ACC and lower semicontinuity for minimal log discrepancies, and termination of log flips: the surviving open core of Birkar-Shokurov's "Mld's vs thresholds and flips"

Minimal log discrepancies (mld's) are the finest standard numerical measure of the singularities of a log pair; Shokurov conjectured that for fixed dimension and boundary coefficients in a set satisfying the descending chain condition, the mld's of log canonical pairs satisfy the ascending chain condition (ACC), and Ambro conjectured that they vary lower semicontinuously. Birkar-Shokurov's "Mld's vs thresholds and flips" (2010) built a web of conditional implications around these two conjectures: together with a boundedness statement for canonical Mori-Fano varieties (weak BAB) and the log minimal model program, they would imply termination of log flips, ACC for all flavors of log canonical thresholds, and finiteness of sequences of log twists. Two pillars are now resolved unconditionally: ACC for log canonical thresholds (Hacon-McKernan-Xu 2014) and weak BAB, which is the $\varepsilon = 1$ case of Birkar's proof of the Borisov-Alexeev-Borisov boundedness conjecture (2021). The surviving core is open: ACC for mld's is unknown even in dimension three, Ambro's semicontinuity conjecture is open in dimension at least four (it is proved in dimension at most three and for toric varieties in all dimensions, by Ambro himself), and termination of log flips is open from dimension four onward. The problem is to settle this core and thereby complete the program.

Background

Minimal model theory studies algebraic varieties through their canonical class. A log pair $(X, B)$ is a normal variety $X$ together with an effective $\mathbb{R}$-divisor $B$ (the boundary) such that $K_X + B$ is $\mathbb{R}$-Cartier, where $K_X$ is the canonical divisor. Pairs carry a numerical measure of singularities: if $\varphi: W \to X$ is a proper birational morphism from a normal variety and one writes $K_W + B_W = \varphi^*(K_X + B)$, the log discrepancy of a divisor $D$ on $W$ is $a(D, X, B) = 1 - \mu_D B_W$, where $\mu_D B_W$ is the coefficient of $D$ in $B_W$. The pair is log canonical (lc) if $a(D, X, B) \ge 0$ for every such $D$, and $\varepsilon$-log canonical ($\varepsilon$-lc) if $a \ge \varepsilon$; a variety is canonical if $(X, 0)$ is 1-lc, and terminal if the inequality is strict for all exceptional $D$. For a point $P$ of $X$, the minimal log discrepancy is $\mathrm{mld}(P, X, B) = \inf a(D, X, B)$, the infimum over all divisors on all birational models whose center on $X$ is exactly the closure of $P$ (so for a closed point $P$ the image must be $P$ itself, and at the generic point of a prime divisor $D$ on $X$ the value recovers $a(D, X, B) = 1 - \mathrm{mult}_D B$). In codimension at least two the value is either a nonnegative real number or $-\infty$, the latter meaning that the pair is not lc at $P$. For example, at a smooth closed point $P$ of a variety of dimension $n$ one has $\mathrm{mld}(P, X, B) = n - \mathrm{mult}_P B$ whenever $\mathrm{mult}_P B \le 1$; in particular $\mathrm{mld}(P, X, 0) = n$. The log canonical threshold $\mathrm{lct}(X, B)$, the supremum of the $t$ such that $(X, tB)$ is lc, is a companion invariant, and the $a$-lc threshold is defined analogously with lc replaced by $a$-lc. A set of real numbers satisfies the descending chain condition (DCC) if it contains no infinite strictly decreasing sequence, and the ascending chain condition (ACC) if it contains no infinite strictly increasing one; a basic empirical principle of the field is that singularity invariants of lc pairs form ACC or DCC sets once the boundary coefficients are restricted to a fixed DCC set. The resolved instance is ACC for log canonical thresholds (Hacon-McKernan-Xu 2014). The outstanding instance is Shokurov's ACC conjecture for mld's: for every dimension $d$ and every DCC set $\Gamma \subseteq [0,1]$, the set of all $\mathrm{mld}(P, X, B)$ with $(X, B)$ lc, $\dim X \le d$, and coefficients of $B$ in $\Gamma$ satisfies the ACC. A second outstanding local conjecture, due to Ambro, predicts lower semicontinuity (LSC): for a fixed pair, the function $x \mapsto \mathrm{mld}(x, X, B)$, defined on the closed points of $X$ with values in $\mathbb{R}_{\ge 0} \cup \{-\infty\}$, is lower semicontinuous, meaning singularities can only worsen, never improve, at special points; it is proved in dimension at most 3 and for toric varieties in all dimensions, but open in general in dimension at least 4. These local questions are coupled to the global log minimal model program (LMMP), the algorithm that produces simpler birational models of a pair by repeatedly contracting or flipping curves on which $K_X + B$ is negative. A log flip replaces a small contraction $X \to Z$ (a birational contraction whose exceptional locus has codimension at least two) by another small modification $X^+ \to Z$ over which the transformed canonical class is ample; each step is an isomorphism in codimension one but changes the variety, and termination, the assertion that every sequence of flips from a given pair is finite, is what guarantees that the program outputs a minimal model or a Mori fiber space (a contraction onto a lower-dimensional base with $-K_X$ relatively ample); it is classical in dimension at most 3 and open in general. A Mori-Fano variety is a $\mathbb{Q}$-factorial lc variety of Picard rank 1 with $-K_X$ ample, and a class of varieties is bounded if its members occur as fibers of finitely many morphisms of finite type. Weak BAB asserts that the canonical $d$-dimensional Mori-Fano varieties are bounded; it is a boundary case of the Borisov-Alexeev-Borisov (BAB) conjecture, which asserts boundedness of $\varepsilon$-lc Fano varieties for every $\varepsilon > 0$. Birkar and Shokurov's paper ties these threads together. Its Main Theorem assumes the LMMP, the ACC conjecture for mld's, and weak BAB in dimension $\le d$, and derives ACC for $a$-lc thresholds in dimension $\le d$, ACC for lc thresholds in dimension $\le d+1$, termination of log flips in dimension $\le d+1$ for effective pairs (pairs $(X/Z, B)$, over a base $Z$, whose log canonical class satisfies $K_X + B \equiv M/Z$ for some effective $\mathbb{R}$-divisor $M$ on $X$; the property is preserved under flips), and existence of pl flips (flips of plt pairs whose boundary contains a component of coefficient 1) in dimension $\le d+2$. The introduction explains that termination reduces to two local, even formal, problems: ACC for mld's and Ambro's semicontinuity conjecture. The paper also studies log twists, elementary modifications of a pair that may be iterated until a final twist is reached, proving that a sequence of nonfinal twists terminates under a uniform upper bound $a_0 < 1$ on the minimal log discrepancies along the sequence (Proposition 3.7), and that such a bound follows from ACC for mld's near 1 (Lemma 3.8 and Corollary 3.9). In the two decades since, the conjectural inputs have been resolved one by one: flip existence for klt pairs (pairs with $\mathrm{mld} > 0$ everywhere) in all dimensions (Birkar-Cascini-Hacon-McKernan 2010), ACC for log canonical thresholds in all dimensions (Hacon-McKernan-Xu 2014), and the BAB conjecture (Birkar 2021), whose $\varepsilon = 1$ case is weak BAB. What survives is the core on mld's and termination itself.

Problem Statement

The problem is the surviving open core of Birkar-Shokurov's "Mld's vs thresholds and flips": the conjectural inputs that remain open after the intervening resolutions, namely the following three linked questions. (1) Shokurov's ACC conjecture for minimal log discrepancies, in the source's own formulation (Conjecture 1.3): for every integer $d \ge 1$ and every set $\Gamma \subseteq [0,1]$ satisfying the DCC, prove that the set of values $\mathrm{mld}(P, X, B)$, over all log canonical pairs $(X, B)$ of dimension at most $d$ with closed point $P \in X$ and all coefficients of $B$ in $\Gamma$, satisfies the ascending chain condition; or refute it by exhibiting such a $d$ and $\Gamma$ together with an infinite strictly increasing sequence of such values. The first open case is $d = 3$; the conjecture is known for surfaces and for toric pairs in all dimensions. (2) Ambro's lower-semicontinuity conjecture for minimal log discrepancies, in the version cited by the source as the second local input to termination: prove that for every pair $(X, B)$ the function $x \mapsto \mathrm{mld}(x, X, B)$, defined on the closed points of $X$ with values in $\mathbb{R}_{\ge 0} \cup \{-\infty\}$, is lower semicontinuous, or refute it by an explicit pair (necessarily of dimension at least 4, since the conjecture is proved in dimension at most 3 and for toric pairs in all dimensions) together with explicitly computed minimal log discrepancies at a point and on a punctured neighborhood demonstrating the failure; the first open case is dimension 4. (3) Termination of log flips: prove that for every dimension $d$, every sequence of log flips starting from an effective pair of dimension $d$ (in the source's terminology, a pair $(X/Z, B)$ over a base $Z$ whose log canonical class satisfies $K_X + B \equiv M/Z$ for some effective $\mathbb{R}$-divisor $M$, a property preserved under flips) is finite, or exhibit an effective pair with an infinite sequence of log flips; this is open from dimension 4 onward. A proof of (1), together with (2) on Shokurov's reduction, would make the source Main Theorem's conclusions unconditional with the now-resolved hypotheses removed: termination in all dimensions, ACC for $a$-lc thresholds with $0 \le a < 1$ (equivalent to an accumulation-point form of (1) by work of J. Liu), and finiteness of sequences of log twists with no uniform-boundedness or near-1 hypothesis (the source's Proposition 3.7 and Corollary 3.9). Since (3) is precisely the termination conclusion of that Main Theorem and the only hypotheses left are (1) and (2), resolving (1) together with (2) resolves (3) as well; correspondingly, a refutation of (3) would necessarily refute (1) or (2). The common objective uniting the three questions is to settle the local, formal conjectures about minimal log discrepancies that control the global finiteness of the log minimal model program in all dimensions, completing the Birkar-Shokurov program.

References

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