Finite generation of log canonical algebras and modules for log canonical pairs beyond the known minimal model program cases, and the induced existence of log minimal models with semi-ample canonical class

For a projective log canonical pair $(X,B)$ with $L=I(K_X+B)$ Cartier, the log canonical algebra $R(L)=\bigoplus_{m\ge0}H^0(X,mL)$ and the log canonical modules $M_G^0(L)=\bigoplus_{m\ge0}H^0(X,G+mL)$ (for very ample $G$ ) control the birational classification of $X$ . Birkar's preprint "Divisorial algebras and modules on schemes" (arXiv:1105.0441, 2011) proved that, when $K_X+B$ is pseudo-effective, finite generation of $R(L)$ together with all the modules $M_G^0(L)$ is equivalent to the existence of a log minimal model on which the log canonical divisor is semi-ample, and gave inductive criteria reducing such finite generation to restrictions of algebras and modules to components of $\operatorname{Supp}L$ . The problem, which remains open, is to establish these finite generation properties for log canonical pairs with $K_X+B$ pseudo-effective beyond the cases in which they are already known — the algebra $R(L)$ for klt pairs in all dimensions, for log canonical pairs in dimension at most four, in dimension five outside log general type and in Kodaira dimension at most two, and the algebra $R(L)$ together with the modules $M_G^0(L)$ for pairs whose augmented base locus contains no log canonical centre — in particular for log canonical or purely log terminal pairs of log general type in dimension at least five. By the equivalence theorem, any such advance simultaneously produces log minimal models with semi-ample canonical class for the corresponding class of pairs.

Background

In birational geometry one studies a variety $X$ together with a boundary, meaning an effective $\mathbb{Q}$ -divisor $B$ with coefficients in $[0,1]$ ; the pair $(X,B)$ is called log canonical (lc) if its singularities, measured by discrepancies along birational valuations, are no worse than a prescribed threshold, and Kawamata log terminal (klt) or purely log terminal (plt) if the discrepancies satisfy stronger positivity. The central machine is the minimal model program (MMP, also called the LMMP for pairs): a sequence of birational surgeries (divisorial contractions and flips) intended to replace $X$ by a log minimal model, a pair $(Y,B_Y)$ on which the log canonical divisor $K_Y+B_Y$ is nef, and ideally a good log minimal model on which $K_Y+B_Y$ is semi-ample, meaning some positive multiple defines a morphism. The abundance conjecture predicts semi-ampleness of nef log canonical divisors; the minimal model conjecture predicts the existence of log minimal models whenever $K_X+B$ is pseudo-effective, that is, numerically a limit of effective $\mathbb{Q}$ -divisors. Both are open for lc pairs in dimension at least five. A basic invariant is the section ring $R(L)=\bigoplus_{m\ge0}H^0(X,\mathcal{O}_X(mL))$ of a Cartier divisor $L$ (a divisorial algebra in the language of the source paper); when $L=I(K_X+B)$ for an lc pair this is the log canonical ring, and its finite generation as an algebra is equivalent to the existence of the log canonical model. The growth of the spaces $H^0(X,mL)$ is measured by the Kodaira dimension $\kappa(L)$ , the integer $d$ such that $h^0(X,mL)$ grows like $m^d$ (with $\kappa(L)=-\infty$ if all these spaces vanish); the pair $(X,B)$ is of log general type when $\kappa(K_X+B)=\dim X$ , the maximal case. For klt pairs the finite generation question was settled by Birkar–Cascini–Hacon–McKernan (BCHM, 2010) in the log general type case, but for lc pairs the corresponding statement — the finite generation conjecture for log canonical rings of lc pairs — is a well-known open problem (stated as Conjecture 1.1 in Fujino–Liu 2020), closely tied to abundance: Fujino–Gongyo (2013/2017) showed it is essentially equivalent to the existence of good minimal models in lower dimension, and reduced it to plt pairs of log general type. The source paper (Caucher Birkar, arXiv:1105.0441, 2011) introduces, for a projective scheme $X$ over a Noetherian ring $A$ and a Cartier divisor $L$ , the divisorial module $M^p_{\mathcal{F}}(L)=\bigoplus_{m\ge p}H^0(X,\mathcal{F}\otimes\mathcal{O}_X(mL))$ for a coherent sheaf $\mathcal{F}$ ; when $\mathcal{F}=\mathcal{O}_X(G)$ for a very ample divisor $G$ one writes $M_G^0(L)$ . Its Section 3 proves inductive criteria (Theorems 3.3–3.5): writing $L=\sum l_iL_i$ with $l_i>0$ and the $L_i$ nonzero effective Cartier divisors, and assuming $H^0(X,-L_1)=0$ , finite generation of $R(L)$ together with that of the modules $M^p_{\mathcal{F}}(L)$ for every reflexive coherent sheaf $\mathcal{F}$ and every $p$ is equivalent to finite generation of the restricted algebras $R(L)|_S$ and modules $M^0_{\mathcal{O}_X(l)}(L)|_S$ (images of the restriction maps to sections on $S$ ) for each component $S=L_j$ and any $l\gg0$ (Theorem 3.5; Theorem 3.3 gives the module conclusion alone from finitely generated restricted modules, and Theorem 3.4 the equivalence for the algebra). The restrictions $R(L)|_S$ are quotients of $R(L)$ , not intrinsic algebras of divisors on $S$ , and Remark 3.9 of the paper (stated for a projective log smooth dlt pair over $\mathbb{C}$ with $B$ rational) identifies the key difficulty: relating $R(L)|_S$ to the intrinsic algebra $R(L|_S)$ is hard for arbitrary components $S$ of $\operatorname{Supp}L$ , and is only expected to be tractable when $S$ is a component of $\lfloor B\rfloor$ (an lc centre); the ideal situation, in which every component of $L$ is a component of $\lfloor B\rfloor$ , corresponds to the vanishing of the invariant $\theta(X,B,L)$ defined in Birkar's weak Zariski decomposition paper, and for $\theta(X,B,L)\neq0$ the remark suggests enlarging $B$ to a boundary $\overline{B}\geq B$ with $\theta(X,\overline{B},\overline{L})=0$ , proving finite generation for the enlarged algebra, and chasing it back to $R(L)$ . Its Section 4 proves the equivalence theorem (Theorems 4.1 and 4.5, stated as Theorem 1.3): if $(X/Z,B)$ is lc, $Z=\operatorname{Spec}A$ , and $L=I(K_X+B)$ is Cartier and pseudo-effective over $Z$ , then $(X/Z,B)$ has a log minimal model on which $K_Y+B_Y$ is semi-ample over $Z$ if and only if $R(L)$ is a finitely generated $A$ -algebra and $M_G^0(L)$ is a finitely generated $R(L)$ -module for any very ample over $Z$ divisor $G$ . The paper's introduction suggests that combining the restriction criteria with available extension theorems (it cites Hacon–McKernan) should give a new proof of finite generation of log canonical rings for klt pairs without the MMP, and expresses the hope that the criteria prove it for lc pairs; the numerical abundance statement $\kappa(L)=\kappa_\sigma(L)$ (equality of the Kodaira dimension with the numerical Kodaira dimension, a birational invariant defined via the growth of $H^0(X,mL+G)$ over all ample $G$ ) follows from module finite generation in Theorem 4.2. Finite generation of log canonical rings is known: for klt pairs in all dimensions (Fujino–Mori reduction plus BCHM); for lc pairs in dimension at most four (Fujino 2010, via the four-dimensional LMMP); trivially when the Kodaira dimension is at most one; for lc pairs of dimension five that are not of log general type (Hashizume 2018, via relatively trivial pairs and lower-dimensional LMMP); and for lc pairs of Kodaira dimension two (Fujino–Liu for plt pairs and Liu for lc pairs, 2018–2020, via lc-trivial fibrations and Hodge theory rather than the MMP). On the minimal model side, log flips for lc pairs exist (Birkar 2011/2012, made unconditional using Hacon–Xu's existence of log canonical closures), and lc pairs whose augmented base locus — the locus where a divisor fails to remain ample after subtracting a small ample class — avoids all lc centres have good log minimal models (Birkar–Hu 2014); but termination of log flips, and the existence of log minimal models for general lc pairs of log general type in dimension at least five, remain open, so by the equivalence theorem the corresponding finite generation statements remain open as well.

Problem Statement

Let $(X,B)$ be a projective log canonical pair over $\mathbb{C}$ with $B$ a $\mathbb{Q}$ -divisor, let $I>0$ be an integer such that $L=I(K_X+B)$ is Cartier, and for a very ample divisor $G$ let $M_G^0(L)=\bigoplus_{m\ge0}H^0(X,\mathcal{O}_X(G+mL))$ be the associated log canonical module over the log canonical algebra $R(L)=\bigoplus_{m\ge0}H^0(X,\mathcal{O}_X(mL))$ . The problem, in the generality of the source paper, is twofold and the two halves are linked by the equivalence theorem (Theorems 4.1 and 4.5 of arXiv:1105.0441):

(1) Prove that $R(L)$ is a finitely generated $\mathbb{C}$ -algebra, and that $M_G^0(L)$ is a finitely generated $R(L)$ -module for every very ample divisor $G$ , for log canonical pairs with $K_X+B$ pseudo-effective that are not among the classes in which both properties are already known. Finite generation of the algebra $R(L)$ alone is known for klt pairs in all dimensions, for log canonical pairs of dimension at most four, for log canonical pairs of dimension five that are not of log general type, and for log canonical pairs of Kodaira dimension at most two. By the equivalence theorem, finite generation of the algebra $R(L)$ together with all the modules $M_G^0(L)$ is equivalent to the existence of a log minimal model with semi-ample log canonical divisor, and is known precisely for the classes in which such a model is known — for instance for log canonical pairs whose augmented base locus contains no log canonical centre (Birkar–Hu 2014). It suffices to establish both properties for any single class of log canonical pairs with $K_X+B$ pseudo-effective in which they are not already known, or in full generality. (When $K_X+B$ is not pseudo-effective, no positive multiple of $K_X+B$ is effective, so $R(L)=\mathbb{C}$ and the finite generation properties hold trivially, while a log minimal model with semi-ample log canonical divisor need not exist; such classes are therefore excluded from (1), and the equivalence theorem, whose hypotheses include pseudo-effectivity of $L=I(K_X+B)$ , does not apply to them. The substantive case is therefore $K_X+B$ pseudo-effective.) The central unsettled cases are log canonical (and already purely log terminal) pairs of log general type in dimension at least five, and log canonical pairs of Kodaira dimension at least three in dimension at least six. By the equivalence theorem this is the same problem as (2).

(2) Produce, for the same classes of log canonical pairs with $K_X+B$ pseudo-effective, a log minimal model $(Y,B_Y)$ on which $K_Y+B_Y$ is semi-ample; the equivalence theorem states that such a model exists exactly when the finite generation properties of (1) hold, so any new verification of (1) in a class where good log minimal models are unknown is automatically a new existence result of type (2), and conversely.

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