is required in this branch — and satisfying $a(D,X,B) \le a(D,Y,B+E)$ for every prime divisor $D$ on a birational model of $X$, with strict inequality whenever $D$ is on $X$ and is contracted on $Y$ (Definitions 2.4 and 2.5 of the paper). Conjecture 1.1 of Birkar's 2010 Compositio paper, the minimal model conjecture, asserts that any lc pair $(X/Z,B)$ has one of the two.

The problem is classical in low dimensions: surfaces go back to the Italian school, dimension three was settled in the 1980s-90s, and Birkar's 2010 paper records the general four-dimensional case as a theorem of Shokurov; Hashizume's 2018 account likewise states that log minimal models exist for pseudo-effective lc pairs through dimension four. Birkar-Cascini-Hacon-McKernan settled the klt case with boundary big over the base in all dimensions (varieties of log general type). The paper proves that the LMMP with scaling in dimension $d-1$ (running the program while successively subtracting a fixed ample divisor) implies the conjecture for effective lc pairs in dimension $d$, yielding it unconditionally for effective lc pairs in dimension five, via termination of flips with scaling in dimension four, which the paper derives from results of Shokurov and from Alexeev-Hacon-Kawamata's termination of many four-dimensional log flips; it also gives new proofs for dimension four and for the Shokurov reduction theorem, which reduces existence of flips to special termination (termination of flips disjoint from the floor $\lfloor B \rfloor$ of the boundary after finitely many steps) and to the existence of pl flips (flips for which a component of $\lfloor B \rfloor$ is relatively mobile). Beyond these cases the conjecture is open in all dimensions.

The paper identifies two conjectural routes. First, weak non-vanishing (Conjecture 1.4 of the paper): every pseudo-effective $\mathbb{Q}$-factorial dlt pair is effective, a purely positivity-theoretic statement about adjoint divisors. Second, termination of log flips; the paper's introduction records reductions of termination to the ascending chain condition (ACC, absence of infinite strictly increasing sequences) for log canonical thresholds (the supremal boundary coefficient preserving lc-ness) and to properties of minimal log discrepancies (mlds, the infimum of log discrepancies at a point, a fine measure of singularity). A third interface emerged later: Birkar proved that, assuming the LMMP in dimension $d-1$, an lc pair has a log minimal model if and only if $K_X+B$ admits a weak Zariski decomposition over $Z$, that is, a birational numerical decomposition into a nef and an effective part, making such decompositions the exact currency of the problem.

Problem Statement

Prove or refute the minimal model conjecture for log canonical pairs in full generality (Conjecture 1.1 of Birkar 2010): show that every lc pair $(X/Z,B)$ — with $X, Z$ normal quasi-projective over an algebraically closed field of characteristic zero, $X \to Z$ projective, $B$ an $\mathbb{R}$-boundary with $K_X+B$ $\mathbb{R}$-Cartier, and with no restriction on dimension, effectivity, pseudo-effectivity, or $\mathbb{Q}$-factoriality — admits a log minimal model or a Mori fibre space in the sense recalled in the Background (either a $\mathbb{Q}$-factorial dlt birational model $(Y/Z, B+E)$ with $K_Y+B+E$ nef over $Z$ and $a(D,X,B) < a(D,Y,B+E)$ for every prime divisor $D$ on $X$ exceptional over $Y$; or a $\mathbb{Q}$-factorial dlt birational model $(Y/Z, B+E)$ carrying a $K_Y+B+E$-negative extremal contraction $Y \to T/Z$ of fibre type with $\dim Y > \dim T$, with $a(D,X,B) \le a(D,Y,B+E)$ for every prime divisor $D$ on a birational model of $X$ and strict inequality for every such $D$ that is on $X$ and contracted on $Y$). A proof must establish this disjunction for every lc pair in every dimension, whether or not $K_X+B$ is pseudo-effective; a refutation must exhibit an lc pair admitting neither a log minimal model nor a Mori fibre space, with a complete argument for the non-existence of both.

References

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