Extension of Birkar's equivalence between good log minimal models and finite generation of log canonical algebras and modules to arbitrary (non-affine) base varieties

Caucher Birkar's 2013 note 'Log canonical algebras and modules' proves that for a log canonical pair $(X/Z,B)$, with $X\to Z$ projective over an affine base $Z=\operatorname{Spec}A$ and $K_X+B$ pseudo-effective over $Z$, the pair has a good log minimal model if and only if its log canonical algebra and the associated log canonical modules are finitely generated. The author notes that the affine-base assumption was made 'for simplicity of notation' and that the general case 'can be formulated and proved in a similar way', but the note contains neither that formulation nor its proof. This problem asks for the missing extension: a correct formulation of the relative finite-generation hypotheses over an arbitrary base variety and a complete proof of the resulting equivalence with the existence of a good log minimal model.

Background

The minimal model program (MMP, or LMMP) is the organizing framework of the birational geometry of complex algebraic varieties: it seeks to replace a given variety or pair, up to birational equivalence, by one on which the canonical class has controlled positivity. A pair $(X,B)$ consists of a normal variety $X$ together with a $\mathbb{Q}$-divisor $B$, the boundary, with coefficients in $[0,1]$; the pair is log canonical (lc) — the weakest of the standard singularity classes — or kawamata log terminal (klt) — a stronger condition — according to the sign of the discrepancies, which measure the singularities via a log resolution, that is, a proper birational morphism from a smooth variety on which the pullback of the boundary has simple normal-crossing support. For a projective morphism $f\colon X\to Z$, a $\mathbb{Q}$-Cartier divisor $D$ on $X$ (one whose positive multiple is Cartier, that is, locally principal) is called nef over $Z$, pseudo-effective over $Z$, or semi-ample over $Z$ when the corresponding property holds relatively over the base; semi-ample over $Z$ means that some positive multiple of $D$ is generated by its sections relative to $f$ and therefore defines a contraction over $Z$. A log minimal model of an lc pair $(X/Z,B)$ is a birational model $(Y/Z,B_Y)$ on which $K_Y+B_Y$ is nef over $Z$; it is a good log minimal model if $K_Y+B_Y$ is semi-ample over $Z$. The abundance conjecture predicts that minimal models of lc pairs are good, so 'good minimal model' is the standard shorthand for 'minimal model plus abundance'. For a Cartier divisor $L$ on $X$, the section ring $R(L)=\bigoplus_{m\ge0}H^0(X,\mathcal{O}_X(mL))$ is a graded algebra, and when $Z=\operatorname{Spec}A$ is affine it is a graded $A$-algebra. For a divisor $G$ on $X$ that is very ample over $Z$ (that is, $G$ relatively embeds $X$ into a projective space over $Z$) and an integer $p$, one has the graded $R(L)$-module $M^p_G(L)=\bigoplus_{m\ge p}H^0(X,\mathcal{O}_X(mL+G))$. When $L=I(K_X+B)$ for an lc pair $(X,B)$ and an integer $I>0$, $R(L)$ is called the log canonical algebra and the modules $M^p_G(L)$ are the log canonical modules. Finite generation of such rings is a central theme of the subject: Birkar–Cascini–Hacon–McKernan proved finite generation of log canonical rings for klt pairs of log general type (that is, of maximal logarithmic Kodaira dimension), and Demailly–Hacon–Păun — to whom Birkar credits the klt case of the theorem below — used extension theorems to reach good minimal models for klt pairs. In the short note 'Log canonical algebras and modules', Birkar proved the following equivalence (Theorem 1.1). Let $(X/Z,B)$ be an lc pair with $Z=\operatorname{Spec}A$ affine and $K_X+B$ pseudo-effective over $Z$, and let $I>0$ be an integer such that $L:=I(K_X+B)$ is Cartier. Then $(X/Z,B)$ has a good log minimal model if and only if $R(L)$ is a finitely generated $A$-algebra and, for any very ample$/Z$ divisor $G$ and any integer $p$, the module $M^p_G(L)$ is finitely generated over $R(L)$. The proof combines a truncation principle for graded algebras and modules, a stable moving/fixed-part decomposition for multiples of $L$ on a resolution, Nakayama's numerical Kodaira dimension $\kappa_\sigma$ (a numerical analogue of the Kodaira dimension — the asymptotic growth rate of the pluricanonical section spaces $H^0(X,mK_X)$ — defined through the asymptotics of sections after adding fixed ample divisors) together with a base-locus theorem of Nakayama from his monograph on Zariski decompositions and abundance, and a run of the relative minimal model program with scaling (a standard variant of the program in which the coefficient of an auxiliary ample boundary is scaled down as the program runs, so as to control the extremal rays encountered). That argument uses the affineness of $Z$ at several points — the identification of the generic-fibre section ring with $R(L)\otimes_A K$ by base change, the Noetherianity of the section ring, the globalization of the stable moving/fixed-part decomposition, and the fact that the existence of a good log minimal model over $Z$ is not manifestly local on $Z$ — and it relies on the auxiliary statements of the note, namely Theorem 2.2, Lemmas 3.1–3.3, and the Nakayama base-locus theorem restated there as Theorem 3.4. Immediately after the theorem Birkar writes: 'Note that we have assumed $Z$ to be affine for simplicity of notation; the general case can be formulated and proved in a similar way.' The note presents no other open problems, and the promised general-base formulation and proof do not appear in it.

Problem Statement

Let $f\colon X\to Z$ be a projective morphism of varieties over $\mathbb{C}$, let $(X/Z,B)$ be an lc pair such that $K_X+B$ is pseudo-effective over $Z$, and let $I>0$ be an integer such that $L:=I(K_X+B)$ is Cartier. Birkar's Theorem 1.1 ('Log canonical algebras and modules', Journal of the Mathematical Society of Japan 65 (2013), 1319–1328) establishes, under the assumption that $Z=\operatorname{Spec}A$ is affine, the equivalence between (i) the existence of a good log minimal model of $(X/Z,B)$ and (ii) finite generation of the log canonical algebra $R(L)$ as an $A$-algebra together with finite generation of the modules $M^p_G(L)$ over $R(L)$ for every very ample$/Z$ divisor $G$ and every integer $p$. The source states that $Z$ was assumed affine 'for simplicity of notation' and that 'the general case can be formulated and proved in a similar way', but presents neither that formulation nor its proof. The problem is to carry out that extension in full. (1) Formulation: identify the correct general-base analogue of the finite-generation hypotheses — over a non-affine base the algebra of global sections over the ground field is not the adequate object, so one must fix the right relative hypothesis together with the matching relative module conditions for very ample$/Z$ divisors — chosen so that, when $Z$ is affine, the statement specializes exactly to Birkar's Theorem 1.1. (2) Proof: prove the resulting equivalence in both directions for an arbitrary base variety $Z$ in the source's setting (varieties over $\mathbb{C}$, projective morphism, lc pair, $K_X+B$ pseudo-effective over $Z$, $L=I(K_X+B)$ Cartier). The single scientific objective is the general-base equivalence; the auxiliary statements of the source note are relevant only insofar as a complete, expert-checkable proof of that equivalence requires them. Greater generality (arbitrary ground fields, schemes, or complex spaces) is welcome but not required, and the problem is not to be weakened, for example by restricting to projective or quasi-affine bases only.

References

1. Caucher Birkar, 'Log canonical algebras and modules', Journal of the Mathematical Society of Japan 65 (2013), no. 4, 1319–1328. DOI: 10.2969/jmsj/06541319, https://doi.org/10.2969/jmsj/06541319.
2. Caucher Birkar, 'Divisorial algebras and modules on schemes', preprint arXiv:1105.0441 (2011), https://arxiv.org/abs/1105.0441 (preprint precursor of the published note).
3. Caucher Birkar, Paolo Cascini, Christopher Hacon, James McKernan, 'Existence of minimal models for varieties of log general type', Journal of the American Mathematical Society 23 (2010), 405–468. DOI: 10.1090/S0894-0347-09-00649-3, https://doi.org/10.1090/S0894-0347-09-00649-3.
4. Jean-Pierre Demailly, Christopher D. Hacon, Mihai Păun, 'Extension theorems, non-vanishing and the existence of good minimal models', Acta Mathematica 210 (2013), 203–259. DOI: 10.1007/s11511-013-0094-x, https://doi.org/10.1007/s11511-013-0094-x (preprint arXiv:1012.0493).
5. Osamu Fujino, Yoshinori Gongyo, 'Log pluricanonical representations and the abundance conjecture', Compositio Mathematica 150 (2014), 593–620. DOI: 10.1112/S0010437X13007495, https://doi.org/10.1112/S0010437X13007495 (preprint arXiv:1104.0361).
6. Osamu Fujino, 'Some remarks on the minimal model program for log canonical pairs', preprint arXiv:1309.3015 (2013, final version 2014; final version to appear in the Kodaira centennial issue of the Journal of Mathematical Sciences, the University of Tokyo), https://arxiv.org/abs/1309.3015.
7. Caucher Birkar, Zhengyu Hu, 'Log canonical pairs with good augmented base loci', Compositio Mathematica 150 (2014), 579–592. DOI: 10.1112/S0010437X13007628, https://doi.org/10.1112/S0010437X13007628 (preprint arXiv:1305.3569).
8. Osamu Fujino, Jinsong Xu, 'Remarks on log pluricanonical representations', preprint arXiv:2501.16568 (2025), https://arxiv.org/abs/2501.16568.
9. Nikolaos Tsakanikas, Lingyao Xie, 'Remarks on the existence of minimal models of log canonical generalized pairs', preprint arXiv:2301.09186 (2023, revised 2024), https://arxiv.org/abs/2301.09186.
10. Noboru Nakayama, 'Zariski-decomposition and Abundance', MSJ Memoirs 14, Mathematical Society of Japan, Tokyo, 2004. DOI: 10.2969/msjmemoirs/014010000, https://doi.org/10.2969/msjmemoirs/014010000 (available at https://projecteuclid.org/ebooks/mathematical-society-of-japan-memoirs/Zariski-decomposition-and-Abundance/toc/10.2969/msjmemoirs/014010000; cited by the source note for $\kappa_\sigma$ and the base-locus theorem restated there as Theorem 3.4).
11. Caucher Birkar, 'Generalised pairs in birational geometry', preprint arXiv:2008.01008 (2020), https://arxiv.org/abs/2008.01008.
12. Caucher Birkar, 'Birational geometry of algebraic varieties', preprint arXiv:1801.00013 (2017), https://arxiv.org/abs/1801.00013.