Termination of terminal log flips in dimensions at least five and the minimal model conjecture for klt pairs

The Log Minimal Model Program (LMMP) predicts that every Kawamata log terminal (klt) pair has a log minimal model or is birational to a Mori fibre space (the minimal model conjecture). Birkar's paper 'Log minimal models according to Shokurov' (Algebra & Number Theory 3 (2009), 951-958) reduces this conjecture in each dimension $d$ to a single termination statement: its Theorem 1.2 shows that the minimal model conjecture holds in dimension $d$ if every sequence of log flips starting from a $d$-dimensional klt pair that is terminal in codimension at least 2 is finite. The reduction is unconditional in its method — it runs the LMMP with scaling of a suitable big divisor and uses the finite-generation results of Birkar-Cascini-Hacon-McKernan to reduce to a limiting regime where only termination of terminal log flips is still needed. Termination of terminal log flips is known through dimension 4 (Fujino; Shokurov; see also Alexeev-Hacon-Kawamata), which yields the minimal model conjecture in dimension 4, but it is open in every dimension $d \geq 5$: no proof or refutation of termination of terminal log flips in any dimension $d \geq 5$ is known, and the minimal model conjecture in dimensions $d \geq 5$ is established unconditionally only for pairs of log general type (those with $K_X+B$ big) in all dimensions and, in dimension five, for effective pairs (those whose $K_X+B$ is $\mathbb{R}$-linearly equivalent to an effective divisor).

Background

Central to birational geometry is the Minimal Model Program (MMP), which classifies projective algebraic varieties up to birational equivalence by repeatedly simplifying them until the canonical class becomes nef (numerically effective, that is, it has non-negative intersection with every curve) or the variety acquires a Fano-type fibration (a contraction to a lower-dimensional base whose fibers have positive anti-canonical class). The modern language works with pairs: a pair $(X/Z, B)$ is a normal quasi-projective variety $X$ with a projective morphism $X \to Z$ and an $\mathbb{R}$-divisor $B$ (a divisor with real coefficients) with coefficients in $[0,1]$ such that $K_X+B$ is $\mathbb{R}$-Cartier (an $\mathbb{R}$-linear combination of Cartier divisors), where $K_X$ is the canonical divisor. For a prime divisor $D$ on a birational model of $X$, the log discrepancy $a(D, X, B)$ measures the singularity of the pair along $D$; the pair is Kawamata log terminal (klt) if all log discrepancies are positive, and it is terminal in codimension at least 2 if $a(D, X, B) > 1$ for every prime divisor $D$ exceptional over $X$.

The MMP proceeds by elementary operations: given a $(K_X+B)$-negative extremal contraction (the contraction associated with an extremal ray of the cone of curves on which $K_X+B$ is negative), one contracts a divisor, takes a fibration, or — when the contraction is small, i.e. contracts no divisor — performs a flip, a birational surgery $X \to Z \leftarrow X^+$ replacing the flipped locus by a $(K_{X^+}+B^+)$-positive one. A log flip is the flip of a $(K_X+B)$-negative extremal flipping contraction. A log minimal model of $(X/Z, B)$ is a $\mathbb{Q}$-factorial (every Weil divisor has a positive integral multiple that is Cartier) klt birational model $(Y/Z, B_Y)$ on which $K_Y+B_Y$ is nef over $Z$ and on which log discrepancies have not decreased; a Mori fibre space is a $(K_Y+B_Y)$-negative extremal contraction to a base of smaller dimension. The minimal model conjecture asserts that every klt pair has a log minimal model or a Mori fibre space. In dimension 2 this is classical; dimension 3 was settled in the 1980s-90s through the efforts of Mori, Shokurov, Kawamata and others. A major advance by Birkar-Cascini-Hacon-McKernan (building on Shokurov's construction of log flips) established, in all dimensions, the existence of log flips for klt pairs and, unconditionally, the existence of minimal models for pairs of log general type (that is, pairs with $K_X+B$ big, so that the divisor has enough sections to define a birational map).

The remaining obstacle is termination: an MMP produces a sequence of flips, and one must know that no infinite sequence arises. Birkar's paper 'Log minimal models according to Shokurov' proves, as its Theorem 1.2, that if terminal log flips terminate in dimension $d$ — meaning every sequence of log flips over $Z$ starting from a $d$-dimensional klt pair terminal in codimension at least 2 is finite — then the minimal model conjecture holds in dimension $d$: running the LMMP over $Z$ on $K_X+B$ with scaling of a suitable big divisor (a variant of the program in which the ray to contract is selected using an auxiliary nef divisor, keeping the process numerically controlled) produces a log minimal model or a Mori fibre space. The proof follows Shokurov's method for log canonical (lc) pairs (the larger singularity class with all log discrepancies non-negative; where he additionally needed the LMMP in dimension $d-1$, see his 'Ordered termination' paper) and uses the finite-generation results of Birkar-Cascini-Hacon-McKernan to reduce to a limiting regime in which only the termination assumption remains. Termination of terminal log flips is known through dimension 4 — cited in the source paper to Fujino's termination of 4-fold canonical flips (with its addendum) together with Shokurov's Steklov paper on minimal log discrepancies and termination of log flips, with Alexeev-Hacon-Kawamata's termination of many 4-dimensional log flips as companion evidence — and this yields the minimal model conjecture in dimension 4 (Corollary 1.3 of the source). In every dimension $d \geq 5$, termination of terminal log flips is open.

Problem Statement

Work over an algebraically closed field of characteristic zero. Following the source paper's convention, a sequence of terminal log flips in dimension $d$ is a sequence $X_1$, $X_2$, $X_3$, ... of log flips over a base $Z$, starting from a $d$-dimensional klt pair $(X/Z, B)$ that is terminal in codimension at least 2 (i.e. $a(D, X, B) > 1$ for every prime divisor $D$ exceptional over $X$), where each step $X_i \to Z_i \leftarrow X_{i+1}$ is a $(K_{X_i}+B_i)$-flip and $B_i$ is the birational transform of $B$; termination means that every such sequence is finite.

The problem is to settle, for every dimension $d \geq 5$ (the first open case is $d = 5$), the following linked questions, which share one scientific objective — completing the minimal model program for klt pairs in dimensions at least five:

(a) Termination: prove that terminal log flips terminate in dimension $d$. By Theorem 1.2 of Birkar's 'Log minimal models according to Shokurov' this immediately implies the minimal model conjecture in dimension $d$: every klt pair $(X/Z, B)$ of dimension $d$ has a log minimal model or a Mori fibre space.

(b) Alternatively, prove the minimal model conjecture for klt pairs of dimension $d \geq 5$ unconditionally, without assuming termination of terminal log flips.

A refutation of the termination component — an explicit klt pair of some dimension $d \geq 5$, terminal in codimension at least 2, together with a construction of an infinite sequence of its log flips — would settle (a) in the negative and leave the minimal model conjecture (b) open.

Proving (a) in every dimension $d \geq 5$ also yields (b), by Theorem 1.2 above, and (b) may also be sought directly, in every dimension $d \geq 5$, without assuming termination of terminal log flips. Known unconditional partial results: the case where $K_X+B$ is big in all dimensions (Birkar-Cascini-Hacon-McKernan) and, in dimension five, the effective case (pairs whose $K_X+B$ is $\mathbb{R}$-linearly equivalent to an effective divisor, e.g. of nonnegative Kodaira dimension; Birkar's Compositio 2010 paper); the novel content therefore lies in the remaining regime, in particular pairs of negative Kodaira dimension for which a Mori fibre space outcome is expected.

References

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