On finiteness of relative log pluricanonical representations

Osamu Fujino

REVIEW 3 major objections 3 minor 12 references

On finiteness of relative log pluricanonical representations

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that relative log pluricanonical representations are finite for projective morphisms of complex analytic spaces, and uses this finiteness to establish the abundance theorem for semi-log canonical pairs, log canonical…

desk verdict Fujino delivers a mostly solid analytic MMP extension with two sketched lemmas and heavy reliance on an unpublished preprint; deserves refereeing. read the letter →

arxiv 2506.00760 v2 pith:6RCZLURO submitted 2025-06-01 math.AG

Osamu Fujino This is my paper

classification math.AG MSC 14E3014E0732C15

keywords logpluricanonicalrepresentationscomplexanalyticspacessemi-logcanonicalpairsabundanceconjectureminimalmodelprogramflipsdltblow-upsrelativebundle

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that, for a projective morphism between complex analytic spaces, the group of B-bimeromorphic self-maps of a log canonical pair acts on the relative pluricanonical bundle through a finite group, provided the canonical divisor is relatively semiample. From this finiteness it derives the relative abundance theorem for semi-log canonical pairs—the possibly non-normal analogue of log canonical pairs glued from normal pieces along divisors: if the pullback of the canonical divisor to the normalization is relatively semiample over a neighborhood of a compact set, then some fixed multiple becomes relatively generated over a possibly smaller neighborhood. The paper also establishes freeness for nef and log abundant log canonical bundles, existence of log canonical flips in the analytic category, and a reduction of the abundance conjecture for projective morphisms of complex analytic spaces to the classical abundance conjecture for projective log canonical pairs. This matters because abundance is the step that turns minimal models into canonically polarized families, and the paper extends its known scope from algebraic varieties to complex analytic base spaces.

What carries the argument

The central object is the relative log pluricanonical representation rho_m: Bim(X/Y,$\Delta$) -> Aut_{O_Y}(pi_*O_X(m(K_X+$\Delta$))), which records how B-bimeromorphic self-maps of the pair act on sections of multiples of the canonical divisor. The proof that its image is finite runs through Theorem 3.2: over a polydisc, after a dlt blow-up and resolution, the evaluation of rho_m at a fiber is a representation of a projective divisorial log terminal pair, where the algebraic finiteness theorem of [FG] bounds element orders; uniformity across the base is supplied by base change to general hyperplane slices, and the bounded-order criterion for subgroups of GL(n,C) then forces the whole image to be finite. The other load-bearing mechanism is the admissible-section formalism: sections that are invariant, in a precise sense, under the finite group of B-bimeromorphic maps and restrict correctly to log canonical strata, and whose existence over the boundary strata is propagated up to the whole space by Lemma 4.10, giving pi-generation of some multiple of the canonical divisor.

What would settle it

Take a flat family of nodal curves over a disc with smooth total space, $\Delta$ equal to zero, and K_X relatively semiample after normalization, for example a family of elliptic curves degenerating to a nodal rational curve. Compute the base locus of O_X(mK_X) for all positive integers m over a neighborhood of the central fiber. The theorem predicts that some single multiple is pi-generated; finding a basepoint common to all m would refute Theorem 1.1, while verifying generation for one explicit m would corroborate it.

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Extended reading notes

Core claim

The central theorem (Theorem 1.1) is an analytic analogue of the algebraic abundance theorem for semi-log canonical pairs, proved without the gluing theory in [K]. The engine behind it is Theorem 1.2: for a log canonical pair (X,$\Delta$) over a complex variety Y, with K_X+$\Delta$ pi-semiample, the natural representation of the B-bimeromorphic automorphism group on the finite rank module pi_*O_X(m(K_X+$\Delta$)) has finite image. The finiteness is proven fiberwise: after base change to a point, the representation factors through a projective log canonical pair on the fiber, where a prior algebraic finiteness theorem bounds the order of every element; a classical group-theoretic criterion then forces the whole image to be finite. The paper then uses admissible sections, obtained by averaging over this finite group, to piece together sections of multiples of the canonical divisor on the normalization and descend them to the semi-log canonical space. Along the way it proves the existence of log canonical flips and good dlt blow-ups in the complex analytic setting, and shows that the relative abundance conjecture for analytic morphisms is no harder than the classical abundance conjecture for projective varieties.

Load-bearing premise

Everything rests on the minimal model program for projective morphisms between complex analytic spaces as developed in the cited preprints—especially dlt blow-ups, termination of flips with scaling, and existence of log canonical models; if any of those foundational results has a gap, the main theorems would weaken or collapse.

Editorial extensions

If this is right

If Theorem 1.1 is correct, the algebraic theorem of [HX] on semi-log canonical abundance follows without the gluing theory in [K], since Lemma 4.11 passes from analytic generation to Zariski generation.
Theorem 1.4 makes every pi-nef and pi-log abundant log canonical bundle relatively free over a neighborhood, so minimal models of analytic families become morphisms to a base with canonically polarized fibers.
Theorem 1.10 and Corollary 1.11 reduce the abundance conjecture for projective morphisms between complex analytic spaces to the classical conjecture for projective log canonical pairs; proving the projective case in dimension n would settle relative abundance in dimension n for analytic families.
Theorem 1.7 gives log canonical flips in the analytic category, so the birational part of the minimal model program for log canonical pairs over analytic bases is now available.
Theorem 1.8 and Theorem 6.2 give good dlt blow-ups and the ACC for log canonical thresholds for complex analytic singularities, extending singularity-theoretic tools from algebraic to analytic settings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The finite-group averaging that replaces the gluing theory for semi-log canonical spaces suggests the same mechanism may produce canonical bundle formulas or adjunction statements for other quotients or pinched analytic spaces where a normalization splits the pair into several birationally identified components.
Because Theorem 1.10 reduces relative abundance to the classical projective conjecture, any proof of that projective conjecture—or any counterexample to it—would transfer automatically to complex analytic families, collapsing a separate analytic version of the conjecture into the classical problem.
The paper's reliance on the strict support condition in vanishing theorems indicates that the genuinely analytic content is concentrated in Lemma 4.2; if a simpler proof of that lemma were found, the whole abundance theorem would follow from the algebraic finiteness theorem plus formal minimal model program arguments.
Theorem 1.5's hypothesis that W is a Stein compact set with noetherian ring of global functions suggests that the cleanest analytic formulation of abundance goes through Stein neighborhoods; testing whether the nefness-spreading statement of Conjecture 5.2 holds for three-dimensional analytic families would make Theorem 1.5 unconditional over arbitrary compact sets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: Fujino has pushed the finiteness of relative log pluricanonical representations and the abundance theorem for semi-log canonical pairs into the complex analytic setting. If the supporting results hold up, this is a real step: it reduces analytic abundance to the classical projective abundance conjecture, and it gives log canonical flips analytically. The writing is careful and honest.

What's new: Theorem 1.1 is the abundance statement for slc pairs, Theorem 1.2 the finiteness of relative B-pluricanonical representations, Theorem 1.7 log canonical flips. None of these appear in the literature for analytic spaces. The paper is also self-aware: it spells out which preprints depend on it and avoids citing them, and Section 7 corrects issues in [Fuj1] and [FG]. That is the right kind of citizenship.

The soft spots are real but proportionate. Lemma 2.29 is load-bearing—it is used in Theorem 3.2 and Lemma 4.7—yet the proof is only a sketch, and Lemma 4.7 omits details entirely. In a paper whose main theorems rest on delicate gluing of sections, those are not cosmetic gaps. A referee should ask for full arguments or exact references. There is also a minor heading typo ("Proof of Theorem 4.5" for Lemma 4.5).

The bigger external issue is that Proposition 4.6 uses [EH2, Theorem 1.2] to run and terminate an MMP with scaling around a compact subset; Theorem 1.7 and Theorem 1.8 depend on [EH2, Theorem 1.3]. [EH2] is a recent, not-yet-refereed preprint. If it has a gap, the analytic results here weaken substantially. That is not circularity—Fujino is transparent that [EH2] is independent—but it does mean this paper is part of a preprint web. A careful referee should cross-check the cited statements in [EH2] rather than taking them at face value.

Overall, I do not see a fatal flaw. The main theorems are plausible and the internal arguments mostly detailed. The central reasoning is coherent, and the reduction of analytic abundance to the algebraic conjecture is a genuine contribution. The right response is a serious peer review with a request to complete the sketched lemmas. It is not a paper to desk reject.

Referee Report

3 major / 3 minor

Summary. The paper proves finiteness of relative log pluricanonical representations in the complex analytic setting (Theorem 1.2 and Corollary 1.3), and uses these results to prove the abundance theorem for semi-log canonical pairs under a semiampleness assumption on the normalization (Theorem 1.1), freeness for nef and log abundant log canonical bundles (Theorem 1.4), and the existence of log canonical flips for complex analytic spaces (Theorem 1.7). The proof follows the strategy of [Fuj1] and [FG], replacing algebraic MMP inputs by the complex analytic MMP developed in [Fuj12], [EH1], and [EH2], and using an admissible-sections gluing argument to descend from a dlt blow-up to the semi-log canonical pair.

Significance. If the foundational MMP inputs are valid, the main results are substantial: Theorem 1.1 extends [HX, Theorem 2] to projective morphisms of complex analytic spaces without Kollár's gluing theory, and Theorem 1.7 provides log canonical flips in that setting. The paper also gives a clean reduction of the analytic abundance problem to the classical abundance conjecture for projective varieties (Theorem 1.10 and Corollary 1.11). The author is careful about circularity: Remark 1.13 explicitly refrains from citing works that depend on the present paper. The main weakness is that several load-bearing technical steps are only sketched or omitted, and the central argument relies on unpublished preprints.

major comments (3)

[§2, Lemma 2.29] Lemma 2.29 is stated with only a 'Sketch of Proof', yet it is used in the proof of Theorem 3.2 and again, through Lemma 4.7, in the gluing argument for Theorem 1.1. In particular, part (ii) is reduced to 'a direct check' after blowing up Z along the center S, but that check is not written out. Since Theorem 3.2 is the engine behind Corollary 1.3 and hence behind the finiteness of relative pluricanonical representations, the proof of Theorem 1.2 is not complete as written.
[§4, Lemma 4.7] Lemma 4.7 is a key step in producing admissible sections: it asserts that averaging over the finite group G preserves preadmissibility and that the product over G restricts to the |G|-th power on the boundary. The proof is omitted entirely ('we omit the details here'), and the assertion is not immediate because the behavior of arbitrary B-bimeromorphic maps on strata requires Lemma 2.29. Since Lemma 4.7 feeds directly into Lemmas 4.8 and 4.10 and hence into Theorem 1.1, this is a load-bearing gap in the written argument.
[§4, Proposition 4.6 and Proposition 4.9] Step 1 of Proposition 4.6 invokes [EH2, Theorem 1.2] to run and terminate a (K_X+Delta-epsilon floor Delta)-MMP with ample scaling over Z around W_z, and Step 4 of Proposition 4.9 repeats this dependence through Proposition 4.6; Theorem 1.7 likewise relies on [EH2, Theorem 1.2]. Since [EH2] is an unpublished preprint, the correctness of Theorem 1.1 is contingent on the full validity of that MMP in the required generality of projective morphisms between complex analytic spaces. The paper would be substantially strengthened by a precise statement of the hypotheses and termination/finiteness properties assumed from [EH2], together with a check that every invocation in this paper falls within those hypotheses.

minor comments (3)

[§4, proof of Corollary 1.3] The proof refers to 'Theorem 1.3', but no Theorem 1.3 exists in the paper; the intended reference is presumably Theorem 1.2.
[§4, proof of Lemma 4.5] The proof heading says 'Proof of Theorem 4.5' and should say 'Proof of Lemma 4.5'.
[§4, Proposition 4.6(i)] There is a typo in 'B-bimermorphic involution'; it should be 'B-bimeromorphic involution'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation reduces to independent algebraic finiteness ([FG]) and the external analytic MMP ([EH2]/[Fuj12]), not to its own target.

full rationale

The paper's central derivation chain is not circular. Theorem 1.1 is proved by reducing to the normalization and a dlt blow-up ([Fuj12, Theorems 1.21 and 1.27]), then to Lemma 4.10, which is established by induction on dimension: Lemma 4.8 and Proposition 4.9 lift generation of admissible sections from the boundary S = floor(Delta) to X, and the base case is kawamata log terminal, where every section is preadmissible (Remark 2.28). None of these steps assumes Theorem 1.1. Theorem 1.2 uses the algebraic finiteness theorem [FG, Theorem 1.1], and Theorem 3.2 uses Burnside's theorem plus the strict support condition from [Fuj13]; both are independent prior results that do not contain Theorem 1.1 as an input. The load-bearing analytic MMP is imported from [EH2] and [Fuj12]; the paper explicitly states that '[EH1] and [EH2] do not rely on the results of the present paper' and, conversely, excludes [EH3] and [H5] because they depend on this paper. The internal Lemma 2.29 is only sketched and Lemma 4.7 omits its proof, but these are completeness or correctness risks, not circularity: neither is equivalent to the target theorem by construction. Section 7 openly corrects issues in the author's prior [Fuj1] and [FG], citing [Bir] for the missing ingredient, and explicitly remarks 'There is no circular reasoning even if one uses [Bir, Theorem 5.2] in the context of [FG].' The only notable textual slip is 'by Theorem 1.3' in the proof of Corollary 1.3, which is a typographical reference error (the argument cites Theorem 1.2) and does not create a loop. In sum, the paper reduces its analytic claims to the classical projective abundance conjecture and to independently established MMP ingredients; it does not rename its inputs as predictions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper does not introduce free parameters or invented entities. It relies on a body of prior literature, primarily the author's own MMP for analytic spaces and the algebraic finiteness theorem of Fujino-Gongyo, as axioms of the field.

assumptions (5)

domain assumption The minimal model program for projective morphisms between complex analytic spaces, including dlt blow-ups, flips, contractions, and termination with scaling, as established in [Fuj12], [EH1], and [EH2].
Used throughout the paper, e.g., in the proofs of Theorem 1.2, Proposition 4.6, Theorem 1.8. The paper explicitly states these results are used without proof.
domain assumption Relative Kawamata-Viehweg vanishing theorem and the strict support condition for projective morphisms between complex analytic spaces, proved in [Fuj13] (see also [Fuj17] and [FF]).
Invoked in Lemma 2.29, the proof of Lemma 4.2, and the proof of Theorem 3.2. These are nontrivial analytic analogues of algebraic vanishing theorems.
standard math The finiteness of log pluricanonical representations for projective log canonical pairs (Theorem 2.23), proved in [FG].
This is the algebraic counterpart of Theorem 1.2 and is used as a key input in the proof of Theorem 3.2.
standard math Burnside's theorem (Theorem 2.24): a subgroup of GL(n,C) with uniformly bounded element orders is finite.
Used in the proof of Theorem 3.2 to conclude finiteness of the representation image.
domain assumption Siu's theorem on the noetherianity of Γ(W, O_Y) for Stein compact subsets (Remark 1.6).
Used to ensure the noetherian condition for Stein compact subsets, which is a key technical requirement in the analytic framework.

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Pith. "Pith review of On finiteness of relative log pluricanonical representations." pith.science (2026). https://pith.science/paper/6RCZLURO

@misc{pith2026250600760,
  author       = {Pith},
  title        = {Pith review of: On finiteness of relative log pluricanonical representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6RCZLURO}},
  note         = {Machine review of arXiv:2506.00760}
}

read the original abstract

We prove the finiteness of relative log pluricanonical representations in the complex analytic setting. As an application, we discuss the abundance conjecture for semi-log canonical pairs within this framework. Furthermore, we establish the existence of log canonical flips for complex analytic spaces. Roughly speaking, we reduce the abundance conjecture for semi-log canonical pairs to the case of log canonical pairs in the complex analytic setting. Moreover, we show that the abundance conjecture for projective morphisms of complex analytic spaces can be reduced to the classical abundance conjecture for projective varieties.

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Reference graph

Works this paper leans on

12 extracted references · 7 canonical work pages

[1]

Abramovich, L.-Y

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Mathematical Society of Japan, Tokyo, 2024 . [Fuj15] O. Fujino, Quasi-log structures on complex analytic spaces , preprint (2022). arXiv:2209.11401 [math.AG] [Fuj16] O. Fujino, Log canonical inversion of adjunction, Proc. Ja pan Acad. Ser. A Math. Sci. 100 (2024), no. 2, 7–11. [Fuj17] O. Fujino, On vanishing theorems for analytic spaces, Proc . Japan Acad...

arXiv 2022
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Bierstone, P

[BieM1] E. Bierstone, P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes ´Etudes Sci. Publ. Math. No. 67 (1988), 5–42. [BieM2] E. Bierstone, P. D. Milman, Canonical desingularization in char acteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. [Bir] C. Birkar, Existence of log canonica...

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[LM] S. Lyu, T. Murayama, The relative minimal model program for ex cellent algebraic spaces and analytic spaces in equal characteristic zero, preprint (2022). ar Xiv:2209.08732 [math.AG] [M] N. Moriyama, Remarks on the minimal model theory for log surfac es in the analytic setting, preprint (2025). arXiv:2505.18055 [math.AG] [N] N. Nakayama, Zariski-deco...

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Prill, The divisor class groups of some rings of holomorphic funct ions, Math

36 OSAMU FUJINO [P] D. Prill, The divisor class groups of some rings of holomorphic funct ions, Math. Z. 121 (1971), 58–80. [RR V] J. P. Ramis, G. Ruget, J. L. Verdier, Dualit´ e relative en g´ eom ´ etrie analytique complexe, Invent. Math. 13 (1971), 261–283. [U] K. Ueno, Classiﬁcation theory of algebraic varieties and compact co mplex spaces, Notes writ...

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On isolated log canonical singularities with index one

[Fuj4] O. Fujino, Fundamental theorems for the log minimal model p rogram, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789. [Fuj5] O. Fujino, On isolated log canonical singularities with index one, J. Math. Sci. Univ. Tokyo 18 (2011), no. 3, 299–323. [Fuj6] O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. ...

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[H1] K. Hashizume, Remarks on special kinds of the relative log minimal model program, Manuscripta Math. 160 (2019), no. 3-4, 285–314. [H2] K. Hashizume, A class of singularity of arbitrary pairs and log can onicalizations, Asian J. Math. 24 (2020), no. 2, 207–238. [H3] K. Has...

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[Fuj12] O. Fujino, Minimal model program for projective morphisms between complex analytic spaces, preprint (2022). arXiv:2201.11315 [math.AG] [Fuj13] O. Fujino, Vanishing theorems for projective morphisms be tween complex analytic spaces, to appear in Math. Res. Lett. [Fuj14]...

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[FF] O. Fujino, T. Fujisawa, Variation of mixed Hodge structure and its applications, to appear in Internat. J. Math. [FG] O. Fujino, Y. Gongyo, Log pluricanonical representations and the abundance conjecture, Com- pos. Math. 150 (2014), no. 4, 593–620. [FH] O. Fujino, K. Hash...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Abramovich, L.-Y

[AFKM] D. Abramovich, L.-Y. Fong, J. Koll´ ar, J. M cKernan, Semi log canonical surface, Flips and Abundance for Algebraic Threefolds , Ast´ erisque 211, Soc. Math. France, Montrouge, 1992, 139–154. [BS] C. B˘ anic˘ a, O. St˘ an˘ a¸ sil˘ a,Algebraic methods in the global theory of complex spaces , Translated from the Romanian. Editura Academiei, Bucharest...

work page 1992

[2] [4]

Fujino, Abundance theorem for semi log canonical three folds, Duke Math

[Fuj1] O. Fujino, Abundance theorem for semi log canonical three folds, Duke Math. J. 102 (2000), no. 3, 513–532. [Fuj2] O. Fujino, The indices of log canonical singularities, Amer. J. M ath. 123 (2001), no. 2, 229–253. [Fuj3] O. Fujino, On Kawamata’s theorem, Classiﬁcation of algebraic varieties , 305–315, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨ urich,

work page 2000

[3] [42]

[Fuj15] O

Mathematical Society of Japan, Tokyo, 2024 . [Fuj15] O. Fujino, Quasi-log structures on complex analytic spaces , preprint (2022). arXiv:2209.11401 [math.AG] [Fuj16] O. Fujino, Log canonical inversion of adjunction, Proc. Ja pan Acad. Ser. A Math. Sci. 100 (2024), no. 2, 7–11. [Fuj17] O. Fujino, On vanishing theorems for analytic spaces, Proc . Japan Acad...

arXiv 2022

[4] [1976]

Bierstone, P

[BieM1] E. Bierstone, P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes ´Etudes Sci. Publ. Math. No. 67 (1988), 5–42. [BieM2] E. Bierstone, P. D. Milman, Canonical desingularization in char acteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. [Bir] C. Birkar, Existence of log canonica...

work page 1988

[5] [1998]

[LM] S. Lyu, T. Murayama, The relative minimal model program for ex cellent algebraic spaces and analytic spaces in equal characteristic zero, preprint (2022). ar Xiv:2209.08732 [math.AG] [M] N. Moriyama, Remarks on the minimal model theory for log surfac es in the analytic setting, preprint (2025). arXiv:2505.18055 [math.AG] [N] N. Nakayama, Zariski-deco...

arXiv 2022

[6] [2004]

Prill, The divisor class groups of some rings of holomorphic funct ions, Math

36 OSAMU FUJINO [P] D. Prill, The divisor class groups of some rings of holomorphic funct ions, Math. Z. 121 (1971), 58–80. [RR V] J. P. Ramis, G. Ruget, J. L. Verdier, Dualit´ e relative en g´ eom ´ etrie analytique complexe, Invent. Math. 13 (1971), 261–283. [U] K. Ueno, Classiﬁcation theory of algebraic varieties and compact co mplex spaces, Notes writ...

work page 1971

[7] [2006]

[DHP] O. Das, C. Hacon, M. P˘ aun, On the 4-dimensional minimal mode l program for K¨ ahler varieties, Adv. Math. 443 (2024), Paper No. 109615. [EH1] M. Enokizono, K. Hashizume, Semistable reduction for complex analytic spaces, to appear in Trans. Amer. Math. Soc. Ser. B. [EH2] M. Enokizono, K. Hashizume, Minimal model program for log can onical pairs on ...

arXiv 2024

[8] [2011]

On isolated log canonical singularities with index one

[Fuj4] O. Fujino, Fundamental theorems for the log minimal model p rogram, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789. [Fuj5] O. Fujino, On isolated log canonical singularities with index one, J. Math. Sci. Univ. Tokyo 18 (2011), no. 3, 299–323. [Fuj6] O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. ...

work page 2011