REVIEW 4 major objections 4 minor 9 references
On endomorphisms of varieties which are injective on open subsets
T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper proves the Miyanishi conjecture for threefolds meeting the hypotheses of a recent theorem, and for open subsets of projective varieties that admit a sequence of divisorial contractions to a model whose birational automorphism…
desk verdict New cases of Miyanishi's conjecture, with sound proofs and only minor presentational terseness; deserves serious review. 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 object is the resolution of indeterminacy $(\tilde{X},p,q)$ of the birational map $\varphi$, a smooth projective variety $\tilde{X}$ with birational morphisms $p:\tilde{X}\to\bar{X}$ and $q:\tilde{X}\to\bar{X}$ satisfying $q=\varphi\circ p$. A divisorial contraction is a birational morphism whose exceptional locus is a prime divisor; the proof watches how the exceptional prime divisors of the contractions $\bar{X}\to X_n$ move under $\varphi$, and by the rigid-model assumption these divisors are permuted. Comparing the pullbacks of $K_{\bar{X}}$ and $K_{\bar{X}}+E$ by $p$ and $q$ in the Néron–Severi group (the group of numerical divisor classes), and using discrepancy coefficients $a(\cdot,\bar{X})$ together with the monotonicity inequality of [8, Lemma 2.27], the proof shows that a curve contracted by $\varphi$ has non-negative intersection with $K_{\bar{X}}$ and non-positive intersection with each relevant $E$. This intersection control is exactly what allows the curve to be pushed through each divisorial contraction while preserving the inequalities.
What would settle it
Find a Q-factorial normal projective variety with canonical singularities, a sequence of divisorial contractions to a model with $\operatorname{Bir}(X_n)=\operatorname{Aut}(X_n)$, an open subset, and an endomorphism that is injective off a codimension-two set but not an automorphism; such an example would directly refute Theorem 1.7. More locally, compute the two log discrepancies in Proposition 3.6 for a boundary divisor $E$ disjoint from the open subset and look for a reversal of the inequality $a(p^{-1}_*D_j,X,E)\le a(p^{-1}_*D_j,X)$; a single reversal would break the proof.
Extended reading notes
Core claim
The central claim is Theorem 1.7: if an open subset $X$ of a Q-factorial normal projective variety $\bar{X}$ has canonical singularities, and $\bar{X}$ admits a sequence of divisorial contractions $\bar{X}=X_0\to X_1\to\cdots\to X_n$ to a model $X_n$ with $\operatorname{Bir}(X_n)=\operatorname{Aut}(X_n)$, then the Miyanishi conjecture holds for $X$. In particular, when $X_n$ is the canonical model of $\bar{X}$ or a birationally superrigid Mori fiber space, every endomorphism of $X$ that is injective off a closed subset of codimension at least two is an automorphism. The proof assumes such an endomorphism is not an automorphism, uses the birationality of injective dominant maps to resolve it as a pair of birational morphisms, and derives that the exceptional divisors of the divisorial contractions are permuted by $\varphi$. It then follows that a curve contracted by the endomorphism survives through every contraction with non-negative intersection against the canonical divisor and non-positive intersection against each relevant exceptional divisor, so it is still contracted in the rigid model $X_n$, contradicting $\operatorname{Bir}(X_n)=\operatorname{Aut}(X_n)$. Before this, Theorem 1.4 shows the exceptional set can be replaced by one of codimension at least three, which yields the threefold statement (Corollary 1.5) and the low-dimensional-singularity statement (Theorem 1.6).
Load-bearing premise
The argument stands on a technical monotonicity property: adding a divisor as a boundary should not increase the log discrepancy (a measure of singularities along exceptional divisors) of any exceptional divisor, even when that boundary divisor lies outside the open subset. The proof cites [8, Lemma 2.27] for this; if the property fails for the singular varieties allowed in Theorem 1.7, the estimate that contracted curves have non-positive intersection with the contraction's exceptional divisor would collapse and the whole contraction argument would not go through.
Editorial extensions
If this is right
- For every threefold satisfying any of the four conditions in Theorem 1.3, the Miyanishi conjecture holds: injective off a codimension-at-least-two set implies automorphism.
- Under the same conditions, a codimension-two exceptional set can always be replaced by one of codimension at least three, so the map is an isomorphism outside a smaller set.
- If the singular locus has dimension at most one, then the corresponding endomorphisms are automorphisms (Theorem 1.6) under surjectivity, Q-factoriality, local complete intersection, or codimension-at-least-three exceptional set.
- Any open subset of a projective variety that divisorial-contracts to a model with $\operatorname{Bir}(X_n)=\operatorname{Aut}(X_n)$ satisfies the conjecture; this includes open subsets of canonical models and of birationally superrigid Mori fiber spaces.
- In all these cases, the endomorphism is birational with finite fibers, so by Zariski's main theorem the automorphism conclusion follows once the exceptional set is small enough.
Reading between the lines
- Extending the argument, the codimension-reduction step suggests an induction-on-dimension strategy for the full conjecture: once an exceptional set of codimension at least three is reached, endomorphisms have finite fibers away from a small set, so the main difficulty is pushing the automorphism property back from a rigid model.
- A concrete testable extension is to verify the log-discrepancy monotonicity of [8, Lemma 2.27] for boundary divisors disjoint from the open subset; if it holds for all canonical singularities, Theorem 1.7 would cover arbitrary sequences of divisorial contractions, not just those ending in canonical models or superrigid fiber spaces.
- The permutation action on the exceptional divisors of the divisorial contractions suggests that counterexamples, if any, would have to respect the birational-geometry chamber structure while still contracting a curve, which may constrain searches for counterexamples.
- For threefolds, the proof depends on the smooth-locus isomorphism theorem [3]; if surjectivity could be derived directly from injectivity off a codimension-two set, the conjecture would follow for all normal threefolds, not only those satisfying the four listed conditions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Miyanishi conjecture, which predicts that an endomorphism of a variety over an algebraically closed field of characteristic zero that is injective off a closed subset of codimension at least two must be an automorphism. Building on a theorem of Biswas and Das, the author proves that in the setting of that theorem one can enlarge the exceptional set so that the endomorphism is an isomorphism away from a subset of codimension at least three (Theorem 1.4). This yields the conjecture for threefolds satisfying the Biswas–Das conditions (Corollary 1.5) and for varieties whose singular locus has dimension at most one (Theorem 1.6). The second half of the paper develops an MMP-based argument: Theorem 1.7 proves the conjecture for any open subset of a Q-factorial normal projective variety with canonical singularities that admits a sequence of divisorial contractions to a variety whose birational automorphism group equals its automorphism group, in particular to a canonical model or a birationally superrigid Mori fiber space.
Significance. If the proofs are correct, these results constitute a meaningful advance on an open conjecture. The threefold cases and the higher-dimensional criterion via divisorial contractions are new, and the paper introduces an intersection-theoretic technique involving boundary divisors and log discrepancies that generalizes the author's previous work. The paper is not machine-checked, but it uses standard tools (Zariski's main theorem, resolutions, discrepancy formulas, intersection theory) in a plausible way. I view the main claims as defensible, but several key steps are too compressed and need to be filled in before the paper can be considered complete.
major comments (4)
- [§2.2, Lemma 2.1] The proof that after replacing Y by A we have φ(Y) ⊂ Sing(X) is not justified as written. The sentence 'Since φ|φ^{-1}(W) does not contract any divisor, we have φ(Y) ⊂ Sing(X)' is not a consequence of Theorem 2.2: a positive-dimensional fiber component through a point y ∈ Y mapping to a point of W need not be a divisor, and Theorem 2.2 only describes the image of the exceptional locus. A direct argument is needed: if y ∈ A and φ(y) ∈ W, then the positive-dimensional component of the fiber through y would meet the dense open subset W on which φ|_W is an isomorphism, contradicting its injectivity. Since Lemma 2.1 is load-bearing for Theorems 1.4–1.6, this step must be rewritten.
- [§3.2, Proposition 3.6] The inequality a(p^{-1}_*D_j, X, E) ≤ a(p^{-1}_*D_j, X) is cited to [8, Lemma 2.27] without checking its hypotheses. The pair (X, E) need not be log canonical, and p is not necessarily a log resolution of (X, E). The inequality is nevertheless true by the elementary formula a(E_i; X, E) = a(E_i; X) − mult_{E_i}(p^*E) ≤ a(E_i; X), since p^*E is effective. The manuscript should state this verification explicitly rather than leaving the reader to infer which form of the cited lemma applies.
- [§3.2, proof of Theorem 1.7] In the induction step, the case where E is not an element of EP_D(f)_X is dismissed with 'by Proposition 3.5 and Remark 3.7, we can show (K_{X_1}, C_1) ≥ 0'. This is a gap in a central induction. The intended argument appears to be that E ∩ X = ∅ forces (E, C) = 0, because the closure of a curve contained in X is disjoint from a closed divisor contained in X̄ \ X. If so, this should be stated and proved; as written, the reader cannot verify the induction step.
- [§2.4, proof of Theorem 1.6] The step 'this contradicts Z ⊂ Sing(X) since π|π^{-1}(W) : π^{-1}(W) → W is an isomorphism' is too terse. It relies on the fact that a resolution π of X that is an isomorphism over the smooth locus W must have the property that the image of the π-exceptional locus contains every irreducible component of Sing(X). This is standard, but it should be stated explicitly because it is essential to the contradiction.
minor comments (4)
- [Throughout §3] The symbol X is used both for the open variety and for its projective completion in Theorem 1.7 and Proposition 3.1. This makes statements such as 'X = X_0 → X_1 → ⋯ → X_n' confusing; please use a different notation for the completion, e.g., X̄.
- [§3.2, displayed equations in Propositions 3.5 and 3.6] In the formula for p^*K_X − q^*K_X, the second sum is written with q^{-1}_*D_i but the coefficient and the subsequent intersection estimates concern q^{-1}_*E_i. This appears to be a typo and should be corrected.
- [§2.2, Lemma 2.4] The expression '(π ∘ φ̃^{-1})(E_0)' should be written as π(φ̃^{-1}(E_0)) for clarity, since φ̃^{-1} maps subsets of X̃ to subsets of X̃, not to subsets of X.
- [§3.2, line after Lemma 3.4] The statement 'A and B is contained in {D_1, …, D_m}' is missing a plural verb; more importantly, the reason that every φ-exceptional divisor lies among the boundary components D_i should be made explicit: a divisor contracted by φ would intersect X and would then be contained in the codimension-two set Y, contradicting the defining assumption that the contracted locus has codimension at least two.
Circularity Check
No significant circularity: the only self-citations (Proposition 3.2 and Proposition 1.13 attributed to the author's own preprint [1]) are minor and non-load-bearing, since their proofs are reproduced in the text or are immediate from external references; the conditional MMP argument in Theorem 1.7 is self-contained and does not reduce to its inputs.
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other
[Section 3.1, Proposition 3.2. The statement is attributed to the author's prior paper [1]; the full proof follows in the same section.]
"Proposition 3.2 ([1, Theorem 1.3]) . Let X be a Q-factorial normal projective variety, and X be an open subset of X with codim(X \ X) ≥ 2. If the canonical divisor KX of X is ample or anti-ample, then Conjecture 1.2 holds for X."
This is a self-citation: [1] is the author's own preprint arXiv:2505.12416, so Proposition 3.2 is formally imported from the author's prior work rather than from an external theorem. However, the full proof is reproduced in the present text (resolution of indeterminacy, Lemma 1.9 showing that phi and phi^{-1} contract no divisor, discrepancy expansion, and the intersection-number contradiction), and the cited theorem's hypotheses (Q-factorial with K_X ample or anti-ample) do not include the target result. Similarly, Proposition 1.13 ([1, Proposition 1.11]) is a one-line consequence of the externally cited Remark 1.11 (Das) and Theorem 1.12 (Mumford's Zariski main theorem).
full rationale
The paper is a pure theorem-proof paper, so the empirical circularity patterns (fitted input called prediction, normalization-forced output) cannot apply: no parameter is fitted and nothing is predicted from data. Walking the derivation chain reveals no step equivalent to its inputs by construction. Section 2 (Theorem 1.4, Corollary 1.5, Theorem 1.6) rests on the external Biswas-Das Theorem 1.3, Kaliman's Lemmas 1.8-1.9, Das's birationality Lemma 1.10, and Mumford's Zariski main theorem; Lemmas 2.1 and 2.4 derive the Y-replacement and the codimension estimates without assuming Conjecture 1.2. Theorem 1.7 (with Xbar the Q-factorial normal projective closure and X ⊂ Xbar the open subset) is conditional on Bir(X_n) = Aut(X_n) and is proved by contradiction: a non-automorphism endomorphism yields a contracted curve C; Proposition 3.5 shows (K_Xbar, C) ≥ 0 from canonical singularities, and Proposition 3.6 shows (E, C) ≤ 0 from the discrepancy comparison p*(K_Xbar+E) - q*(K_Xbar+E) with the monotonicity a(p^{-1}_*D_j, X, E) ≤ a(p^{-1}_*D_j, X) cited to the external textbook Kollar-Mori [8, Lemma 2.27]; C is pushed through the contractions via (K_{X_{i+1}}, C_{i+1}) = (K_{X_i}, C_i) - a(E, X_i)(E, C) ≥ 0, and at X_n the assumption Bir(X_n) = Aut(X_n) contradicts the fact that phi contracts C. The conclusion is never assumed; the assumption only forces birational automorphisms of the model to be regular. The only self-citations are Propositions 1.13 and 3.2 attributed to the author's preprint [1]; both are proven in the text or immediate from external references, so they are minor and non-load-bearing. Separate concerns that are correctness risks rather than circularity: the log-discrepancy monotonicity of [8, Lemma 2.27] (if it failed for a boundary prime divisor E intersecting X but not contained in X, the estimate (E, C) ≤ 0 and the contraction step would collapse); an omitted argument in Section 3.2 for the case E not in EPD(f)_X ('we can show (K_{X_1}, C_1) ≥ 0'); and the abstract's formulation ('any open subset of a projective variety') being stronger than the Q-factorial and canonical-singularity hypotheses of Theorem 1.7. None of these reduces a claimed result to its inputs. Verdict: no circular step; score 2 reflects only the presence of minor, non-load-bearing self-citations.
Assumptions & free parameters
assumptions (7)
- standard math Work over an algebraically closed field k of characteristic zero; varieties are separated integral schemes of finite type over k.
- standard math Resolution of singularities exists for varieties over k (Hironaka).
- standard math Zariski's main theorem in its original form applies to birational morphisms with finite fibers onto normal varieties.
- domain assumption Minimal model program: existence of divisorial contractions and the theory of discrepancies and log discrepancies as in Kollár and Mori.
- domain assumption Bir(X_n) = Aut(X_n) in Theorem 1.7.
- domain assumption The endomorphism φ on the open subset X extends to a birational automorphism of the projective closure X-bar.
- standard math Intersection theory on Q-factorial varieties and computations in the Neron-Severi group.
Cite this review
Pith. "Pith review of On endomorphisms of varieties which are injective on open subsets." pith.science (2026). https://pith.science/paper/ZSNRPBBA
@misc{pith2026250521145,
author = {Pith},
title = {Pith review of: On endomorphisms of varieties which are injective on open subsets},
year = {2026},
howpublished = {\url{https://pith.science/paper/ZSNRPBBA}},
note = {Machine review of arXiv:2505.21145}
}
abstract
We consider conditions under which endomorphisms of varieties become automorphisms. For example, there is a remarkable theorem, called Ax-Grothendieck theorem, which states that any injective endomorphism of a variety is bijective. Over an algebraically closed field of characteristic zero, bijectivity of endomorphisms of varieties implies that the endomorphisms are automorphisms, thus Ax-Grothendieck theorem gives one of the conditions we considering. There is also a conjecture, called Miyanishi conjecture, which claims that for any endomorphism of a variety over an algebraically closed field of characteristic zero, if it is injective outside a closed subset of codimension at least $2$, then it is an automorphism. Recently, I. Biswas and N. Das prove that any endomorphism which satisfies the conditions of Miyanishi conjecture induces an automorphism of the singular locus of the variety with some conditions. In this paper, we prove that Miyanishi conjecture holds for any threefold which satisfies the conditions of I. Biswas and N. Das. For higher dimensional varieties, we also observe how divisorial contractions affect endomorphisms by using minimal model program theory. We can prove that Miyanishi conjecture holds for any open subset of a projective variety which has a sequence of divisorial contractions to the canonical model or a birationally superrigid Mori fiber space.
Reference graph
Works this paper leans on
-
[1]
On Miyanishi conjecture for quasi-projective varieties
T. Asano, “On Miyanishi conjecture for quasi-projective varieties” , arXiv:2505.12416v1
-
[2]
Injective endomorphisms of varieties and schemes
J. Ax, “Injective endomorphisms of varieties and schemes” , Pacific J. Math., Vol. 31, 1–7, 1969
work page 1969
-
[3]
On the injective self-maps of algebraic varieties
I. Biswas and N. Das, “On the injective self-maps of algebraic varieties” , to appear in J. Pure Appl. Algebra, Vol. 229, Issue 6, 2025
work page 2025
-
[4]
N. Das, “On endomorphisms of varieties” , International Mathematics Re- search Notices, Vol. 2022, No. 22, 17534–17545, 2022
work page 2022
-
[5]
Open problems in affine algebraic geome- try
G. Freudenburg and P. Russell, “Open problems in affine algebraic geome- try”, Contemporary Math., Vol. 369, 1–30, 2005
work page 2005
- [6]
-
[7]
S. Kaliman, “On a theorem of Ax” , Proc. Amer. Math. Soc., Vol. 133, 975–977, 2004
work page 2004
-
[8]
Birational Geometry of Varieties
J. Koll´ ar and S. Mori,“Birational Geometry of Varieties”, Cambridge Uni- versity Press, 1998
work page 1998
Show all 9 references
-
[9]
The Red Book of Varieties and Schemes (Second, Expanded Edition)
D. Mumford, “The Red Book of Varieties and Schemes (Second, Expanded Edition)”, Springer-Verlag, 1999. 11
1999
Reviewed August 7, 2026 · model on record in the stance chip above.
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