Boundedness of the index of Calabi-Yau varieties and pairs: uniform pluricanonical non-vanishing when the canonical class is numerically trivial

The introduction of Birkar's 'Anti-pluricanonical systems on Fano varieties' (2019) proves effective non-vanishing for anti-pluricanonical systems on klt Fano varieties and then remarks that for a projective variety with numerically trivial canonical class, although some multiple of the canonical class gives a non-empty linear system, 'it is not clear whether we can choose m depending only on the dimension'. Since a numerically trivial canonical class is torsion, this asks whether the index — the least m with $mK_X$ linearly trivial — of klt Calabi-Yau varieties of fixed dimension is bounded. Later literature consolidates this into the index conjecture for Calabi-Yau pairs: for lc pairs $(X,B)$ with $K_X+B$ $\mathbb{Q}$-linearly trivial and boundary coefficients in a fixed finite (equivalently DCC) set, the index is bounded in terms of the dimension and the coefficient set. The problem remains open: known are dimension at most 3 (including klt threefolds by Jiang and lc threefold pairs by Y. Xu), non-klt lc fourfold pairs and semi-log-canonical threefold pairs (Jiang-Liu), smooth Calabi-Yau fourfolds (Masamura), and non-canonical klt Calabi-Yau fourfolds (Jiao); the canonical fourfold case and all dimensions at least 5 are open, with doubly exponential lower bounds known, including for smooth varieties (Singh). Two questions often mentioned alongside it — the Borisov-Alexeev-Borisov conjecture and boundedness of complements for general boundary coefficients — are settled in later work and lie outside the problem posed here.

Background

The minimal model program (MMP) of higher-dimensional birational geometry decomposes an arbitrary projective variety, up to birational equivalence, into building blocks of three types: Fano varieties, on which the anti-canonical class $-K_X$ is ample; Calabi-Yau varieties, on which the canonical class is trivial in a suitable sense; and varieties of general type. Here $X$ is a normal projective variety, $K_X$ its canonical class, and a $\mathbb{Q}$-divisor is a finite rational linear combination of codimension-one subvarieties; a divisor class is numerically trivial, written $K_X \equiv 0$, if it pairs to zero with every curve, and linearly trivial, written $D \sim 0$, if it is the divisor of a rational function. The linear system $|D|$ is the set of effective divisors linearly equivalent to $D$. Throughout, we work over an algebraically closed field of characteristic zero, the standard convention of the minimal model program and of the sources cited here.

The source paper, Birkar's 'Anti-pluricanonical systems on Fano varieties', settled the effectivity question on the Fano side: for each dimension $d$ there is an integer $m$ such that $|-mK_X|$ is non-empty for every Fano variety of dimension $d$ with mild singularities (Theorem 1.1 there), and its companion paper proved the Borisov-Alexeev-Borisov (BAB) conjecture. 'Mild singularities' is made precise via pairs: a pair $(X,B)$ is a normal variety $X$ with an effective $\mathbb{R}$-divisor $B$ such that $K_X+B$ is $\mathbb{R}$-Cartier; the log discrepancy of a divisor $E$ over $X$ is a numerical invariant of the singularities of the pair along $E$, and $(X,B)$ is log canonical (lc) if all log discrepancies are at least $0$, Kawamata log terminal (klt) if all are strictly positive, $\varepsilon$-lc if all are at least $\varepsilon>0$, canonical if all are at least $1$, and terminal if all are at least $2$. The BAB conjecture — that the $\varepsilon$-lc Fano varieties of fixed dimension form a bounded family, i.e. are covered by finitely many flat families of bounded degree over bases of bounded degree — is now a theorem, but it gives no information in the Calabi-Yau direction, where no $\varepsilon$ lower bound is available.

A klt variety with $K_X \equiv 0$ is its own minimal model — the MMP terminates on it with the identity map — so no boundedness input analogous to the Fano side is available. Call a projective klt variety $X$ with $K_X \equiv 0$ a Calabi-Yau variety in the weak sense. A basic result of the MMP gives that a numerically trivial canonical class is torsion: there is a positive integer $m$ with $mK_X \sim 0$. The index of such $X$ is the least such $m$. For quotients of abelian varieties the index is governed by the induced action on differential forms; for the simplest Calabi-Yau manifolds (hypersurfaces of degree $n+1$ in $\mathbb{P}^n$) it is 1, and small dimensions can be classified by hand, so the difficulty of bounding the index is a genuinely high-dimensional phenomenon: Calabi-Yau varieties do not form a bounded family in any dimension at least 2 (K3 surfaces alone vary in infinitely many components with unbounded degree), and no structure theory short of the conjecture itself is known to control uniformly the torsion order of the canonical class inside the Picard group, which is what the index measures.

The question has a standard formulation for pairs, articulated around Prokhorov-Shokurov's work on complements and now called the index conjecture. An lc Calabi-Yau pair is a projective lc pair $(X,B)$ with $K_X+B \sim_{\mathbb{Q}} 0$ (linearly trivial over the rationals); its index is the least $m \ge 1$ with $m(K_X+B) \sim 0$. Some authors instead call a pair with $K_X+B \equiv 0$ log Calabi-Yau; for klt varieties the two conditions agree, since a numerically trivial canonical class is torsion there, but for lc pairs the numerical version additionally asserts that a numerically trivial log canonical divisor is $\mathbb{Q}$-linearly trivial, a log-abundance-type statement not known in general, so the conjecture is standardly formulated with the $\mathbb{Q}$-linear condition, under which the index is defined. A set of real numbers satisfies the descending chain condition (DCC) if it contains no infinite strictly decreasing sequence; by a theorem of Hacon-McKernan-Xu it suffices to treat finite coefficient sets. The conjecture: for every dimension $d$ and every finite set $\Phi \subset [0,1] \cap \mathbb{Q}$ of allowed boundary coefficients there is $M = M(d,\Phi)$ such that every lc Calabi-Yau pair of dimension $d$ with coefficients of $B$ in $\Phi$ has index dividing $M$. The klt variety case ($B=0$, $X$ klt) is exactly the source paper's remark (for klt varieties, $K_X \equiv 0$ and $K_X \sim_{\mathbb{Q}} 0$ are equivalent), and the klt-pair case reduces to klt varieties by a theorem of Y. Xu. Prokhorov-Shokurov proved boundedness of indices for boundaries with coefficients in the hyperstandard sets $\Phi(\mathfrak{R}) = \{1 - r/m \mid r \in \mathfrak{R},\ m \in \mathbb{N}\}$ with $\mathfrak{R} \subset [0,1] \cap \mathbb{Q}$ finite, conditionally on their complements and adjunction conjectures, and unconditionally in dimension 2. As of 2026 the conjecture is known in dimension at most 3 (the terminal threefold case goes back to 1986 and is compiled by Masamura; klt threefolds by Jiang via his gap theorem for minimal log discrepancies — the infima of log discrepancies — of non-canonical singularities; lc threefold pairs by Y. Xu), for non-klt lc pairs in dimension 4 and semi-log-canonical pairs in dimension 3 (Jiang-Liu; semi-log-canonical is the non-normal analogue of lc), for smooth Calabi-Yau fourfolds (Masamura), and for non-canonical klt Calabi-Yau fourfolds (Jiao). The canonical fourfold case beyond smooth varieties, and all dimensions $\ge 5$ even for varieties with $B=0$, are open. In the opposite direction, Esser-Totaro-Wang constructed Calabi-Yau varieties whose index grows doubly exponentially with the dimension — Singh extended this doubly exponential growth to smooth Calabi-Yau varieties — and Esser-Totaro-Wang conjecture their examples are optimal, so any bound must be large.

Two standard frameworks reduce the question to a finite computation and give the expected shape of the answer. The first is Shokurov's theory of complements: an $n$-complement of $K_X+B$ is a boundary $B^+ \ge B$ with $(X,B^+)$ lc and $n(K_X+B^+) \sim 0$; in the Calabi-Yau situation $B=0$ an $n$-complement of $K_X$ is exactly a trivialization of the index. The second is the canonical bundle formula, which for a fibration $f\colon X \to Z$ with $K_X+B \sim_{\mathbb{Q}} 0$ over a lower-dimensional base writes $K_X+B \sim_{\mathbb{Q}} f^*(K_Z+B_Z+M_Z)$ with the discriminant $B_Z$ and moduli part $M_Z$, reducing the index to the torsion orders of the coefficients and of the moduli part; Birkar-Di Cerbo-Svaldi's boundedness of elliptic Calabi-Yau varieties with a rational section is proved under the hypothesis that the relevant torsion indices are bounded, and Prokhorov-Shokurov's conditional proof of the index conjecture for hyperstandard boundaries runs exactly along this route, reducing to their (still open in general) adjunction conjecture on effectivity of the moduli part.

One name is shared with a different statement: Shokurov's index conjecture for minimal log discrepancies is the local assertion that the Cartier index of $K_X$ at a point of a klt singularity is bounded in terms of the dimension and the minimal log discrepancy there, and the non-klt counterexample of Nakamura-Shibata (reference 20) refutes that local statement and produces no counterexample to the Calabi-Yau pair index conjecture posed here.

Problem Statement

Primary (source-faithful) form. The source paper remarks, immediately after stating its effective non-vanishing theorem for Fano varieties: 'If $K_X$ is numerically trivial, there is $m$ such that $|mK_X|$ is non-empty but it is not clear whether we can choose $m$ depending only on the dimension.' Formally: for each positive integer $d$, does there exist a positive integer $m(d)$ such that $|m(d)K_X|$ is non-empty for every projective klt variety $X$ of dimension $d$ with $K_X \equiv 0$? Since a numerically trivial canonical class on a klt variety is torsion, $|mK_X|$ is non-empty exactly when $mK_X \sim 0$, so this asks whether the index of klt Calabi-Yau varieties of dimension $d$ is bounded in terms of $d$ alone.

General (standard) form — the index conjecture. For each positive integer $d$ and each finite set $\Phi \subset [0,1] \cap \mathbb{Q}$ of allowed boundary coefficients (equivalently, by the Hacon-McKernan-Xu DCC reduction, each DCC set), does there exist $M = M(d,\Phi)$ such that every projective lc pair $(X,B)$ of dimension $d$ with $K_X+B \sim_{\mathbb{Q}} 0$ and all coefficients of $B$ in $\Phi$ satisfies $M(K_X+B) \sim 0$? The primary form is the klt case of the $\Phi = \{0\}$ instance, and the klt-pair case reduces to the variety case by a theorem of Y. Xu.

The general statement asks, for every dimension $d$ and every finite (equivalently DCC) coefficient set $\Phi$, whether there is $M = M(d,\Phi)$ such that $M(K_X+B) \sim 0$ for every projective lc pair $(X,B)$ of dimension $d$ with coefficients of $B$ in $\Phi$; its two logically exhaustive outcomes are the existence of such a bound or, negatively, one fixed dimension $d$ and one fixed finite coefficient set $\Phi$ together with an infinite sequence of projective lc Calabi-Yau pairs of dimension $d$ with coefficients in $\Phi$ whose indices are unbounded. The problem posed here is the general statement; results in a fixed dimension, for example the canonical fourfold case or any fixed range of dimensions below the general one, settle only special cases. No restriction on singularities beyond klt/lc, on dimension, on coefficient sets, on geometry, or on the form of the answer is intended; in particular the expected bound must be allowed to be at least doubly exponential in the dimension, as the known constructions require, including the smooth ones (Singh).

References

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