Good log minimal models and finite generation of log canonical rings for log canonical pairs of log general type without base-locus hypotheses

Birkar and Hu (Compositio Math. 150 (2014) 579-592) proved that a projective log canonical pair $(X,B)$ with $\mathbb{Q}$-divisor boundary has a good log minimal model — and hence a finitely generated log canonical ring — whenever $K_X+B\sim_{\mathbb{Q}}f^*M$ for a big $\mathbb{Q}$-divisor $M$ whose augmented base locus $\mathbf{B}_+(M)$ contains the image of no log canonical centre; in the absolute case (Corollary 1.2), whenever $K_X+B$ is big and $\mathbf{B}_+(K_X+B)$ contains no lc centre. The open question is whether that base-locus hypothesis can be weakened or removed, as the source itself asks. It is a recognized open problem: for log canonical pairs of log general type with no restriction on $\mathbf{B}_+(K_X+B)$ relative to lc centres, neither a good log minimal model nor finite generation is known (open already in dimension 5), and the natural weakening to log bigness would imply abundance-type statements, so it subsumes the abundance conjecture in special situations. As of September 2026 the core remains open: known cases cover klt pairs in all dimensions, lc pairs of dimension at most 4, dimension-5 lc pairs not of log general type, Kodaira dimension at most 2, and conditional results for log abundant pairs.

Background

The minimal model program (MMP, or log minimal model program, LMMP) is birational geometry's systematic attempt to classify projective varieties by repeatedly contracting or flipping curves on which the canonical class is negative. Its basic object is a pair $(X,B)$: a normal projective variety $X$ together with an effective $\mathbb{Q}$-divisor boundary $B$ such that $K_X+B$ is $\mathbb{Q}$-Cartier (some positive multiple is a Cartier divisor), where $K_X$ is the canonical divisor. Singularities of a pair are measured by discrepancies: a pair is Kawamata log terminal (klt) if every divisor $E$ over $X$ has discrepancy $a(E,X,B)>-1$ and $\lfloor B\rfloor=0$, and log canonical (lc) if $a(E,X,B)\geq -1$ for all $E$; an lc centre is the image on $X$ of a divisor of discrepancy exactly $-1$, and these loci are exactly where klt techniques break down. A pair is dlt (divisorial log terminal) if it is lc and the non-klt locus is a union of codimension-one lc centres away from which singularities improve to klt; dlt pairs are the inductive workhorses of the theory. A $\mathbb{Q}$-Cartier divisor is nef if it intersects every curve non-negatively, semi-ample if some multiple is basepoint-free and defines a morphism, and big if its plurisectional growth is maximal, $h^0(X,\lfloor mD\rfloor)\sim cm^{\dim X}$; the Kodaira dimension $\kappa(D)$ records the rate of growth of these spaces of sections, and a pair with $K_X+B$ big is said to be of log general type.

Running the LMMP on $K_X+B$ is expected to produce either a log minimal model — in the sense of Birkar–Hu (§2.3): a log birational model $(X',B_{X'})$, where $B_{X'}=B^{\sim}+E$ is the birational transform of $B$ plus the reduced exceptional divisor, which is a weak lc model (that is, $K_{X'}+B_{X'}$ is nef and $a(D,X,B)\leq a(D,X',B_{X'})$ for every prime divisor $D$ on $X$ exceptional over $X'$) and is moreover $\mathbb{Q}$-factorial dlt with this discrepancy inequality strict — or a Mori fibre space, a model fibred over a lower-dimensional base with $-(K_X+B)$ ample on the fibres. A good log minimal model is a log minimal model on which $K_{X'}+B_{X'}$ is semi-ample; semi-ampleness yields the log canonical model, the target of the pluricanonical maps, and implies in particular that the log canonical ring $R(K_X+B)=\bigoplus_{m\geq 0}H^0(X,\mathcal{O}_X(\lfloor m(K_X+B)\rfloor))$ is a finitely generated algebra. Conversely, finite generation of this ring, combined with the existence of a log minimal model, produces a good log minimal model. The assertion that nefness of the log canonical divisor should always upgrade to semi-ampleness is the abundance conjecture, open even for klt pairs in dimension at least 4.

The foundational theorem of the subject, due to Birkar–Cascini–Hacon–McKernan (BCHM, 2010), establishes finite generation of $R(K_X+B)$ — and, when $K_X+B$ is big, good minimal models — for klt pairs in all dimensions. The klt hypothesis is essential to the method, which exploits finite generation of adjoint rings available only away from the non-klt locus; the passage from klt to lc is the genuine gap. Fujino and Gongyo showed that the finite generation conjecture for log canonical rings of lc pairs reduces to plt pairs (purely log terminal: lc pairs with a single 'worst' boundary component of discrepancy $-1$) of log general type, and that finite generation for lc pairs in dimension $d$ is equivalent to the existence of good minimal models for $\mathbb{Q}$-factorial pseudo-effective dlt pairs of dimension at most $d-1$; this is why the lc problem subsumes abundance.

Birkar and Hu (2014) proved the strongest general result in the lc case. The augmented base locus of a $\mathbb{Q}$-divisor $D$ is $\mathbf{B}_+(D)=\bigcap_A \mathbf{B}(D-A)$, the intersection over ample $\mathbb{Q}$-divisors $A$ of the stable base loci $\mathbf{B}(D-A)$ (the loci of base divisors of all positive multiples); $\mathbf{B}_+(D)$ is the closed locus where $D$ fails to be ample even after an arbitrarily small ample perturbation, and it contains the stable base locus. Their Theorem 1.1 states: if $(X,B)$ is a projective lc pair with $B$ a $\mathbb{Q}$-divisor, $f\colon X\to Z$ a surjective morphism to a normal projective variety with $K_X+B\sim_{\mathbb{Q}}f^*M_Z$ for a big $\mathbb{Q}$-divisor $M_Z$, and $\mathbf{B}_+(M_Z)$ does not contain the image of any lc centre of $(X,B)$, then $(X,B)$ has a good log minimal model and $R(K_X+B)$ is finitely generated. Taking $f=\mathrm{id}$ gives Corollary 1.2: an lc pair of log general type with $\mathbf{B}_+(K_X+B)$ containing no lc centre has a good log minimal model. The hypothesis is what drives the proof: it lets one write $M_Z\sim_{\mathbb{Q}}A_Z+L_Z$ with $A_Z$ ample and $\mathrm{Supp}(A_Z+L_Z)$ avoiding the lc-centre images, pull back to a semi-ample divisor $G$ avoiding the lc centres together with a divisor $P$ supported exactly on $\lfloor B\rfloor$, and run an LMMP with scaling (a run of the program controlled by a decreasing ample coefficient) that terminates near $\lfloor B\rfloor$ by special termination — the theorem that the LMMP eventually stops modifying a neighbourhood of $\lfloor B\rfloor$ — and by induction on the lc centres, using Birkar's existence theorem for log canonical flips and a special LMMP.

The question left open by this theorem is what survives when the hypothesis is weakened or dropped. The natural weakening, log bigness, asks only that $K_X+B$ be big and that its pullback to the normalization of every lc centre be big. Birkar–Hu's Example 5.2 shows this is not accessible by current methods: for a blow-up $X$ of the vertex of a projective cone over a smooth variety $S$ with $\kappa(K_S)\geq 0$, with a suitable boundary $B$, the pair $(X,B)$ is log big with a unique lc centre, but any run of the LMMP must work over the cone, and in the expected situation its termination is essentially equivalent to the existence of a minimal model of $S$; even when $K_S$ is nef, semi-ampleness on the resulting model amounts to semi-ampleness of $K_S$ — the abundance problem for $S$. Hashizume and Hu later phrased this precisely: the minimal model theory for log big lc pairs implies the abundance theorem in a special situation.

The known cases delimit a clear frontier. Finite generation of log canonical rings holds for all lc pairs of dimension at most 4 (Fujino, via four-dimensional log minimal models and a four-dimensional log abundance input); for lc pairs of dimension 5 that are not of log general type (Hashizume, whose theorem also gives good minimal models when $\kappa(K_X+B)\leq 4$ or the pair is klt, assuming a good minimal model on the general fibre of the Iitaka fibration, i.e. the fibration defined by the pluricanonical linear series when $\kappa\geq 1$); and for lc pairs of arbitrary dimension with Kodaira dimension at most 2 (the $\kappa=2$ case due to Haidong Liu, building on Fujino–Liu for plt pairs). Hashizume–Hu establish conditional minimal model theory for log abundant lc pairs (log abundant: the log canonical divisor and its pullbacks to lc centres are abundant, i.e. have semi-ample positive parts on suitable models), assuming the klt theory in a fixed dimension and bounds on the numerical dimension (the intersection-theoretic measure of positivity of a nef divisor, never exceeding its Kodaira dimension) along lc centres; Hu proves that lc pairs whose boundary contains an ample divisor have a good log minimal model or a Mori fibre space. Beyond these, for lc pairs of log general type in dimension at least 5 with no restriction on $\mathbf{B}_+(K_X+B)$, nothing is known in either direction. In the neighbouring category of generalized pairs — pairs $(X,B,M)$ augmented by a nef divisor part $M$ traced from a higher birational model — the finite generation analogue is false, even for klt generalized pairs, so the conjecture is genuinely a statement about ordinary pairs.

Problem Statement

Let $(X,B)$ be a projective log canonical pair over an algebraically closed field of characteristic zero, with $B$ an effective $\mathbb{Q}$-divisor and $K_X+B$ $\mathbb{Q}$-Cartier and big, i.e. $(X,B)$ is of log general type. Make no restriction on the augmented base locus $\mathbf{B}_+(K_X+B)$ relative to the log canonical centres of $(X,B)$.

(1) Does every such pair $(X,B)$ have a good log minimal model — a log minimal model in the sense above (a $\mathbb{Q}$-factorial dlt weak lc model $(X',B')$ with $B'=B^{\sim}+E$ and strictly increasing discrepancies for divisors exceptional over $X'$) on which $K_{X'}+B'$ is semi-ample? Equivalently, in the range where log minimal models are known to exist (e.g. dimension at most 5, by Birkar's theorem on existence of log minimal models for effective lc pairs): is the log canonical ring $R(K_X+B)=\bigoplus_{m\geq 0}H^0(X,\mathcal{O}_X(\lfloor m(K_X+B)\rfloor))$ of every such pair a finitely generated algebra? A good log minimal model always yields finite generation; conversely, finite generation together with a log minimal model yields a good log minimal model.

(2) In particular, does the conclusion of Birkar–Hu's Corollary 1.2 remain true when the hypothesis '$\mathbf{B}_+(K_X+B)$ contains no lc centre' is weakened to '$(X,B)$ is log big', that is, $K_X+B$ is big and the pullback of $K_X+B$ to the normalization $S^{\nu}$ of every lc centre $S$ of $(X,B)$ is big (the formal definition in Birkar–Hu, §2.2)? By Birkar–Hu's Example 5.2, an affirmative answer implies abundance-type statements (semi-ampleness of $K_S$ for the smooth varieties $S$ with $\kappa(K_S)\geq 0$ arising in their cone construction), so this subsumes the abundance conjecture in special situations.

For a pair as in (1), the existence of a good log minimal model is equivalent, in the range where log minimal models are known to exist, to finite generation of the log canonical ring $R(K_X+B)$. No counterexample is currently known for ordinary pairs.

References

1. Caucher Birkar and Zhengyu Hu, Log canonical pairs with good augmented base loci, Compositio Mathematica 150 (2014), no. 4, 579-592. DOI: 10.1112/S0010437X13007628. https://doi.org/10.1112/S0010437X13007628 (arXiv:1305.3569)
2. Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan, Existence of minimal models for varieties of log general type, Journal of the American Mathematical Society 23 (2010), no. 2, 405-468. DOI: 10.1090/S0894-0347-09-00649-3. https://doi.org/10.1090/S0894-0347-09-00649-3 (arXiv:math/0610203)
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13. Jihao Liu and Yanze Wang, A klt generalized pair with infinitely generated canonical ring, preprint (2026). arXiv:2608.03258. https://arxiv.org/abs/2608.03258