Images of toric variety and amplified endomorphism of weak Fano threefolds

Technical Summary: Images of Toric Varieties and Amplified Endomorphism of Weak Fano Threefolds

Problem Statement
This paper addresses two related conjectures concerning the classification of smooth projective varieties under specific geometric and dynamical constraints.

1. The Toric Image Conjecture (Occhetta-Wiśniewski): If a smooth projective variety $X$ admits a surjective map from a proper toric variety, is $X$ itself toric? This is known for surfaces, Fano 3-folds, and the case where the Picard rank $\rho(X)=1$, but remains open in higher dimensions and ranks.
2. The Int-Amplified Endomorphism Conjecture (Fakhruddin, Meng, Zhang, Zhong): Is a smooth projective rationally connected variety admitting an int-amplified endomorphism necessarily toric? This is known for curves, surfaces, hypersurfaces, and Fano 3-folds.

The author focuses on the specific case of weak Fano 3-folds (where $-K_X$ is nef and big) with Picard rank $\rho(X)=2$. The goal is to determine if such varieties, under the conditions of being a "toric image" or admitting an "int-amplified endomorphism," must be toric.

Methodology
The core technical tool employed is Bott vanishing (and its logarithmic variant). The paper utilizes the following logical framework:

• Contrapositive Approach: The author establishes that specific classes of weak Fano 3-folds of Picard rank 2 fail to satisfy Bott vanishing.
• Theoretical Link: It is a known result (cited from [30]) that if a variety is a toric image or admits an int-amplified endomorphism, it must satisfy Bott vanishing (or log Bott vanishing).
• Elimination: By proving that non-toric candidates fail Bott vanishing, the author eliminates them from the list of possible counterexamples. Consequently, any variety satisfying the hypotheses (toric image or int-amplified endomorphism) that is not Fano must belong to a very specific list of exceptions which are shown to be toric.

The proofs rely heavily on:

• Hirzebruch-Riemann-Roch (HRR) calculations: Explicit computation of Euler characteristics $\chi(X, \Omega^2_X(L))$ for various line bundles $L$ to detect non-vanishing cohomology.
• Minimal Model Program (MMP): Analyzing the structure of the cone of curves $NE(X)$, specifically the contractions of the two rays (one $K_X$-negative, one $K_X$-trivial) for Picard rank 2.
• Flop Diagrams: Utilizing the behavior of toric images and endomorphisms under flops (Lemma 5.5, Lemma 6.2) to reduce cases where a fibration does not exist to cases where it does.
• Classification Tables: Referencing existing classifications of weak Fano 3-folds (e.g., [5], [13], [14]) to test specific numerical invariants against the derived formulas.

Key Contributions and Results

1. Failure of Bott Vanishing (Section 4)
The author derives explicit formulas for the Euler characteristic $-\chi(X, \Omega^2_X(H-K_X))$ and related terms for weak Fano 3-folds of Picard rank 2.

• Theorem 4.1: Provides formulas for general weak Fano 3-folds, for hypersurfaces in projective bundles over $\mathbb{P}^1$, and for projective bundles over $\mathbb{P}^2$.
• Corollary 4.2: Demonstrates that several important classes of weak Fano 3-folds of Picard rank 2 do not have Bott vanishing. These include:
  • Degree 6 and 8 del Pezzo fibrations over $\mathbb{P}^1$.
  • Conic bundles over $\mathbb{P}^2$ (with specific exceptions).
  • Varieties listed in specific tables of prior literature ([5], [13], [14]).

2. The Toric Image Theorem (Theorem A)

• Statement: Let $X$ be a smooth projective 3-fold with $\rho(X)=2$ and $-K_X$ nef. If $X$ is the image of a proper toric variety under a surjective map, then $X$ is either a toric Fano variety or one of the following three specific weak Fano varieties:
  1. $\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(1)^{\oplus 2})$
  2. $\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}^{\oplus 2} \oplus \mathcal{O}_{\mathbb{P}^1}(2))$
  3. $\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2}(3))$
• Significance: This proves the Occhetta-Wiśniewski conjecture for the case of Picard rank 2 and nef anticanonical bundle. The proof notably avoids heavy combinatorial fan geometry, relying instead on the failure of Bott vanishing to rule out non-toric candidates.

3. The Int-Amplified Endomorphism Theorem (Theorem B)

• Statement: Let $X$ be a smooth projective weak Fano 3-fold with $\rho(X)=2$. If $X$ admits an int-amplified endomorphism, then $X$ is either toric Fano or one of the three specific varieties listed in Theorem A.
• Significance: This proves the Fakhruddin et al. conjecture for weak Fano 3-folds of Picard rank 2. The argument mirrors the proof of Theorem A, utilizing the fact that int-amplified endomorphisms imply log Bott vanishing (Lemma 6.4).

Author's Claims on Significance

• Extension of Previous Work: The results extend previous proofs of these conjectures from Fano 3-folds to the broader class of weak Fano 3-folds.
• Independence and Complementarity: The author notes that while a separate preprint [4] recently proved the int-amplified conjecture for rationally connected 3-folds of Picard rank 2 using different methods (without Bott vanishing), the method of this paper is distinct. The author claims their arguments for Theorem B do not extend to the general case in [4], but conversely, the method in [4] does not prove Theorem A (the toric image case).
• Classification Insight: As a byproduct, the paper provides a proof that the three specific varieties listed in Theorem A are the only toric weak Fano 3-folds of Picard rank 2 that are not Fano, without relying on fan combinatorics.
• Generalization (Remark 6.5): The author suggests that the proof technique implies a more general statement: any $F$-liftable weak Fano 3-fold of Picard rank 2 is toric, which is a special case of a conjecture by [1].

Conclusion
The paper successfully resolves the two conjectures for the specific but non-trivial case of weak Fano 3-folds with Picard rank 2. By establishing precise numerical obstructions (failure of Bott vanishing) for non-toric candidates, the author demonstrates that any variety satisfying the dynamical or mapping conditions must be toric (or one of three explicitly identified toric weak Fano varieties).