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Kawaguchi-Silverman conjecture for certain surjective endomorphisms

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Pith's one-line read The Kawaguchi-Silverman conjecture holds for every surjective endomorphism of any projective surface.

desk verdict Strong paper that proves KSC for all projective surfaces and a new threefold class; the threefold proof has a small but real gap in the final reduction that is likely fixable. read the letter →

arxiv 1908.01605 v2 pith:F32RJ474 submitted 2019-08-05 math.AG math.DSmath.NT

Sheng Meng , De-Qi Zhang This is my paper · ORCID
classification math.AGmath.DSmath.NT MSC 37P5514E3008A35
keywords Kawaguchi-Silvermanconjecturearithmeticdegreedynamicalint-amplifiedendomorphismequivariantminimalmodelprogramtoricvarietiesprojectivesurfacesrationallyconnectedthreefolds
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

This paper resolves the Kawaguchi-Silverman conjecture for all surjective endomorphisms of projective surfaces, including singular ones, by showing that the arithmetic degree of any point with Zariski-dense orbit equals the first dynamical degree. In higher dimensions, it reduces the conjecture to a single special case, called Case TIR, and then proves that this case cannot arise for rationally connected smooth projective threefolds admitting an int-amplified endomorphism. If the reduction holds, the conjecture would follow for a broad class of higher-dimensional varieties from a single remaining scenario. The proof turns on the equivariant minimal model program, the effectiveness of the anti-canonical divisor, and a toric characterization of varieties with totally invariant ramification.

What carries the argument

The central object is 'Case TIR' (totally invariant ramification): a surjective endomorphism $f$ of a $\mathbb{Q}$-factorial klt projective variety $X$ admitting an int-amplified endomorphism, with anti-Iitaka dimension $\kappa(X,-K_X)=0$, with $-K_X$ numerically equivalent to a positive multiple of an irreducible effective divisor $D$ equal to the support of the ramification divisor, and with an $f$-equivariant Fano contraction whose dynamical degree is strictly smaller than $\delta_f$. The machinery that carries the argument is the equivariant minimal model program, which lets the authors repeatedly replace $X$ by lower-dimensional models; the effectiveness theorem for $-K_X$; the anti-Iitaka fibration and Chow reduction, which show KSC follows once $f^*K_X \equiv \delta_f K_X$ and $\kappa(X,-K_X)>0$; and a toric-pair criterion showing that a rationally connected smooth variety with an int-amplified endomorphism of totally invariant ramification and a suitable numerical eigenspace structure is toric.

What would settle it

Exhibit a $\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism whose Albanese morphism is not surjective, or a surjective endomorphism of a projective surface with a Zariski-dense orbit whose arithmetic degree is not the first dynamical degree; either would directly contradict the paper's theorems.

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Extended reading notes

Core claim

The paper establishes that KSC holds for every surjective endomorphism of any projective surface, removing the smoothness assumption that earlier work required. For higher dimensions, it proves that if $X$ is a $\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism and KSC holds for all surjective endomorphisms whose ramification divisor is totally invariant and irreducible (Case TIR), then KSC holds for every surjective endomorphism of $X$. As a consequence, KSC holds for every surjective endomorphism of any rationally connected smooth projective threefold admitting an int-amplified endomorphism. The central mechanism is a reduction: after running an equivariant minimal model program, either KSC follows immediately, or the map falls into Case TIR; the authors then rule out Case TIR in the threefold setting using a characterization of toric pairs.

Load-bearing premise

The higher-dimensional reduction depends, without reproof, on two external results: that a $\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism has a surjective Albanese morphism, and that it admits an equivariant minimal model program; if either fails, the reduction to Case TIR breaks.

Editorial extensions

If this is right

  • For every projective surface and every surjective endomorphism, points with Zariski-dense orbit have arithmetic degree equal to the first dynamical degree.
  • For rationally connected smooth projective threefolds admitting an int-amplified endomorphism, the same equality holds for all surjective endomorphisms.
  • The reduction to Case TIR means that resolving KSC in higher dimensions reduces to eliminating or handling one special configuration, rather than analyzing all endomorphisms individually.
  • The effectiveness of $-K_X$ for varieties admitting an int-amplified endomorphism stands as an independent structural result with applications beyond KSC.
  • The proof recovers and extends the smooth-surface case without relying on it, so the surface statement is unconditional for singular surfaces as well.

Reading between the lines

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

  • If the toric-pair criterion generalizes, a rationally connected smooth projective variety of any dimension admitting an int-amplified endomorphism with totally invariant ramification should be toric, which would eliminate Case TIR in all dimensions unless the anti-Iitaka dimension is positive.
  • The surface proof's identification of the 'troubled' Fano contraction as being of product type suggests that similar product structures may force the desired numerical eigenvector in higher-dimensional fibrations, potentially weakening the need for the full MMP induction.
  • A testable extension is to replace rational connectedness in the threefold theorem by uniruledness or by klt singularities; if the Albanese-surjectivity and equivariant-MMP assumptions remain valid, the same toric argument may apply.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proves the Kawaguchi-Silverman conjecture (KSC) for every surjective endomorphism of a projective surface, including singular surfaces, and reduces KSC in higher dimensions to a well-specified ``Case TIR'' (totally invariant ramification case). Building on this reduction, it proves KSC for every surjective endomorphism of a rationally connected smooth projective threefold admitting an int-amplified endomorphism. The main ingredients are an equivariant minimal model program for surfaces and threefolds, effectivity of the anti-canonical divisor, and a toric characterization of pairs admitting an int-amplified endomorphism with totally invariant ramification.

Significance. If the threefold theorem is correct, the paper resolves KSC for all surfaces and for a substantial class of threefolds, going well beyond the previously known smooth-surface and Mori-dream-space cases. The surface argument is detailed and self-contained modulo standard results, and the product-type characterization in Theorem 5.2 is a genuine structural contribution. The toric criterion in Theorem 10.6 is also an interesting tool that could be useful beyond this paper. The high-dimensional reduction is conditional on the authors' earlier theorems on int-amplified endomorphisms and equivariant MMP, which are cited as published or to-appear; the main caveat is that the final step of the threefold proof contains a nontrivial gap in the preservation of Case TIR under an iterated composition.

major comments (2)
  1. [Section 10, proof of Theorem 1.11 (last paragraph)] The exclusion of Case TIR3 depends on replacing f by f^k∘I and asserting that ``fr still satisfies Case TIR3 (cf. [28, Theorem 1.4])''. This step is not justified. Case TIR is defined in this paper, and [28, Theorem 1.4] states only the existence of a finite-index submonoid equivariance for an MMP; it does not address conditions (A1)–(A4) for the composed map. In particular, condition (A2) for the descended map requires (f_r^k∘I_r)^*D = δ_{f_r^k∘I_r}D. Since f_r^*D = δ_{f_r}D and I_r^*D = λD for some integer λ>1, the displayed equality would force λ = δ_{I_r}. But δ_{I_r} is the spectral radius of I_r^* on N^1(X_r), and there is no reason that the eigenvalue on the extremal ray spanned by D attains this maximum; [28, Theorem 1.4] contains no statement to this effect. Without (A2), Proposition 10.7 and Theorem 10.6 cannot be applied to the modified map, so the contradiction proving Theorem 1.11 is not established. A direct verification of (A1)–(A4) for the iterated composition, or a different argument showing that some int-amplified endomorphism in the monoid realizes its first dynamical degree on D, is needed.
  2. [Section 10, proof of Theorem 1.11 (first paragraph)] The reduction ``By Theorem 1.7, it suffices to show that fr := f|Xr does not satisfy Case TIR3'' is not explained and appears insufficient. Theorem 1.7(2) requires KSC to hold for Case TIR for every f_i : Xi -> Xi appearing in any equivariant MMP starting from X, not only for the final model Xr. The proof rules out Case TIR3 only for fr. If some intermediate f_i (i<r) satisfied Case TIR3 with respect to a Fano contraction Xi -> P1 different from the chosen MMP's final contraction, the argument would not rule it out: Proposition 10.7 is stated only for a smooth rationally connected X, while Xi may be singular after birational MMP steps, and the manuscript does not prove that f_i being Case TIR3 forces fr to be Case TIR3. The reduction to fr must be justified, or the argument must be extended to all intermediate models.
minor comments (3)
  1. [Section 1 and 7] The phrase ``Q-Goreinstein'' in Theorem 1.5 and Proposition 1.6 is a typo for ``Q-Gorenstein''; it appears several times.
  2. [Section 10, proof of Theorem 10.6] The line ``h0(X, ˆΩ1_X(log D) = dim(X)'' is missing a closing parenthesis; it should read ``h0(X, ˆΩ1_X(log D)) = dim(X)''.
  3. [Section 2, Notation] The cones ``NE(X)'' and ``PE1(X)'' are both defined as the pseudo-effective cone; using one symbol would avoid redundancy.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: surface and threefold KSC proofs are derived from prior MMP and toric theorems, not from KSC itself.

full rationale

KSC is an external target, and the paper's unconditional surface and threefold claims are derived rather than assumed. The reduction machinery (Theorems 1.5, 1.7, 8.6, 9.2, 10.6, and 10.7) uses no fitted parameter and no quantity defined in terms of KSC; Case TIR is a list of geometric hypotheses (κ(X, -K_X)=0, f^*D=δ_f D, Supp R_f=D, a Fano contraction, and a dimensional inequality), and Theorem 1.7(2) is an explicit conditional/induction step, not a renaming of the conclusion. The authors' own prior work ([24], [27], [28]) is load-bearing, but it consists of published theorems with stated hypotheses (existence of an int-amplified endomorphism, klt/Q-factorial assumptions) that do not include the Kawaguchi-Silverman conjecture; those citations are therefore legitimate external support under the review rules. The skeptical point about the step 'Replacing f by f^k ◦ I for some k ≫ 1, we may assume f is also int-amplified and fr still satisfies Case TIR 3 (cf. [28, Theorem 1.4])' in the proof of Theorem 1.11 identifies a possible gap or an incomplete verification of preservation of conditions (A2) and (A4); it is not circularity, because Case TIR is not defined in terms of KSC and the cited theorem concerns equivariance of the MMP rather than the KSC equality. The paper also openly states the limitation 'We are not able to show the slope semistability for the general int-amplified case' before Proposition 10.5, confirming that the toric criterion is deliberately restricted rather than assumed. Consequently, no step reduces the target equality to its own input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters and no ad-hoc invented objects. The paper's own reduction introduces the conditional Case TIR as a stated hypothesis, not an entity. All other inputs are standard algebraic geometry results cited from the literature.

assumptions (6)
  • domain assumption Algebraically closed base field of characteristic zero
    Stated at the start of the introduction; the proof uses the MMP and Hodge index theorem over such fields.
  • standard math Wahl's theorem: a normal projective surface with a non-isomorphic surjective endomorphism has log canonical singularities
    Cited as [40, Theorem 2.8] and used throughout Section 5.
  • standard math Existence and termination of the equivariant MMP for Q-factorial klt varieties admitting an int-amplified endomorphism
    Cited as [28, Theorems 1.1 and 1.2], used in Theorem 1.7 and Proposition 10.7.
  • standard math Surjectivity of the Albanese morphism for varieties admitting an int-amplified endomorphism
    Cited as [24, Theorem 1.8], used in Proposition 9.2 and Theorem 1.7.
  • domain assumption KSC holds for Case TIR in the equivariant MMP outputs (hypothesis in Theorem 1.7(2))
    Explicit condition in the reduction theorem, not known unconditionally.
  • standard math Perron-Frobenius type theorem for cone-preserving linear maps
    Cited as [2] and used in Lemma 9.1 and Proposition 9.2.

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Pith. "Pith review of Kawaguchi-Silverman conjecture for certain surjective endomorphisms." pith.science (2026). https://pith.science/paper/F32RJ474

@misc{pith2026190801605,
  author       = {Pith},
  title        = {Pith review of: Kawaguchi-Silverman conjecture for certain surjective endomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F32RJ474}},
  note         = {Machine review of arXiv:1908.01605}
}
abstract

We prove the Kawaguchi-Silverman conjecture (KSC), about the equality of arithmetic degree and dynamical degree, for every surjective endomorphism of any (possibly singular) projective surface. In high dimensions, we show that KSC holds for every surjective endomorphism of any $\mathbb{Q}$-factorial Kawamata log terminal projective variety admitting an int-amplified endomorphism, provided that KSC holds for any surjective endomorphism with the ramification divisor being totally invariant and irreducible. In particular, we show that KSC holds for every surjective endomorphism of any rationally connected smooth projective threefold admitting an int-amplified endomorphism. The main ingredients are the equivariant minimal model program, the effectiveness of the anti-canonical divisor and a characterization of toric pairs.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

    math.AG 2019-08 conditional novelty 7.0 of 10

    Every surjective self-map of a smooth rationally connected projective variety with an int-amplified endomorphism satisfies the Kawaguchi-Silverman conjecture.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    Birkar, P

    C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 23(2):405-468, 2010

    work page 2010

  2. [2]

    Birkhoff, Linear transformations with invariant cones, Amer

    G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274-276

    work page 1967

  3. [3]

    Brown, J

    M. Brown, J. McKernan, R. Svaldi and H. Zong, A geometric char acterisation of toric varieties, Duke Math. J., Volume 167, Number 5 (2018), 923-968

    work page 2018

  4. [4]

    Campana, Connexit´ e rationnelle des vari´ et´ es de Fano, Ann

    F. Campana, Connexit´ e rationnelle des vari´ et´ es de Fano, Ann. Sci. ´Ecole Norm. Sup. (4) 25 (1992), no. 5, 539-545

    work page 1992

  5. [5]

    Cascini, S

    P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of n ormal projective three- folds in arbitrary characteristic, Math. Ann. (to appear), DOI: 10.1007/s00208-019-01877-6 , arxiv:1710.01903

    arXiv

  6. [6]

    Debarre, Higher-dimensional algebraic geometry, Universite xt

    O. Debarre, Higher-dimensional algebraic geometry, Universite xt. Springer-Verlag, New York, 2001

    work page 2001

  7. [7]

    T. -C. Dinh and N. Sibony, Groupes commutatifs d’automorphisme s d’une vari´ et´ e k¨ ahl´ erienne com- pacte, Duke Math. J. 123 (2004), no. 2, 311-328. KA W AGUCHI-SIL VERMAN CONJECTURE 33

    work page 2004

  8. [8]

    T. -C. Dinh and N. Sibony, Equidistribution problems in complex dyna mics of higher dimension, Internat. J. Math. 28 (2017), no. 7, 1750057, 31 pp

    work page 2017

Show all 43 references
  1. [9]

    Fakhruddin, Questions on self-maps of algebraic varieties, J

    N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc., 18(2):109- 122, 2003

    2003

  2. [10]

    Fujino, Fundamental theorems for the log minimal model pro gram, Publ

    O. Fujino, Fundamental theorems for the log minimal model pro gram, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727-789

    2011

  3. [11]

    Fujino, Minimal model theory for log surfaces, Publ

    O. Fujino, Minimal model theory for log surfaces, Publ. Res. In st. Math. Sci. 48 (2012), no. 2, 339-371

    2012

  4. [12]

    D. Greb, S. Kebekus and T. Peternell, ´Etale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties, Duke Math. J. 165 (2016), no. 10, 1965-2004

    2016

  5. [13]

    J. M. Hwang and N. Nakayama, On endomorphisms of Fano manifo lds of Picard number one, Pure Appl. Math. Q, 7(4), pp.1407-1426, 2011

    2011

  6. [14]

    Kawaguchi and J

    S. Kawaguchi and J. Silverman, Examples of dynamical degree e quals arithmetic degree, Michigan Math. J. 63(2014), no. 1, 41-63

    2014

  7. [15]

    Kawaguchi and J

    S. Kawaguchi and J. Silverman, Dynamical canonical heights fo r Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties, Trans. Am er. Math. Soc. 368 (2016), no. 7, 5009-5035

    2016

  8. [16]

    Kawaguchi and J

    S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21-48

    2016

  9. [17]

    Koll´ ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete

    J. Koll´ ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete

  10. [18]

    A Series of Modern Surveys in Mathematics 32

    Folge. A Series of Modern Surveys in Mathematics 32. Springer-Verlag, Berlin, 1996

    1996

  11. [19]

    Koll´ ar, Y

    J. Koll´ ar, Y. Miyaoka and S. Mori, Rationally connected varieties , J. Alg. Geom. 1 (1992), 429-448

    1992

  12. [20]

    Koll´ ar and S

    J. Koll´ ar and S. Mori, Birational geometry of algebraic varieties , Cambridge Tracts in Math., 134 Cambridge Univ. Press, 1998

    1998

  13. [21]

    Lesieutre and M

    J. Lesieutre and M. Satriano, Canonical heights on hyper-K¨ a hler varieties and the Kawaguchi- Silverman conjecture, arxiv.org:1802.07388

    arXiv

  14. [22]

    Matsuzawa, Kawaguchi-Silverman conjecture for endomor phisms on several classes of varieties, arxiv.org:1902.06072

    Y. Matsuzawa, Kawaguchi-Silverman conjecture for endomor phisms on several classes of varieties, arxiv.org:1902.06072

    1902 arXiv

  15. [23]

    Matsuzawa, K

    Y. Matsuzawa, K. Sano and T. Shibata, Arithmetic degrees and dynamical degrees of endomor- phisms on surfaces, Algebra Number Theory 12 (2018), no. 7, 1635-1657

    2018

  16. [24]

    Matsuzawa and S

    Y. Matsuzawa and S. Yoshikawa, int-amplified endomorphisms on normal projective surfaces, arxiv.org:1902.06071

    1902 arXiv

  17. [25]

    Meng, Building blocks of amplified endomorphisms of normal pro jective varieties, Math

    S. Meng, Building blocks of amplified endomorphisms of normal pro jective varieties, Math. Z. (to appear), DOI: 10.1007/s00209-019-02316-7 , arxiv.org:1712.08995

    arXiv

  18. [26]

    Meng, On endomorphisms of projective varieties with numeric ally trivial canonical divisors, arxiv.org:1901.07089

    S. Meng, On endomorphisms of projective varieties with numeric ally trivial canonical divisors, arxiv.org:1901.07089

    1901 arXiv

  19. [27]

    Meng and D

    S. Meng and D. -Q. Zhang, Building blocks of polarized endomorph isms of normal projective vari- eties, Adv. Math. 325 (2018), 243-273

    2018

  20. [28]

    Meng and D.-Q

    S. Meng and D.-Q. Zhang, Characterizations of toric varieties v ia polarized endomorphisms, Math. Z. 292 (2019), no. 3-4, 1223-1231

    2019

  21. [29]

    Meng and D

    S. Meng and D. -Q. Zhang, Semi-group structure of all endomo rphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. (to appear) , 2019, arxiv:1806.05828. 34 SHENG MENG, DE-QI ZHANG

    2019 arXiv

  22. [30]

    Meng and D

    S. Meng and D. -Q. Zhang, Normal projective varieties admittin g polarized or int-amplified endo- morphisms, Acta Math. Vietnam. (to appear), 2019, arxiv:1806.07747

    2019 arXiv

  23. [31]

    Mori, Flip theorem and the existence of minimal models for 3-fo lds, J

    S. Mori, Flip theorem and the existence of minimal models for 3-fo lds, J. Amer. Math. Soc. 1 (1988), no. 1, 117-253

    1988

  24. [32]

    Nakayama, Ruled surfaces with non-trivial surjective endo morphisms, Kyushu J

    N. Nakayama, Ruled surfaces with non-trivial surjective endo morphisms, Kyushu J. Math. 56 (2002), no. 2, 433-446

    2002

  25. [33]

    Nakayama, On complex normal projective surfaces admittin g non-isomorphic surjective endo- morphisms, Preprint 2 September 2008

    N. Nakayama, On complex normal projective surfaces admittin g non-isomorphic surjective endo- morphisms, Preprint 2 September 2008

    2008

  26. [34]

    Nakayama, Intersection sheaves over normal schemes, J

    N. Nakayama, Intersection sheaves over normal schemes, J . Math. Soc. Japan 62 (2010), no. 2, 487-595; also RIMS1614

    2010

  27. [35]

    Nakayama and D.-Q

    N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of comp lex normal varieties, Math. Ann. 346 (2010), no. 4, 991-1018

    2010

  28. [36]

    Sano, Dynamical degree and arithmetic degree of endomorp hisms on product varieties, arxiv:1604.04174

    K. Sano, Dynamical degree and arithmetic degree of endomorp hisms on product varieties, arxiv:1604.04174

    arXiv

  29. [37]

    Seidenberg, The hyperplane sections of normal varieties, T rans

    A. Seidenberg, The hyperplane sections of normal varieties, T rans. Amer. Math. Soc., 69 (1950), pp. 357-386

    1950

  30. [38]

    Shokurov, 3-fold log models, Algebraic geometry, 4

    V. Shokurov, 3-fold log models, Algebraic geometry, 4. J. Math . Sci. 81 (1996), no. 3, 2667-2699

    1996

  31. [39]

    Silverman, Arithmetic and dynamical degrees on abelian variet ies, J

    J. Silverman, Arithmetic and dynamical degrees on abelian variet ies, J. Th´ eor. Nombres Bordeaux 29 (2017), no. 1, 151-167

    2017

  32. [40]

    Ueno, Classification theory of algebraic varieties and compac t complex spaces, Lecture Notes in Mathematics, Vol

    K. Ueno, Classification theory of algebraic varieties and compac t complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin, 1975, Notes written in collaboration with P . Cherenack

    1975

  33. [41]

    Wahl, A characteristic number for links of surface singularitie s, J

    J. Wahl, A characteristic number for links of surface singularitie s, J. Amer. Math. Soc. 3 (1990), no. 3, 625-637

    1990

  34. [42]

    Zhang, Polarized endomorphisms of uniruled varieties, Com pos

    D.-Q. Zhang, Polarized endomorphisms of uniruled varieties, Com pos. Math. 146 (2010), no. 1, 145-168

    2010

  35. [43]

    Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans

    D.-Q. Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8849-8872. Max-Planck-Institut f ¨ur Mathematik, Vivatsgasse 7, Bonn 53111, Germany; and Korea Institute for Advanced Study, Seoul 130...

    2016

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