REVIEW 1 major objections 4 minor 32 references
Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism
T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Kawaguchi-Silverman conjecture proved for rationally connected varieties with an int-amplified endomorphism
desk verdict The main theorem is an important all-dimensional KSC result and the proof is largely sound; the one real soft spot is a compressed base-Q-factoriality step that needs a citation but is almost certainly fillable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central mechanism is the equivariant minimal model program (MMP) developed by Meng and Zhang, together with a covering theorem (Theorem 4.4) proving that certain fiber-type contractions force the existence of a quasi-etale cover dominating an abelian variety. The covering theorem uses the index-1 cover of a log Calabi-Yau pair (Y, $\Delta$) to construct a finite cover of the base, then forms the main component of the fiber product with X, and shows the resulting cover is quasi-etale using the horizontality of the ramification divisor. This cover then ascends along the steps of the MMP via Lemma 4.12, so if any step produced a Q-abelian output, the original rationally connected X would admit a quasi-etale cover dominating an abelian variety, which is impossible since X is algebraically simply connected.
What would settle it
Construct a smooth projective rationally connected variety X over Q admitting an int-amplified endomorphism, and a surjective endomorphism f with a Zariski dense point x, such that the arithmetic degree alpha_f(x) is strictly less than the dynamical degree delta_f. According to the theorem, no such example can exist; finding one would disprove the claim. More locally, an explicit example where the base of a fiber-type contraction in the equivariant MMP is not Q-factorial or fails Pic = $N^{1}$ would test the weakest assumption: if such a base can occur while the rest of the proof's hypotheses hold, the reduction step would be invalid.
Extended reading notes
Core claim
Theorem 1.1 states: if X is a smooth projective rationally connected variety over Q admitting an int-amplified endomorphism, then for every surjective endomorphism f of X and every x in X(Q) whose f-orbit is Zariski dense, the arithmetic degree alpha_f(x) equals the dynamical degree delta_f. The proof runs a KX-MMP equivariantly with respect to a power of f, and at each step applies a dichotomy: either the Kawaguchi-Silverman conjecture holds for the induced map on a birational model, or a certain fibration appears with controlled dynamical degrees. The authors show the problematic second case cannot occur for rationally connected X, because the fibration would produce a quasi-etale cover dominating an abelian variety, contradicting algebraic simple connectedness. The key technical ingredient is a covering theorem showing that when a fiber-type contraction has a log Calabi-Yau base, the whole fibration admits a quasi-etale cover by a variety that fibers over an abelian variety; this covering ascends along divisorial contractions, flips, and fiber-type contractions, so the contradiction propagates backward through the MMP.
Load-bearing premise
The base Y' obtained after the MMP step (and in the reduction to exclude Q-abelian outputs) is assumed to be Q-factorial klt with Pic(Y')_Q = $N^{1}$(Y')_Q, but the paper does not justify that the constructed fiber-type contraction has a base with these properties; a Q-factorialization step that preserves the reduction is not provided.
Editorial extensions
If this is right
- The Kawaguchi-Silverman conjecture now holds for all surjective endomorphisms on smooth projective rationally connected varieties that admit an int-amplified endomorphism, a class that includes many Fano-type and rationally connected threefolds and higher-dimensional examples.
- The proof gives a template for proving the conjecture on broader classes: one only needs to rule out the appearance of fiber-type contractions with Q-abelian bases during the equivariant MMP.
- Any future counterexample to the conjecture on a rationally connected variety must fail to admit an int-amplified endomorphism, since the theorem removes all such cases.
- The covering theorem itself is a standalone structural result: certain fiber-type contractions with log Calabi-Yau bases are necessarily covered by fibrations over abelian varieties, which may be useful in other dynamical and birational questions.
Reading between the lines
- If the equivariant MMP can be run on varieties that are not Q-factorial or have Pic(X)_Q != N^1(X)_Q, the same strategy could extend the conjecture beyond the rationally connected case, provided the abelian-covering obstruction can be ruled out.
- The proof suggests that for any variety admitting an int-amplified endomorphism, the only potential obstruction to Kawaguchi-Silverman is the existence of a fiber-type contraction to a Q-abelian base; one could test this by looking for rationally connected threefolds whose MMP outputs are Q-abelian but which do not admit int-amplified endomorphisms.
- The quasi-etale covering construction may be adaptable to prove statements about the non-existence of int-amplified endomorphisms on varieties that are algebraically simply connected, since it forces a cover by an abelian variety.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves the Kawaguchi-Silverman conjecture (KSC) for every surjective endomorphism on a smooth projective rationally connected variety over Q that admits an int-amplified endomorphism. The proof combines the equivariant minimal model program of Meng–Zhang with a new covering theorem (Theorem 4.4) that, under a horizontal ramification condition, produces a quasi-étale abelian cover. The main theorem is obtained by running an equivariant K_X-MMP, applying a key trichotomy (Proposition 5.1) to each Mori fiber space, and descending either to a Q-abelian variety or to a lower-dimensional base; properties of arithmetic/dynamical degrees (Remarks 2.10 and 2.11) then transfer KSC back to the original variety.
Significance. If the proof is correct, this is a substantial advance: it establishes KSC for a large class of higher-dimensional rationally connected varieties, going well beyond the surface case and the threefold results of Meng–Zhang. The paper is well structured and contains detailed arguments for the covering theorem and the MMP reduction, and the main theorem is a clear falsifiable statement. The strategy of excluding problematic fibrations via a covering theorem is original and likely to be influential.
major comments (1)
- [§5, Theorem 5.6, case (3), second subcase] When δ_{f^m|X'_r} = δ_{f^m|Y'}, the proof runs the K_{Y'}-MMP and invokes Theorem 3.4. This requires Y' to be Q-factorial klt. The paper does not prove that Y', the base of the fiber type contraction π1: X'_r → Y' arising from Proposition 5.1(3), is Q-factorial klt. Lemma 4.10 gives klt only under the hypothesis that −K_{X'_r} is π1-ample; in the δ_f > δ_g branch of Proposition 5.1, π1 is a (K_{X'_r}+Δ_r)-negative extremal ray contraction, and −K_{X'_r} is not shown to be π1-ample. The same issue occurs when the argument excludes Q-abelian outputs after running K_{Y'}-MMP. Since the induction on ρ and dim depends on passing to Y', this is a load-bearing gap. Please add a proof or a precise reference that Y' is Q-factorial klt in this setting, or modify the descent to avoid running an MMP on Y'.
minor comments (4)
- [§5, Theorem 5.6] When the MMP on X_r or Y' terminates in a Q-abelian variety that is a point, the statement 'Q-abelian case does not occur because endomorphisms on Q-abelian varieties satisfy (*)' is not correct: a point satisfies property (*) only vacuously, and the contradiction argument does not apply. This case is harmless (it leads to KSC via Proposition 5.1(1) with dim Y = 0), but it should be stated explicitly.
- [§5, Theorem 5.6] The assertion that every variety X' appearing in an MMP starting from a rationally connected variety satisfies N^1(X')_Q = Pic(X')_Q is used without proof or reference. This is standard but should be cited, for example to Debarre's book or a suitable statement on rationally connected varieties and the Picard group.
- [§4, Theorem 4.4] Theorem 4.4 is introduced as being taken from an upcoming paper by the second author, but a proof is included in the text. To avoid concerns about reliance on unpublished work, please add a sentence clarifying that the proof is self-contained and that the reference is only for provenance.
- [§5, Proposition 5.1] There are several typographical and notational slips in the text (e.g., 'there exits' in the statement of Proposition 5.1, and inconsistent subscripts X_1/X_r). The authors should proofread the manuscript carefully before publication.
Circularity Check
No circularity: the proof reduces KSC to independent known cases and supplies a full proof of its new covering theorem.
full rationale
The derivation chain runs the Kawaguchi-Silverman conjecture for f on a smooth rationally connected variety through equivariant MMP to previously established cases, including abelian varieties, rank-one Fano contractions, and prior KSC results cited from independent work. The new geometric input, Theorem 4.4, is attributed to the second author's upcoming paper but is not used as an equivalent of the target conjecture: the paper actually gives a complete proof of Theorem 4.4 in Section 4.2, using standard MMP facts, Ambro's canonical bundle formula, and a quasi-etale cover argument. Proposition 5.1, though sourced from Meng-Zhang [29], is an independent reduction statement and is proved in the text; none of its cases presuppose equality of arithmetic and dynamical degrees. No parameter is fitted, and no prediction is renamed from an input: the equalities such as delta_{f^n|X'} = delta_{f^n|Y'} and the reductions in Remark 2.11 are derived from the geometry of the MMP rather than imposed by construction. The reader-flagged point that Y' should be justified to be Q-factorial klt before running K_{Y'}-MMP in Theorem 5.6 is a rigor gap in the written proof, not a circularity: it does not assume KSC and is plausibly fillable by standard MMP facts. The authors' self-citations, such as [21] and [22], supply independent lemmas on arithmetic degrees and are not the load-bearing equivalent of the main conjecture. Thus no circular step is exhibited under the required standard.
Assumptions & free parameters
assumptions (8)
- standard math BCHM minimal model program: for a Q-factorial klt pair over Q, a K_X-MMP with scaling terminates with a Mori fiber space when K_X is not pseudo-effective (cite [2]).
- domain assumption Meng-Zhang equivariant MMP (Theorem 3.4): for a Q-factorial klt projective variety admitting an int-amplified endomorphism, every surjective endomorphism has an iterate equivariant with an MMP sequence ending in a Q-abelian variety.
- domain assumption Existing KSC cases: KSC for abelian varieties and cited classes (surfaces, hyperkähler, etc.) is available ([15,31,24,19,20,22,23]).
- standard math Gongyo abundance theorem [12, Theorem 1.2]: for an lc pair with numerically trivial log canonical divisor, K_Y+Delta is Q-linearly equivalent to 0.
- standard math Ambro canonical bundle formula [1, Theorem 4.1]: used in Lemma 4.10 to conclude the base of a Fano contraction of a klt variety is klt.
- domain assumption Meng-Zhang [26, Theorem 5.2]: a Q-abelian variety with int-amplified endomorphism is finitely quasi-étale covered by an abelian variety compatibly.
- standard math Smooth rationally connected projective varieties over characteristic zero are algebraically simply connected and admit no dominant morphism to a positive-dimensional abelian variety.
- domain assumption [29, Lemma 9.1]: a ray generated by a nef eigen-divisor D with f^*D = delta_f D is extremal in the effective cone under the stated numerical conditions.
Cite this review
Pith. "Pith review of Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism." pith.science (2026). https://pith.science/paper/L3NTS7KB
@misc{pith2026190811537,
author = {Pith},
title = {Pith review of: Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism},
year = {2026},
howpublished = {\url{https://pith.science/paper/L3NTS7KB}},
note = {Machine review of arXiv:1908.11537}
}
read the original abstract
We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.
Reference graph
Works this paper leans on
-
[1]
Ambro, F., The moduli b-divisor of an lc-trivial fibration , Compos. Math. 141 (2005), no. 2, 385–403
work page 2005
-
[2]
D., and McKernan, J., Existence of minimal models for varieties of log general type , J
Birkar, C., Cascini, P., Hacon, C. D., and McKernan, J., Existence of minimal models for varieties of log general type , J. Amer. Math. Soc., 23(2):405–468, 2010
work page 2010
-
[3]
Bombieri, E., Gubler, W., Heights in Diophantine geometry , Cambridge uni- versity press, 2007
work page 2007
-
[4]
Broustet, A., Gongyo, Y., Remarks on log Calabi-Yau structure of varieties admitting polarized endomorphisms , Taiwanese J. Math. 21 (2017), no. 3, 569– 582
work page 2017
-
[5]
Broustet, A., H¨ oring, A.,Singularities of varieties admitting an endomorphism , Math. Ann. 360 (2014), no. 1-2, 439–456
work page 2014
-
[6]
Cascini, P., Meng, S., Zhang, D-Q., Polarized endomorphisms of normal pro- jective threefolds in arbitrary characteristic , arXiv:1710.01903v2
-
[7]
Dang, N-B., Degrees of iterates of rational maps on normal projective va rieties, arXiv:1701.07760
-
[8]
Springer- Verlag, New York, 2001
Debarre, O., Higher-dimensional algebraic geometry , Universitext. Springer- Verlag, New York, 2001. xiv+233 pp
work page 2001
Show all 32 references
-
[9]
Diller, J., Favre, C., Dynamics of bimeromorphic maps of surfaces , Amer. J. Math. 123 no. 6, 1135–1169, (2001). 24 YOHSUKE MATSUZA W A AND SHOU YOSHIKA W A
2001
-
[10]
Dinh, T-C., Nguyˆ en, V-A., Comparison of dynamical degrees for semi- conjugate meromorphic maps , Comment. Math. Helv. 86 (2011), no. 4, 817– 840
2011
-
[11]
Dinh, T-C., Sibony, N., Equidistribution problems in complex dynamics of higher dimension , Internat. J. Math. 28, 1750057 (2007)
2007
-
[12]
Algebraic Geom
Gongyo, Y., Abundance theorem for numerically trivial log canonical di visors of semi-log canonical pairs , J. Algebraic Geom. 22 (2013), no. 3, 549–564
2013
-
[13]
H., Diophantine geometry
Hindry, M., Silverman, J. H., Diophantine geometry. An introduction , Gradu- ate Text in Mathematics, no. 20, Springer-Verlag, New York, 2000
2000
-
[14]
Jonsson, M., Wulcan, E., Canonical heights for plane polynomial maps of small topological degree, Math. Res. Lett., 19(6):1207–1217, 2012
2012
-
[15]
H., Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical hei ghts on abelian vari- eties, Trans
Kawaguchi, S., Silverman, J. H., Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical hei ghts on abelian vari- eties, Trans. Amer. Math. Soc. 368 (2016), 5009–5035
2016
-
[16]
H., On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties , J
Kawaguchi, S., Silverman, J. H., On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties , J. Reine Angew. Math. 713 (2016), 21–48
2016
-
[17]
Koll´ ar, J., Mori, S.,Birational geometry of algebraic varieties, 1998, Cambridge Univ. Press
1998
-
[18]
Lang, S., Fundamentals of Diophantine Geometry , Springer-Verlag, New York, 1983
1983
-
[19]
Lesieutre, J., Satriano, M., Canonical heights on hyper-K¨ ahler varieties and the Kawaguchi-Silverman conjecture , preprint, 2018, arXiv:1802.07388
2018 arXiv
-
[20]
Lin, J-L., On the arithmetic dynamics of monomial maps , arXiv:1704.02661
-
[21]
Matsuzawa, Y., On upper bounds of arithmetic degrees , to appear in Amer. J. Math
-
[22]
Matsuzawa, Y., Kawaguchi-Silverman conjecture for endomorphisms on sev- eral classes of varieties , preprint
-
[23]
Matsuzawa, Y., Sano, K., Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties , to appear in Ergodic Theory Dynam. Systems
-
[24]
7, 1635–1657
Matsuzawa, Y., Sano, K., Shibata, T., Arithmetic degrees and dynamical de- grees of endomorphisms on surfaces , Algebra Number Theory 12 (2018), no. 7, 1635–1657
2018
-
[25]
Matsuzawa, Y., Yoshikawa, S., Int-amplified endomorphisms on normal pro- jective surfaces, preprint
-
[26]
Meng, S., Building blocks of amplified endomorphisms of normal projec tive varieties, Math. Z. (2019). https://doi.org/10.1007/s00209-019-02316 -7
2019 doi
-
[27]
and Zhang, D.-Q., Building blocks of polarized endomorphisms of normal projective varieties , Adv
Meng, S. and Zhang, D.-Q., Building blocks of polarized endomorphisms of normal projective varieties , Adv. Math. 325 (2018), 243–273
2018
-
[28]
and Zhang, D.-Q., Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism , arXiv:1806.05828
Meng, S. and Zhang, D.-Q., Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism , arXiv:1806.05828
-
[29]
and Zhang, D.-Q., Kawaguchi-Silverman conjecture for surjective endomorphisms, arXiv:1908.01605
Meng, S. and Zhang, D.-Q., Kawaguchi-Silverman conjecture for surjective endomorphisms, arXiv:1908.01605
1908 arXiv
-
[30]
H., Dynamical degree, arithmetic entropy, and canonical heigh ts for dominant rational self-maps of projective space , Ergodic Theory Dynam
Silverman, J. H., Dynamical degree, arithmetic entropy, and canonical heigh ts for dominant rational self-maps of projective space , Ergodic Theory Dynam. Systems 34 (2014), no. 2, 647–678
2014
-
[31]
H., Arithmetic and dynamical degrees on abelian varieties , J
Silverman, J. H., Arithmetic and dynamical degrees on abelian varieties , J. Th´ eor. Nombres Bprdeaux29 (2017), no. 1, 151–167
2017
-
[32]
T., (Relative) dynamical degrees of rational maps over an algeb raic closed field , arXiv:1501.01523v1
Truong, T. T., (Relative) dynamical degrees of rational maps over an algeb raic closed field , arXiv:1501.01523v1. 25 Graduate school of Mathematical Sciences, the University o f Tokyo, Komaba, Tokyo, 153-8914, Japan Graduate school of Mathematical Sciences, the University o f ...
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.