Sharp Bounds for Totally Invariant Cycles of Projective Varieties

Wentao Chang; Yujie Luo

REVIEW 2 major objections 5 minor 1 cited by

Sharp Bounds for Totally Invariant Cycles of Projective Varieties

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For any self-map of projective n-space of degree q>1, the number of totally-invariant codimension-r cycles is at most the binomial coefficient n+1 choose r, and this bound is sharp.

desk verdict Resolves Favre's question with a sharp P^2 bound and proves a clean binomial bound in all codimensions for P^n; the main proof is solid, with one load-bearing unproved characteristic-free lemma to fix. read the letter →

arxiv 2607.27738 v2 pith:TUAXY6LW submitted 2026-07-30 math.AG

Wentao Chang , Yujie Luo This is my paper · ORCID

classification math.AG MSC 14E0514C25

keywords int-amplifiedendomorphismtotallyinvariantsubvarietyprojectivespacedegreeboundsHilbertfunctiontracesplittingtotally-periodicpolynomialcompletelyramifiedpoints

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 establishes explicit, sharp upper bounds on how many subvarieties of a projective variety can be swept into themselves by an int-amplified endomorphism, i.e. a self-map whose pullback amplifies a line bundle. For projective n-space, it proves that the sum of degrees of all totally-invariant prime cycles of codimension r is at most binom(n+1,r), and that the number of such cycles is at most the same number; both are attained by the coordinate power map. The argument works in every codimension, for cycles that become invariant only after iteration, and over any algebraically closed field provided the map's degree is invertible. A corollary settles a question of Favre by showing that any endomorphism of the projective plane of degree at least two has at most three totally-invariant points.

What carries the argument

The paper's working object is the normalized trace of a finite locally free endomorphism. When a reduced subscheme Z satisfies (h^{-1}(Z))_red = Z, Lemma 2.1 splits the inclusion I_Z -> h_*I_Z by d^{-1}Tr_h, making H^i(X, I_Z ⊗ A) a direct summand of H^i(X, I_Z ⊗ h^*A). Combined with Lemma 2.2 (iterates of an int-amplified map make h^*L ⊗ L^{-1} ample) and uniform Fujita vanishing, this yields the vanishing H^i(X, I_Z ⊗ L^t) = 0 for all i,t>0 (Proposition 2.3). Successive very ample hyperplane sections then express the degree of Z as the length of a zero-dimensional scheme, bounded by the Hilbert function of a complete intersection, which in P^n evaluates to the binomial coefficient.

What would settle it

An endomorphism of P^n of degree q>1 invertible in k with total degree of its totally-invariant codimension-r cycles exceeding binom(n+1,r), or a smooth projective X with an int-amplified endomorphism of degree invertible in k where the Hilbert-function inequality in Theorem 1.1 fails, would refute the central claim. A simpler place to look: verify Lemma 2.2 in characteristic p>0 with p ∤ deg(f); if the ampleness of (f^s)^*L ⊗ L^{-1} fails for some large s, Proposition 2.3 collapses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: for an endomorphism f of P^n with f^*O(1) = O(q), q>1 invertible in the field, the set of totally-periodic prime cycles of codimension r is finite and the sum of their degrees is at most binom(n+1,r), so there are at most binom(n+1,r) of them. This bound is sharp, realized by the coordinate power map, whose coordinate linear subspaces are the extremal cycles. The same mechanism yields a general bound on any smooth projective variety X: for an ample line bundle L, the total L-degree of codimension-r totally-periodic cycles is controlled by a finite difference of the Hilbert function of L, given in Theorem 1.1. The proof uses a normalized trace to split the id

Load-bearing premise

The proof assumes that the cone-theoretic eigenvalue criterion for int-amplified endomorphisms, which the cited reference states in characteristic zero, remains valid over any algebraically closed field; the paper asserts this without proof.

Editorial extensions

If this is right

Favre's question for P^2 is answered: |T_∞(f)| ≤ 3 for every degree-q>1 endomorphism of the projective plane, with equality for the coordinate power map.
In every codimension 1 ≤ r ≤ n of P^n, the coordinate power map attains the bound, so the binomial constants cannot be improved; any endomorphism achieving the full binomial polynomial must, after an iterate, be projectively conjugate to that map.
For completely ramified points, when q ≥ n+1, the paper obtains the sharp bound n+1 and projective linear independence, improving the previous plane-curve bound of nine to three for cubic and higher degree maps of P^2.
The general Hilbert-function bound applies to all smooth projective varieties admitting int-amplified endomorphisms of degree invertible in k, giving degree and cardinality bounds that are uniform over all such endomorphisms once L and a are fixed.
Applications include: blow-ups of P^n along smooth codimension-two totally-invariant cycles are Fano; log canonical centers of invariant boundaries satisfy the same binomial bound; and the number of contractible extremal rays is linearly controlled.

Reading between the lines

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

If the paper's one-sentence characteristic-free assertion of the cone-theoretic criterion is correct, Theorem 1.1's bounds hold verbatim over every algebraically closed field of characteristic not dividing deg(f); checking a single int-amplified map over F_p whose degree is coprime to p would be a cheap test.
Remark 4.2 shows that the polynomial identity Φ = (1+t)^{n+1} does not characterize P^n as an underlying variety; a natural next step would be to classify all smooth projective X admitting an int-amplified f with maximal totally-periodic polynomial.
Because the proof of Proposition 2.3 only needs iterates to amplify L, the same Hilbert-function bound should apply to any self-map satisfying that amplification condition, not only to int-amplified endomorphisms.
The sharp bound on completely ramified points when q ≥ n+1 suggests that for low-degree maps (q < n+1) the linear-independence statement may fail; computing examples in degree 2 on P^2 would map the boundary of the sharp result.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

This paper deserves a serious referee. The headline result is the sharp bound |T∞(f)| ≤ 3 for endomorphisms of P^2, answering Favre's question, and more generally the binomial bound ≤ binom(n+1,r) for totally invariant codimension-r cycles in P^n. Theorem 1.1, a Hilbert-function bound for arbitrary smooth projective X, is new and cleanly proved via trace splitting, uniform Fujita vanishing, and a Koszul restriction. The toric refinement and the surjectivity result for evaluation maps are nice bonuses. The paper is honest: it cites prior work, and the bounds are derived, not fitted.

The main thing to fix is Lemma 2.2. The theorem is stated over any algebraically closed field, and the proof leans on Meng's cone-theoretic eigenvalue criterion, which is stated in characteristic zero. The paper says the proof is purely cone-theoretic and works in general, but gives no argument. That is load-bearing: if it fails, Theorem 1.1 and the P^n bound have no positive-characteristic proof. I suspect the extension is true—the criterion is about N^1 and eigenvalues, and a finite-dimensional cone argument should give it—but the authors need to spell that out or restrict to characteristic zero.

The reader flagged Proposition 1.5(4)'s exponent q^{ar} as an error. On reading, that is correct: for a codimension-r cycle, f^a pulls back to degree q^{ar}. So that concern disappears. There is a garbled projection formula in Corollary 5.6 that looks like a typo and should be rewritten. Proposition 1.5(4)'s equality characterization is plausible and the proof is dense but coherent.

One minor frustration: the paper packs in a lot of applications (regularity, log canonical thresholds, Fano blow-ups, extremal rays). Some of these read as sketches, and if the core is accepted, they should be checked separately. But they do not undermine the main result.

For a reading group: yes, the main theorem and its proof are worth studying. I would cite it if I worked in arithmetic dynamics or endomorphisms. Send it to peer review; the referee should insist on a proof or a characteristic-zero caveat for Lemma 2.2.

Referee Report

2 major / 5 minor

Summary. The paper proves explicit upper bounds on the total degree and number of totally-periodic prime cycles of codimension r for int-amplified endomorphisms of smooth projective varieties, under the assumption that deg(f) is invertible in the ground field. The main mechanism is a normalized trace splitting for ideal sheaves of totally periodic subschemes (Lemma 2.1), followed by uniform Fujita-type vanishing (Proposition 2.3) and a finite-difference/Hilbert-function computation. In projective space the bound becomes binom(n+1,r), with sharpness attained by the coordinate power map; this answers Favre's question for P^2 by giving |T_∞(f)|≤3. The paper also proves toric analogues, studies the totally-periodic polynomial, and derives applications to completely ramified points, log canonical centers, and contractible extremal rays.

Significance. If the proof is completed at the points noted below, the paper is a significant contribution. Theorem 1.1 gives a closed-form, parameter-free bound that is uniform over all int-amplified endomorphisms once L and a are fixed, and the projective-space specialization is sharp with an explicit extremal example. The method is transparent, uses standard intersection-theoretic and vanishing-theoretic tools, and reproduces known codimension-one results as special cases. The paper also gives falsifiable numerical predictions and explicit constructions, including the coordinate power map and the F_1 example in Remark 4.2. The positive-characteristic scope is an intended feature, so the missing justification for the cone-theoretic criterion is a genuine gap rather than a stylistic issue.

major comments (2)

[Section 2, Lemma 2.2] This lemma is the only bridge between the int-amplified hypothesis and the numerical eigenvalue statement used in Proposition 2.3. The cited criterion [26, Theorem 1.1 and Remark 1.2] is stated in characteristic zero, and the manuscript's one-sentence assertion that the proof is 'purely cone-theoretic' and works over any algebraically closed field is not a proof. Since Theorem 1.1 is stated over an arbitrary algebraically closed field and only deg(f) invertible is imposed, positive characteristic is explicitly in scope. If this characteristic-free extension cannot be justified, Proposition 2.3, and hence Theorem 1.1 and the P^n bounds, do not follow in positive characteristic. Please provide a self-contained proof or a citation to a characteristic-free statement.
[Section 4, proof of Proposition 1.5(4)] As printed, the proof uses the equality (f^a)_*V = q^{ar}V and defines the Chow morphism τ_{r,e,a} with target Chow_{n-r,q^{ar}e}. For an integral V of codimension r (dimension m=n-r) that is totally invariant, flat pushforward satisfies (f^a)_*[V] = q^{a m}[V] = q^{a(n-r)}[V], not q^{ar}[V]; for points (r=n) one gets 1·[V], not q^{an}[V]. The factor q^{ar} is the correct factor for flat pullback (f^a)^*[V]. Thus the displayed equality and the subsequent properness argument for C_{r,e,a} are invalid as written. If the intended operation is pullback, replace the pushforward stars with pullback stars throughout and check the parametrized family; if pushforward is intended, the degree factor and the target Chow degree must be changed. This does not affect Theorems 1.1–1.4, but it is load-bearing for the generic left-equality statement in Proposition 1.5(4).

minor comments (5)

[Section 4, Proposition 1.5(3)] The expression Φ_{X,f}(t)-t^{n+1} includes the constant term 1. This is correct for the product inequality in (3), but it is easy to misread; a parenthetical reminder that the constant term is retained would help.
[Section 3, Proposition 2.3 proof] The sentence 'The divisor h*H−H is a sum of pullbacks of the ample divisor f*H−H' is terse. Since h=f^b, the telescoping sum is immediate, but spelling it out would improve readability.
[Remark 3.2] For finite fields, 'q is invertible in k' means q is not divisible by p. Stating this explicitly would remove ambiguity in the example k=F_p.
[Section 5.3, Corollary 5.6] The identity 'projection formula gives c deg(f|_Z)=deg(f)' is stated without derivation. A one-line computation of the cycle classes involved would help the reader verify the contradiction.
[General] There are minor typographical issues, including inconsistent spacing in formulas such as T^r_∞(X,f) and Chow_{n-r,e}(P^n). These should be cleaned up during revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; the only flagged gap is an external-support issue in Lemma 2.2, not a circular reduction.

full rationale

I walked the proof chain: Theorem 1.3 uses Theorem 1.1; Theorem 1.1 is proved from Proposition 2.3; Proposition 2.3 is proved from Lemma 2.1 (trace splitting, proved locally) and Lemma 2.2. Lemma 2.2 invokes Meng's cone-theoretic eigenvalue criterion [26, Theorem 1.1 and Remark 1.2] and explicitly asserts, without proof, that the criterion extends from characteristic zero to any algebraically closed field. This is a missing verification / potential correctness gap, but it is not circularity: the cited criterion is an external theorem, not an input equivalent to the paper's conclusion, and the paper's derivation from it is explicit. The bound B_{X,L,a,r} is a computed alternating sum of Hilbert functions, not a fitted parameter, and no quantity from T^r_∞(f) is inserted into the bound. Sharpness is demonstrated by the coordinate power map, whose totally invariant linear subspaces are counted directly. I found no self-definitional step, no fitted-input-called-prediction, no load-bearing self-citation, and no renaming of a known result as a new derivation. The unsupported characteristic-free extension in Lemma 2.2 should be verified, but it does not make the argument circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central result rests on standard algebraic geometry facts and several external theorems in algebraic dynamics. No free parameters are fitted. The most fragile import is the unproved positive-characteristic extension of Meng's criterion. No new entities are postulated.

assumptions (7)

domain assumption Meng's cone-theoretic eigenvalue criterion for int-amplified endomorphisms holds over any algebraically closed field.
Invoked in Lemma 2.2 to obtain f^*L⊗L^{-1} ample after a power. The cited statement is characteristic zero; the paper asserts without proof that the proof is characteristic-free.
domain assumption Fujita's uniform vanishing theorem (Keeler's form) gives H^i(X,F⊗L^m⊗N)=0 for m>>0 and N nef.
Used in Proposition 2.3 to vanish the high cohomology of I_Z twisted by (h^s)^*L^t. Cited to [15] and [22].
domain assumption Fakhruddin's theorem [12, Theorem 1.2] that a generic degree-q endomorphism of P^n has no positive-dimensional totally invariant subvarieties.
Used in Proposition 1.5(4) to prove the image of C_{r,e,a} is proper for r<n.
domain assumption Broustet-Höring [3, Theorem 1.4 and Lemma 2.10] on non-log-canonical loci and log canonical centers of invariant boundaries.
Used in Corollary 5.6.
domain assumption Meng-Zhang [27, Lemma 2.6] states that for an int-amplified endomorphism, a proper totally-invariant subvariety has pushforward c[Z] with c>1.
Used in Corollary 5.6.
standard math Chow scheme representability and flat pullback properties (Kollár, Stacks Project).
Used to define morphisms on Chow schemes in Proposition 1.5(4).
standard math Miracle flatness: a finite morphism from a smooth variety is locally free.
Used in Lemma 2.1 and Proposition 2.3 to obtain rank d = deg(f).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp Bounds for Totally Invariant Cycles of Projective Varieties." pith.science (2026). https://pith.science/paper/TUAXY6LW

@misc{pith2026260727738,
  author       = {Pith},
  title        = {Pith review of: Sharp Bounds for Totally Invariant Cycles of Projective Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUAXY6LW}},
  note         = {Machine review of arXiv:2607.27738}
}

abstract

Let $X$ be a smooth projective variety, $f:X\to X$ an int-amplified endomorphism, and $L$ any ample line bundle on $X$. We prove that, in every codimension, the total degree of prime cycles that become totally invariant under an iterate of $f$ satisfies an explicit Hilbert-function bound on their total $L$-degree. In particular, when $X=\mathbf{P}^n$, the number of totally invariant prime $(n-r)$-cycle is bounded by $\binom{n+1}{r}$, and this bound is optimal.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

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

Log Calabi--Yau structure for endomorphisms on $\mathbf{P}^n$
math.AG 2026-08 conditional novelty 7.0 of 10

For every q-polarized endomorphism of P^2, some iterate (P^2, R_{f^s}/(q^s-1)) is log canonical, settling Gongyo's conjecture for smooth projective surfaces.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · cited by 1 Pith paper

[1]

Amerik and F

E. Amerik and F. Campana,Exceptional points of an endomorphism of the projective plane, Math. Z.249 (2005), no. 4, 741–754

work page 2005
[2]

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(2010), no. 2, 405–468

work page 2010
[3]

Broustet and A

A. Broustet and A. Höring,Singularities of varieties admitting an endomorphism, Math. Ann.360(2014), no. 1–2, 439–456

work page 2014
[4]

Cerveau and A

D. Cerveau and A. Lins Neto,Hypersurfaces exceptionnelles des endomorphismes deCP(n), Bol. Soc. Brasil. Mat. (N.S.)31(2000), no. 2, 155–161

work page 2000
[5]

D. A. Cox, J. B. Little, and H. K. Schenck,Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011

work page 2011
[6]

de Fernex, L

T. de Fernex, L. Ein, and M. Mustaţă,Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett.10(2003), no. 2–3, 219–236

work page 2003
[7]

Dinh and N

T.-C. Dinh and N. Sibony,Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9)82 (2003), no. 4, 367–423

work page 2003
[8]

Dinh and N

T.-C. Dinh and N. Sibony,Equidistribution towards the Green current for holomorphic maps, Ann. Sci. Éc. Norm. Supér. (4)41(2008), no. 2, 307–336

work page 2008

Show all 34 references

[9]

Dinh and N

T.-C. Dinh and N. Sibony,Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math.203(2009), no. 1, 1–82

2009
[10]

Dinh and N

T.-C. Dinh and N. Sibony,Equidistribution speed for endomorphisms of projective spaces, Math. Ann.347 (2010), no. 3, 613–626

2010
[11]

Fakhruddin,Questions on self maps of algebraic varieties, J

N. Fakhruddin,Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc.18(2003), no. 2, 109–122

2003
[12]

Fakhruddin,The algebraic dynamics of generic endomorphisms ofPn, Algebra Number Theory8(2014), no

N. Fakhruddin,The algebraic dynamics of generic endomorphisms ofPn, Algebra Number Theory8(2014), no. 3, 587–608. SHARP BOUNDS FOR TOTALLY INV ARIANT CYCLES OF PROJECTIVE V ARIETIES 19

2014
[13]

Favre,Equidistribution problems in holomorphic dynamics inP2, inDynamical systems

C. Favre,Equidistribution problems in holomorphic dynamics inP2, inDynamical systems. Part II, Publ. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, pp. 79–111

2003
[14]

Favre and M

C. Favre and M. Jonsson,Brolin’s theorem for curves in two complex dimensions, Ann. Inst. Fourier (Grenoble) 53(2003), no. 5, 1461–1501

2003
[15]

Fujita,Vanishing theorems for semipositive line bundles, inAlgebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol

T. Fujita,Vanishing theorems for semipositive line bundles, inAlgebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol. 1016, Springer, Berlin, 1983, pp. 519–528

1982
[16]

Fulton,Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol

W. Fulton,Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2, Springer-Verlag, Berlin, 1998

1998
[17]

J. E. Fornæss and N. Sibony,Complex dynamics in higher dimension. I, Astérisque222(1994), 201–231

1994
[18]

Hartshorne,Algebraic geometry, Graduate Texts in Mathematics, vol

R. Hartshorne,Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York– Heidelberg, 1977

1977
[19]

Höring,Totally invariant divisors of endomorphisms of projective spaces, Manuscripta Math.153(2017), no

A. Höring,Totally invariant divisors of endomorphisms of projective spaces, Manuscripta Math.153(2017), no. 1–2, 173–182

2017
[20]

Hu and S

Y. Hu and S. Keel,Mori dream spaces and GIT, Michigan Math. J.48(2000), 331–348

2000
[21]

Kawakami and B

T. Kawakami and B. Totaro,Endomorphisms of varieties and Bott vanishing, J. Algebraic Geom.34(2025), no. 2, 381–405

2025
[22]

D. S. Keeler,Ample filters of invertible sheaves, J. Algebra259(2003), no. 1, 243–283; corrigendum, J. Algebra507(2018), 592–598

2003
[23]

Kollár,Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol

J. Kollár,Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32, Springer-Verlag, Berlin, 1996

1996
[24]

Lazarsfeld,Positivity in algebraic geometry

R. Lazarsfeld,Positivity in algebraic geometry. I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 48, Springer-Verlag, Berlin, 2004

2004
[25]

Mabed,Totally invariant divisors of non trivial endomorphisms of the projective space, Geom

Y. Mabed,Totally invariant divisors of non trivial endomorphisms of the projective space, Geom. Dedicata 217(2023), no. 5, Paper No. 79, 12 pp

2023
[26]

Meng,Building blocks of amplified endomorphisms of normal projective varieties, Math

S. Meng,Building blocks of amplified endomorphisms of normal projective varieties, Math. Z.294(2020), no. 3–4, 1727–1747

2020
[27]

Meng and D.-Q

S. Meng and D.-Q. Zhang,Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett.27(2020), no. 2, 523–549

2020
[28]

Milnor,Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol

J. Milnor,Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006

2006
[29]

Mumford,Abelian varieties, 2nd ed., with appendices by C

D. Mumford,Abelian varieties, 2nd ed., with appendices by C. P. Ramanujam and Y. Manin, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Oxford University Press, London, 1974

1974
[30]

Nakayama and D.-Q

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

2010
[31]

The Stacks Project Authors,The Stacks Project,https://stacks.math.columbia.edu
[32]

Yoshikawa,Structure of Fano fibrations of varieties admitting an int-amplified endomorphism, Adv

S. Yoshikawa,Structure of Fano fibrations of varieties admitting an int-amplified endomorphism, Adv. Math. 391(2021), Paper No. 107964

2021
[33]

Zhang,Invariant hypersurfaces of endomorphisms of projective varieties, Adv

D.-Q. Zhang,Invariant hypersurfaces of endomorphisms of projective varieties, Adv. Math.252(2014), 185–203

2014
[34]

Zhong,Totally invariant divisors of int-amplified endomorphisms of normal projective varieties, J

G. Zhong,Totally invariant divisors of int-amplified endomorphisms of normal projective varieties, J. Geom. Anal.31(2021), no. 3, 2568–2593. 20 WENTAO CHANG AND YUJIE LUO School of Mathematical Sciences, Fudan University, People’s Republic of China Email address:wtchang21@m.fu...

2021

Pith tools

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

[1] [1]

Amerik and F

E. Amerik and F. Campana,Exceptional points of an endomorphism of the projective plane, Math. Z.249 (2005), no. 4, 741–754

work page 2005

[2] [2]

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(2010), no. 2, 405–468

work page 2010

[3] [3]

Broustet and A

A. Broustet and A. Höring,Singularities of varieties admitting an endomorphism, Math. Ann.360(2014), no. 1–2, 439–456

work page 2014

[4] [4]

Cerveau and A

D. Cerveau and A. Lins Neto,Hypersurfaces exceptionnelles des endomorphismes deCP(n), Bol. Soc. Brasil. Mat. (N.S.)31(2000), no. 2, 155–161

work page 2000

[5] [5]

D. A. Cox, J. B. Little, and H. K. Schenck,Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011

work page 2011

[6] [6]

de Fernex, L

T. de Fernex, L. Ein, and M. Mustaţă,Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett.10(2003), no. 2–3, 219–236

work page 2003

[7] [7]

Dinh and N

T.-C. Dinh and N. Sibony,Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9)82 (2003), no. 4, 367–423

work page 2003

[8] [8]

Dinh and N

T.-C. Dinh and N. Sibony,Equidistribution towards the Green current for holomorphic maps, Ann. Sci. Éc. Norm. Supér. (4)41(2008), no. 2, 307–336

work page 2008