Images of toric variety and amplified endomorphism of weak Fano threefolds

Supravat Sarkar

REVIEW 5 major objections 3 minor 30 references

Images of toric variety and amplified endomorphism of weak Fano threefolds

T0 review · 5 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A smooth projective threefold of Picard rank 2 with nef anticanonical bundle that is the image of a projective toric variety is itself toric, with exactly three non-Fano exceptions.

desk verdict The Bott-vanishing formulas in Theorem 3.1 are the real contribution; the classification theorems are credible but rest on unshown table numerics in Corollary 3.2, so the paper needs referee checks rather than desk reject. read the letter →

arxiv 2506.16325 v3 pith:CGWWAUGQ submitted 2025-06-19 math.AG

Supravat Sarkar This is my paper · ORCID

classification math.AG MSC 14M2514E20

keywords toricimageweakFanothreefoldPicardrank2Bottvanishingint-amplifiedendomorphismvarietydelPezzofibrationconicbundle

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 proves a dichotomy for smooth projective threefolds of Picard rank 2 whose anticanonical bundle is nef and big (the weak Fano condition): if such a variety is the image of a projective toric variety, or if it carries an int-amplified endomorphism, then it is toric, with the same three explicit exceptions in both cases. An int-amplified endomorphism is a finite surjective self-map that pulls some ample divisor back to a divisor whose difference from the original is again ample. The proof eliminates every other family in the known classification by showing it fails Bott vanishing, a cohomology vanishing property that toric images and varieties with such endomorphisms must satisfy. These results are special cases of two open conjectures previously settled for surfaces and for Fano threefolds. If correct, the paper gives the full list of rank-2 toric weak Fano threefolds that are not Fano.

What carries the argument

The workhorse is Bott vanishing, the cohomology vanishing property that toric varieties and their images satisfy and that varieties with an int-amplified endomorphism inherit. In the form used here, it forces $\chi(X,\Omega_X^2\otimes L)\ge 0$ for every ample line bundle $L$. The paper proves three Hirzebruch–Riemann–Roch formulas (Theorem 3.1) that turn Bott vanishing into a numerical test for rank-$2$ weak Fano threefolds, for example $$-\chi(X,\$Omega_X^{2}$(H-K_X))=16+h-\frac{$c_1^{3}$}{2}-\frac54($c_1^{2}$H+$c_1H^{2}$)+\frac34c_2H-\$frac12H^{3}$.$$ Substituting the Chern numbers of each family from the classification tables into these formulas, Corollary 3.2 shows the test excludes every family except the three listed bundles.

What would settle it

A concrete way to disprove the central claim would be to exhibit a smooth projective weak Fano threefold $X$ with $\rho(X)=2$, not Fano and not one of the three listed projective bundles, for which Bott vanishing holds, or equivalently for which the formula of Theorem 3.1(1) gives a nonnegative value after substituting the true Chern numbers. If any entry of the cited classification tables has a wrong Chern number, re-computation would reveal it as a surviving case.

Watch

Extended reading notes

Core claim

The paper establishes that the only smooth projective weak Fano threefolds of Picard rank $2$ that can be images of projective toric varieties, and the only ones that can carry an int-amplified endomorphism, are the same three varieties: the projective bundles $\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(1)^2)$, $\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}^2\oplus\mathcal{O}_{\mathbb{P}^1}(2))$, and $\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. Every other rank-$2$ weak Fano threefold fails Bott vanishing, an obstruction that toric images and varieties with an int-amplified endomorphism cannot have.

Load-bearing premise

The argument depends on the published lists of all weak Fano threefolds of Picard rank 2 being complete and numerically accurate; if a list misses a family or contains a wrong Chern number, an excluded case could survive the Bott-vanishing test and the conclusions would fail.

Editorial extensions

If this is right

The three varieties $\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(1)^2)$, $\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}^2\oplus\mathcal{O}_{\mathbb{P}^1}(2))$, and $\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$ are the complete classification of rank-2 toric weak Fano threefolds that are not Fano.
Any rank-2 weak Fano threefold with an int-amplified endomorphism is one of those three bundles, hence toric, settling the amplified-endomorphism conjecture in this class.
The families covered by Corollary 3.2—degree 6 and degree 8 del Pezzo fibrations over $\mathbb{P}^1$, most conic bundles over $\mathbb{P}^2$, and the birational cases of parts (4)–(6)—are all shown not to satisfy Bott vanishing.
A Frobenius-liftable weak Fano threefold of Picard rank 2 is toric, a generalization stated in Remark 5.5 that follows by the same Bott-vanishing argument.

Reading between the lines

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

The same table-driven Bott test could be repeated for weak Fano threefolds of higher Picard rank once complete numerical classification lists exist, potentially proving both conjectures beyond rank 2.
A reader could independently verify the paper's exclusions by recomputing the Chern numbers of each listed family from its defining fibration and substituting them into Theorem 3.1(1), which would also check the internal consistency of the cited tables.
The three surviving exceptions are all projective bundles over $\mathbb{P}^1$ or $\mathbb{P}^2$; in higher rank one might expect any surviving non-Fano toric image to be a toric projective bundle as well.

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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

Two things to know before you read it. The genuinely new material is Theorem 3.1: explicit formulas for -chi(X, Omega_X^2(L)) for three classes of weak Fano threefolds, derived by Hirzebruch-Riemann-Roch and toric Euler-Jaczewski computations. These look solid and are reusable. The two main theorems, A and B, are special cases of Occhetta-Wisniewski and Fakhruddin-Meng-Zhang-Zhong, and the strategy is a credible extension of Totaro's Bott-vanishing program. But the proof is not self-contained at the point where it matters: Corollary 3.2 rules out many families by citing classification tables without displaying the substituted Chern numbers. That is the load-bearing step. If any row in Fukuoka's table, Jahnke-Peternell-Radloff's tables, or Cutrone-Marshburn's tables is missing or mistyped, the exclusion fails and both theorems lose their conclusion.

What it does well: Theorem 3.1's computations are detailed; in 3.1(3) when chi = 0 it checks h^0 separately rather than assuming vanishing; the author discloses that [4] independently proves the endomorphism statement, so Theorem B's novelty is honestly bounded. The argument in Section 4 is coherent: reduce to fibrations, exclude bad families by Bott non-vanishing, then identify the survivors from tables.

Soft spots: the stress-test note is on target. In Corollary 3.2(1) the proof ends with 'the result follows by looking at [9, Table 1]'; in (3) the same for [12] and [13]; (6) says 'identical' to (4) with no numerics. A referee will need to check those pages and either reproduce the substitutions or have the author include them. Minor but real: Step 2 of Theorem A has an incomplete sentence, 'If -K_X is not spanned, by [12, Corollary 1.5],' and the implication is lost. Lemma 4.7 is taken from the author's coauthored preprint [14]; it is cited, not hidden, but it is a nontrivial dependency.

Summary: this paper is for people working on toric images, Bott vanishing, and endomorphisms of threefolds. The central approach seems sound and the new computations deserve to be in the literature. I would send it to a serious referee and ask for the Corollary 3.2 numerics to be shown and the typos fixed. If the table checks pass, it should be accepted; if a row is wrong, the exclusion step needs repair.

Referee Report

5 major / 3 minor

Summary. This paper studies weak Fano threefolds of Picard rank two. The main results are Theorem A: every smooth projective threefold with rho(X)=2 and -K_X nef that is the image of a projective toric variety is either toric Fano or one of P_{P1}(O_{P1} oplus O_{P1}(1)^2), P_{P1}(O^2_{P1} oplus O_{P1}(2)), P_{P2}(O_{P2} oplus O_{P2}(3)); and Theorem B: the same conclusion holds if the variety admits an int-amplified endomorphism. The proof works by deriving explicit formulas for certain Euler characteristics in Theorem 3.1, using them in Corollary 3.2 to show that all non-toric candidates in the available classification tables fail Bott vanishing, and then running a Mori-theoretic analysis that reduces the possibilities to the listed cases.

Significance. If correct, these results establish two new special cases of the Occhetta-Wisniewski image-of-toric conjecture and the Fakhruddin-Meng-Zhang-Zhong amplified-endomorphism conjecture, extending earlier work from Fano threefolds to weak Fano threefolds of Picard rank two. The Chern-class computations in Theorem 3.1 are explicit and checkable, and the final classification is concrete and falsifiable. The main caveat is that several exclusions are delegated to classification tables whose entries are not displayed; the proof is therefore only as strong as those tables and the reader's ability to verify the substitutions.

major comments (5)

[Sections 4 and 5 (proofs of Theorems A and B)] The proofs repeatedly invoke 'X does not have Bott vanishing, a contradiction' (e.g., Claim 3 in Step 1 of Theorem A and the conic-bundle case in Step 2), but nowhere in the proof of Theorem A is it stated or justified that a toric image has Bott vanishing. The connection is only made in Remark 5.5 via [1, Theorem 4.4.1] and [1, Theorem 3.2.4]. Similarly, the proof of Theorem B uses the same contradiction without stating the implication 'int-amplified endomorphism implies Bott vanishing', presumably from [15]. These are load-bearing premises; without them the contradictions are non-sequiturs. Please add explicit statements and references at the start of Sections 4 and 5.
[Corollary 3.2, cases (1), (3), (6)] The Bott non-vanishing test is the only mechanism that excludes the non-toric weak Fano threefolds of Picard rank two in Theorems A and B. However, in case (1) the string 'the result follows by looking at [9, Table 1]' replaces the numerical check for -chi = 13 + h - c1^3/2; in the conic-bundle part of case (3) the check for -chi = h - c1^3/2 + 2d + 3 is delegated to [12, Table A.3] and [13, Tables 7.2, 7.6, 7.7]; and case (6) is dismissed as 'identical' to case (4) with no numbers. If any table row is omitted, misidentified, or contains a wrong Chern number, the corresponding family survives the Bott test and both main theorems lose an exclusion step. Please display, for every cited row, the values of c1^3, h, d and the resulting sign of the Euler characteristic (or the required H^0 computation).
[Theorem B, Step 1, Claim 1] The proof of Claim 1 says only: 'Replacing f by a power of f, by Lemmas 5.3 and 5.2, we have f^{-1}(D_red)=D, f^{-1}(E_red)=E.' Lemma 5.3 assumes f^{-1}D=D and f^{-1}C=C for all flopping curves, which is exactly the preservation property being asserted; Lemma 5.2 concerns the induced map on the target of a birational contraction and does not establish preservation of both E and D in X. As written, the step appears circular. The author should either prove the invariance directly from the int-amplified condition or cite a precise lemma (with the hypotheses verified) that guarantees that a power of an int-amplified endomorphism preserves these two boundary divisors. This is load-bearing because Claim 1 is used to obtain normal crossing in codimension two via Lemma 5.1 in Claim 2.
[Corollary 3.2(3), Case 1] Theorem 3.1(3) is applied under the hypothesis that the rank-two bundle E on P^2 satisfies E(1) ample. In the application, however, the bundles are only described as E := F(2) (in the first three cases) or E := F or F^+ (in the other three cases), with the statement that they are 'nef but not ample'. Nefness of E does not imply ampleness of E(1), and no check is provided for each of the six cases. Without verifying E(1) is ample, the formulas of Theorem 3.1(3) cannot be used. Please add the missing ampleness verification or replace the reference with a version of the theorem that does not require it.
[Theorem A, Step 1, Claim 3; Theorem B, Step 1, Claim 3] The transition from the geometric hypotheses to 'X is as in [12, No. 1, Table A.5]' is another unshown classification lookup. The proof does not display the numerical invariants that identify this unique row, and it is not clear which hypotheses rule out the other rows of the table. Since Corollary 3.2(6) then applies to that row, a mistaken identification would again break both proofs. Please either display the relevant invariants or add a sentence explaining the comparison.

minor comments (3)

[Lemma 5.3 and proof of Theorem B] Lemma 5.3 states that the induced map is 'f+ : X -> X', but it should be 'f+ : X+ -> X+'; in the proof of Theorem B, 'by Lemma 5.3 X' has int-amplified endomorphism' should clarify whether X' or X+ is meant, since both appear in the flop diagram.
[Theorem A, Step 2] The sentence 'If -K_X is not spanned, by [12, Corollary 1.5].' is incomplete; it should state the conclusion obtained from [12, Corollary 1.5].
[Theorem B, Step 1, Claim 3] The phrase 'Let and B = psi(D)' contains a typo: it should be 'Let B = psi(D)'.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing self-citation: Lemma 4.7, used to rule out the curve-center blowup in Theorem A, is justified only by "Follows from [14]", a preprint coauthored by the author; the remaining Bott-vanishing/classification argument is otherwise independent.

self citation load bearing [Section 4, Lemma 4.7, used in Claim 2 of the proof of Theorem A]
"Lemma 4.7. If D is a prime divisor in P3 such that (P3, D) is toric image, then D is linear. Proof. Follows from [14]."

The only proof supplied for Lemma 4.7 is a bare reference to [14], which is Kawakami–Sarkar–Witaszek, a preprint coauthored by the present author. The lemma is then load-bearing in Claim 2 of Theorem A: when the K_X-negative contraction is the blow-up of a smooth curve C in Y ≅ P^3, the argument applies Lemma 4.7 to (P^3, φ(D)) to conclude that φ(D) is a hyperplane, and this conclusion is what excludes dim φ(E) = 1. Without an independent proof of Lemma 4.7, that exclusion rests entirely on the author's own unpublished preprint. Since [14] is not machine-checked, code-reproduced, or otherwise verified in the present paper, this step is a localized load-bearing self-citation rather than an independent external input.

full rationale

The derivation is not circular in the sense of fitting a parameter or defining the target into the hypothesis. The Bott-vanishing formulas in Theorem 3.1 are derived from the Hirzebruch–Riemann–Roch theorem and the Euler–Jaczewski sequence, and Corollary 3.2 applies them to external classification tables [5], [9], [12], [13], [27]; no 'prediction' is a renamed input. The only problematic step is Lemma 4.7: its proof is a bare citation to [14], a preprint coauthored by Supravat Sarkar, the present author. This lemma is load-bearing in Theorem A, Claim 2, where it converts the fact that φ(D) is a toric-image divisor in P^3 into the statement that φ(D) is a hyperplane; that statement is used to eliminate the case dim φ(E) = 1. Since [14] is not machine-checked, code-reproduced, or externally verified here, this is a self-citation doing essential work in the proof. The paper also cites [14] again in Remark 5.5, but that remark is a generalization, not part of the main theorem. Because the central classification still rests on independent Bott computations and tables, the overall circularity is moderate (4), not a full collapse.

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

The paper introduces no free parameters and no new postulated objects. It is a theorem-proof paper that relies on standard tools, external classification tables, and one lemma from the author's own recent preprint [14]. The absence of fitted parameters keeps the circularity burden low; the main burden is the black-box use of classifications and a self-cited result.

assumptions (7)

standard math Hirzebruch-Riemann-Roch theorem for vector bundles on smooth projective threefolds and on P2
Used throughout Theorem 3.1 for the Euler-characteristic computations, labeled equations (1) and (24).
standard math Serre duality and Kodaira vanishing on weak Fano threefolds
Used in equations (2), (3), and (13) to convert Ω^2 cohomology into Ω^1 cohomology and to set H^i(O_X)=0 for i>0.
domain assumption Smooth toric varieties have Bott vanishing and toric images are F-liftable
The whole method requires toric spaces to pass the Bott non-vanishing obstruction; the paper cites [1, Theorem 4.4.1] and [15].
domain assumption Classification tables of weak Fano threefolds of Picard rank 2 are complete and numerically correct
Corollary 3.2 enumerates cases from [5, Table 8,9], [12, Table 7.1, A.3], [13, Tables 7.2, 7.5, 7.6, 7.7], and [27, Theorem 2.3 and Section 4]. A wrong or missing table entry could leave an excluded case alive.
ad hoc to paper Lemma 4.7 from the coauthored preprint [14]: a toric image prime divisor in P3 is linear
Used in Claim 2 of the proof of Theorem A to identify φ(D) with a hyperplane; the cited source is an unreviewed preprint by the author and collaborators.
standard math Euler-Jaczewski exact sequence for projective bundles over P1
Used in Theorem 3.1(2), equation (8), to express Ω_W in terms of line bundles on the toric projective bundle.
standard math Kollár's classification of extremal rays on smooth threefolds
Used in Claims 2 and 3 to describe the possible divisorial, small, or fibration contractions and their exceptional loci.

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Cite this review

Pith. "Pith review of Images of toric variety and amplified endomorphism of weak Fano threefolds." pith.science (2026). https://pith.science/paper/CGWWAUGQ

@misc{pith2026250616325,
  author       = {Pith},
  title        = {Pith review of: Images of toric variety and amplified endomorphism of weak Fano threefolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGWWAUGQ}},
  note         = {Machine review of arXiv:2506.16325}
}

abstract

We show that some important classes of weak Fano $3$-folds of Picard rank $2$ do not satisfy Bott vanishing. Using this we show that any smooth projective $3$-fold $X$ of Picard rank $2$ with $-K_X$ nef which is the image of a projective toric variety is toric. This proves a special case of a conjecture by Ochetta-Wisniewski, extending a corresponding previous work for Fano $3$-folds. We also show that a weak Fano $3$-fold of Picard rank $2$ having an int-amplified endomorphism is toric. This proves a special case of a conjecture by Fakhrudding, Meng, Zhang and Zhong, extending corresponding previous work for Fano $3$-folds.

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On smooth rationally connected projective threefolds of Picard number two admitting int-amplified endomorphisms

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