Effective termination of general type MMPs in dimension at most five

Jihao Liu; Jingjun Han; Ziquan Zhuang

REVIEW 3 major objections 5 minor 9 references

Effective termination of general type MMPs in dimension at most five

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Every general type fivefold minimal model program terminates after a bounded number of steps; in dimension four the bound is explicit.

desk verdict A genuinely new fivefold termination result, but the proof leans on an imported Cartier-index black box and an outlined Lemma 3.5; both need attention before the paper is fully established. read the letter →

arxiv 2509.01501 v1 pith:NYP6UJPH submitted 2025-09-01 math.AG

Jingjun Han , Jihao Liu , Ziquan Zhuang This is my paper · ORCID

classification math.AG MSC 14E3014J3014J35

keywords minimalmodelprogramterminationeffectivelogpairsCartierindexdiscrepanciesbirationalgeometrydimensionfive

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 general type minimal model programs (MMPs) in dimension five terminate after a bounded number of steps: for any projective klt pair (X,Δ) with K_X+Δ big, or with K_X+Δ pseudo-effective and Δ big, every (K_X+Δ)-MMP stops, and the number of steps depends only on the pair. In dimension four, and partly in dimension three, the paper turns termination into an explicit bound: under hypotheses bounding the number of strata, Picard numbers, algebraic Betti numbers, and volume, any MMP takes at most M^M steps with M=(2N)^(9!). The paper notes that even termination of such fourfold MMPs was previously unknown, so these are the first explicit length bounds for them. The engine is a uniform bound on the Cartier index of Q-Cartier Weil divisors appearing in general type MMPs; that bound makes the set of possible discrepancies discrete, which lets a refined induction reduce termination to known lower-dimensional cases.

What carries the argument

The load-bearing object is the uniform Cartier-index bound of [HQZ25, Theorem 3.2], stated here as Theorem 2.3: in any MMP type contraction of a klt pair with big boundary, the Cartier index of every Q-Cartier Weil divisor is bounded by a constant depending only on the pair. This turns discrepancies into values from a discrete set (Lemma 3.3). With discreteness in place, the paper refines Birkar's inductive approach: termination for MMPs with discrete discrepancies in dimension n−1 implies termination for MMPs with uniform index bound in dimension n. For the explicit bounds, the mechanism is a lexicographically decreasing sequence of invariants—Picard numbers of strata, weighted difficulty c

What would settle it

A concrete refutation would be a projective klt fivefold (X,Δ) with K_X+Δ big and an infinite (K_X+Δ)-MMP, which contradicts Theorem 1.1. A more targeted test is to exhibit a sequence of MMP type contractions of such a pair in which the Cartier index of Q-Cartier Weil divisors is unbounded; that would contradict Theorem 2.3, the black box on which the proof rests.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if (X,Δ) is a projective klt pair of dimension five and either K_X+Δ is big or K_X+Δ is pseudo-effective with Δ big, then there is an integer m depending only on (X,Δ) such that every (K_X+Δ)-MMP terminates after at most m steps. Theorem 1.2 is the quantitative version in dimension four: for a projective log smooth klt pair with NΔ integral, at most N strata, strata of Picard number at most N, algebraic Betti number h_4^alg(X)≤N, Δ big with vol(Δ)≥1/N and (Δ·H^3)≤N, every MMP stops within M^M steps, M=(2N)^(9!). The same package yields explicit termination for general type fourfold MMPs, terminal fourfold MMPs, and arbitrary threefold MMPs, and derives the s

Load-bearing premise

The load-bearing premise is that in any MMP type contraction of a klt pair with big boundary, the Cartier index of every Q-Cartier Weil divisor is bounded by a constant depending only on the pair (Theorem 2.3, cited from [HQZ25]); if that uniformity fails, so does the discreteness of discrepancies and the whole termination argument.

Editorial extensions

If this is right

Every general type MMP on a projective klt fivefold terminates, closing an open case in dimension five.
MMPs on fivefolds with pseudo-effective canonical class and big boundary also terminate.
Fourfold klt MMPs with big boundary terminate, with an explicit bound on the number of steps in terms of topological and volume data.
The strong Sarkisov program holds in dimension five.
Explicit termination bounds are obtained for terminal fourfold MMPs and for arbitrary threefold MMPs under boundedness assumptions on strata and Picard numbers.

Reading between the lines

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

If the uniform Cartier-index bound (Theorem 2.3) extends to higher dimensions, the same discreteness-plus-induction scheme would give effective termination for general type MMPs in all dimensions, conditional on lower-dimensional termination.
The explicit bounds, with M=(2N)^(9!), are enormous and inherit factorial-sized constants from the index bound; sharper index bounds would directly improve the MMP length bounds.
The threefold bound holds without any bigness assumption, suggesting that in dimension three the discreteness of discrepancies, rather than general type, is what controls MMP length; testing the bound on known examples would indicate how far it is from optimal.

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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

Here's the short version: this paper proves a genuinely new and important result — effective termination of general type MMPs in dimension 5, plus explicit bounds in dimension 4 — and the overall architecture is credible. But the fivefold part currently rests on two things that aren't proved in this text: a uniform Cartier-index bound imported from an unreviewed preprint by two of the same authors, and Lemma 3.5, whose proof is only an outline. Those are exactly the spots to push on if you referee it.

What's genuinely new: Theorem 1.1 (and Theorem 3.9) resolves a long-open special case — termination of fivefold MMPs when K+Δ is big, or more generally when Δ is big. The paper states clearly that even fourfold big-boundary termination was not previously known. Theorem 1.2 gives explicit bounds in dimension 4 under boundedness assumptions; those bounds are gigantic (involving (2N)^{9!} and M^M) but they're real and explicit. The strategy — Birkar's induction plus discretizing discrepancies via a Cartier-index bound, then using termination of terminal fourfold flips — is coherent and well-motivated. The effective part in Section 4 is substantial work with M-adic difficulties and lexicographic invariants; it looks internally consistent.

Where the soft spots are, in proportion: the main one is Theorem 2.3, the index bound, which the paper imports from [HQZ25]. That's a preprint by Han, Qi, and Zhuang, so two of the present authors, and it's not yet refereed. The paper doesn't address whether the proof of [HQZ25, Thm 3.2] uses the termination result being proved here. I don't see a circular reduction in this text, but the independence should be stated explicitly. This is a fixable issue, not fatal.

The second is Lemma 3.5, used in the fivefold termination proof. The proof is an outline — "essentially follows from special termination" — and it's load-bearing. The ingredients in Lemmas 3.3 and 3.4 are proved and look fine, but the passage from truncations to a face-contracting MMP on U needs a complete write-up. For a theorem of this significance, an outline isn't enough; it should be fully proved or stated with a complete proof in an appendix. I'd also want the proof of Lemma 3.8 checked carefully — it's plausible, but the induction on the exceptional locus deserves full detail.

The threefold explicit bounds in Theorem 4.16 appear self-contained apart from known results, so that part looks solid.

Bottom line: this is a serious paper that deserves a serious referee. Send it to review, with an expert on special termination and index bounds, and ask the authors to (1) fill in Lemma 3.5, and (2) state explicitly that [HQZ25, Thm 3.2] is independent of the termination results proved here. After those points are addressed, I'd expect this to be a major paper.

Referee Report

3 major / 5 minor

Summary. The paper proves effective termination of general-type MMPs in dimensions at most 5. Theorem 1.1 states that for a projective klt pair (X,Δ) of dimension 5, if K_X+Δ is big, or if K_X+Δ is pseudo-effective and Δ is big, then every (K_X+Δ)-MMP terminates after a number of steps bounded by a constant depending only on (X,Δ). Theorem 1.2 gives explicit, albeit enormous, bounds in dimension 4 under boundedness assumptions on strata, Picard numbers, and volumes. The paper also gives explicit termination bounds for threefold MMPs, terminal fourfold MMPs, and derives the strong Sarkisov program in dimension 5. The strategy is to combine uniform Cartier index bounds, discrete-discrepancy arguments, Birkar's inductive termination method, and a refined difficulty-function argument; the effective bounds are obtained via König's lemma and an M-adic difficulty formalism.

Significance. If the central claims are correct, this is a substantial advance: termination of general-type fivefold MMPs was open, and explicit bounds for fourfold MMPs are new. The paper is well organized, states its results clearly, and isolates its key new ideas, especially the use of uniform Cartier index bounds to force discrepancies into a discrete set and the subsequent effective counting with König's lemma. The main caveat is that the principal engine, Theorem 2.3, is imported from an unreviewed preprint with overlapping authorship, and Lemma 3.5, which is load-bearing for the fivefold termination, is only outlined. The paper would be a strong contribution once those two points are fully verified or made explicit.

major comments (3)

[§2.2, Theorem 2.3; used in §3.1 (Theorem 3.1, Lemma 3.7)] The uniform Cartier index bound is the main engine of the paper. The proof given is only “This is a special case of [HQZ25, Theorem 3.2]”, where [HQZ25] is an unreviewed preprint by two of the present authors. The bound is used in the fourfold case on p. 10 (“By Theorem 2.3…”) and in the fivefold case through Lemma 3.7 and Lemma 3.5. I do not see a circularity with the paper’s target theorem, but the manuscript’s main theorem is conditional on an unverified external result. The authors should either include a proof of Theorem 2.3, or state clearly that Theorem 1.1 depends on [HQZ25] and give the exact hypotheses and dependence.
[§3.1, Lemma 3.5] Lemma 3.5 is explicitly proved only as an outline, but it is needed for Theorem 3.1(2) via Lemma 3.8. In particular, the passage from the truncated MMP on (Y_i,D_i) to the induced face-contracting MMP on (W_i,B_i), and then to the klt MMP on U_i = W_i \setminus \lfloor B_i\rfloor, is asserted by reference to [Fuj07, Prop. 4.2.14 and Lemma 4.2.15]. The application of Lemmas 3.3 and 3.4 to the sequence (3.2) requires checking that the induced sequence is face-contracting and that its discrepancies lie in a fixed discrete set; neither condition is verified in the manuscript. Since this lemma is load-bearing for the fivefold termination proof, this is a gap, not a presentation issue.
[§4.3, Lemma 4.12 and Theorems 4.14/4.15] The explicit fourfold bounds in Theorems 4.14 and 4.15 rely on [HQZ25, Corollaries 3.18 and 3.20] in the same black-box manner as Theorem 2.3. Moreover, Lemma 4.12 assumes that the negative discrepancies lie in a fixed finite set S, and the proof of Theorems 4.14/4.15 asserts this in one sentence, passing from the Cartier index bound to S = [0,1] \cap (1/N) R. This reduction should be spelled out, because if the index bound has a hidden dependence or a different uniformity, the explicit bounds collapse.

minor comments (5)

[p. 10] Typo: “Cariter” should be “Cartier”.
[§4.2, Theorem 4.8 and Corollary 4.9] The formulas use 1/b_k and 1/(1-b_1), which are undefined when k=0, i.e. when \Delta=0. The terminal case \Delta=0 should be treated separately or the statement should assume k\ge 1.
[§3.1, Lemma 3.8] In the proof of the claim Ex(\pi_i) \subseteq \lfloor D_i\rfloor, the line “a(E,X_i,\Delta_i) = -\operatorname{mult}_E D_i = -\operatorname{mult}_E D_{i-1}” implicitly uses that E is exceptional over both X_i and X_{i-1}; this should be stated explicitly.
[§4.1, Definition 4.5] The definition of \rho(X,\Delta) uses \rho(\tilde B) for the normalization of the boundary; the independence of this from the chosen compactification for singular strata is only implicit in Lemma 4.2. A short clarification would help.
[§3.1, Remark 3.6] Remark 3.6 is helpful, but it would be more useful if it stated exactly which hypotheses of Lemma 3.4 must be checked when passing from the sequence on (Y_i,D_i) to the induced sequence on W_i and U_i.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found; the main dependencies are self-cited but are independent statements, not the theorem being proved.

full rationale

The claimed theorems are not obtained by fitting or by renaming. The central external input is Theorem 2.3 (from [HQZ25, Thm 3.2]), a uniform bound on the Cartier index of Q-Cartier Weil divisors for MMP type contractions of klt pairs with big boundary. This is used to make discrepancy sets discrete (Lemma 3.3), to lift Cartier-index control to dlt modifications (Lemma 3.7), and to enter the explicit-bound machinery in Section 4. Although [HQZ25] is by two of the present authors and is a preprint, its statement is not equivalent to the paper's Theorem 1.1 or 1.2: it assumes klt + big boundary and concludes a Cartier-index bound; it does not state fivefold termination or explicit step bounds. No equation in the paper redefines the target conclusion as an input. Lemma 3.5 is admittedly only outlined ('We only outline the main steps'), and its full verification is a legitimate completeness concern; but the outlined induction is over lower-dimensional MMPs and does not presuppose the fivefold termination being proved. Likewise the use of [HLS24] in Lemma 3.3 is an independent decomposition statement. Accordingly, no circular step can be quoted or exhibited; the appropriate verdict is no significant circularity, with residual verification risk coming from the unproved-in-this-paper self-cited index bound and the sketched Lemma 3.5.

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

No new entities are introduced; the paper's input parameters (N, volume, Picard numbers) are hypotheses, not fitted constants. The 'M-adic difficulty' δ_M is a defined invariant with weights chosen for an inequality to hold, not fitted to data.

assumptions (5)

standard math Cone theorem and existence of flips and minimal models for klt pairs (BCHM10, various)
Used throughout, e.g., Lemma 3.10 uses finitely many K+Δ-negative extremal rays; Corollary 3.2 uses BCHM Theorem D.
standard math ACC for log canonical thresholds [HMX14, Theorem 1.1]
Contradicts the increasing lct sequence in the proof of Theorem 3.1 in dimension 5.
domain assumption Uniform bound on Cartier index in general type MMP [HQZ25, Theorem 3.2, Corollaries 3.18, 3.20]
Imported as Theorem 2.3 and used in §3 and §4; it is the load-bearing input and is from an unreviewed preprint by two of the present authors.
standard math Termination of terminal 4-fold MMP [Fuj04, Fuj05]
Used in Lemma 2.10 and Theorem 4.7 to guarantee strict decrease of the difficulty invariant.
standard math Special termination of MMP [Fuj07, Theorem 4.2.1]
The outline of Lemma 3.5 is based on the same proof.

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Cite this review

Pith. "Pith review of Effective termination of general type MMPs in dimension at most five." pith.science (2026). https://pith.science/paper/NYP6UJPH

@misc{pith2026250901501,
  author       = {Pith},
  title        = {Pith review of: Effective termination of general type MMPs in dimension at most five},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYP6UJPH}},
  note         = {Machine review of arXiv:2509.01501}
}

abstract

We prove the effective termination of general type MMPs in dimension at most $5$, and give explicit bounds (in terms of topological invariants and volume of divisors) on the number of steps in the MMP when the dimension is at most $4$.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

[AHK07] Valery Alexeev, Christopher Hacon, and Yujiro Kawamata, Termination of (many) 4- dimensional log flips , Invent. Math. 168 (2007), no. 2, 433–448. [BCHM10] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), no. 2, 405–

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With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. [KMM87] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 1987, pp. 283–360. [Kol13] J´ anos Koll´ ar,Singularities of the minimal model program , Cambridge Tracts in Mathemat- ic...

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On the strong Sarkisov program

[He24] Yang He, On the strong Sarkisov program (2024). arXiv:2311.08750v2. [HL22] Jingjun Han and Zhan Li, Weak Zariski decompositions and log terminal models for general- ized pairs, Math. Z. 302 (2022), no. 2, 707–741. MR4480208 [HLS24] Jingjun Han, Jihao Liu, and Vyacheslav Shokurov, ACC for minimal log discrepancies of exceptional singularities , Peki...

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[2025]

arXiv:2506.20183v2. 26 J. Han, J. Liu, and Z. Zhuang [HX13] Christopher D. Hacon and Chenyang Xu, Existence of log canonical closures , Invent. Math. 192 (2013), no. 1, 161–195. [Kaw92] Yujiro Kawamata, Termination of log flips for algebraic 3-folds, Internat. J. Math. 3 (1992...

2013 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

[AHK07] Valery Alexeev, Christopher Hacon, and Yujiro Kawamata, Termination of (many) 4- dimensional log flips , Invent. Math. 168 (2007), no. 2, 433–448. [BCHM10] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), no. 2, 405–

work page 2007

[2] [32]

Shokurov, Letters of a bi-rationalist

MR4913095 [Sho04] Vyacheslav V. Shokurov, Letters of a bi-rationalist. V. Minimal log discrepancies and termi- nation of log flips , Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 328–351. [Sho09] , Letters of a bi-rationalist. VII. Ordered termination , Tr. Mat. Inst. Steklova 264 (2009), 184–208. [Sho85] , A nonvanishin...

work page 2004

[3] [257]

Termination of 4-fold canonical flips

MR2563688 [Bir12] , Existence of log canonical flips and a special LMMP , Publications Math´ ematiques de l’IH ´ES 115 (2012), 325–368. [CC24] Paolo Cascini and Hsin-Ku Chen, Chern numbers of terminal threefolds , Int. Math. Res. Not. IMRN 23 (2024), 14464–14478. [CH11] Jungkai A. Chen and Christopher D. Hacon, Factoring 3-fold flips and divisorial contra...

work page 2012

[4] [468]

[Bir07] Caucher Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J. 136 (2007), no. 1, 173–180. MR2271298 [Bir10] , On termination of log flips in dimension four , Math. Ann. 346 (2010), no. 2, 251–

work page 2007

[5] [1992]

211 (1992)

Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Ast´ erisque No. 211 (1992). [Mar22] Diletta Martinelli, Effective bounds for the number of MMP-series of a smooth threefold , Glasg. Math. J. 64 (2022), no. 1, 106–113. [Mor25] Joaqu ´ ın Moraga,Termination of pseudo-effective 4-...

work page 1991

[6] [1998]

With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. [KMM87] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 1987, pp. 283–360. [Kol13] J´ anos Koll´ ar,Singularities of the minimal model program , Cambridge Tracts in Mathemat- ic...

work page 1998

[7] [2007]

On the strong Sarkisov program

[He24] Yang He, On the strong Sarkisov program (2024). arXiv:2311.08750v2. [HL22] Jingjun Han and Zhan Li, Weak Zariski decompositions and log terminal models for general- ized pairs, Math. Z. 302 (2022), no. 2, 707–741. MR4480208 [HLS24] Jingjun Han, Jihao Liu, and Vyacheslav Shokurov, ACC for minimal log discrepancies of exceptional singularities , Peki...

work page Pith review arXiv 2024

[8] [2013]

[Kol21] J´ anos Koll´ ar,Relative MMP without Q-factoriality, Electron

With a collaboration of S´ andor Kov´ acs. [Kol21] J´ anos Koll´ ar,Relative MMP without Q-factoriality, Electron. Res. Arch. 29 (2021), no. 5, 3193–3203. [Kol89] , Flops, Nagoya Math. J. 113 (1989), 15–36. [K+92] J´ anos Koll´ ar et al.,Flips and abundance for algebraic threefolds , Soci´ et´ e Math´ ematique de France, Paris,

work page 2021