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Families of smooth Fano fourfolds of Picard rank 1 without Bott vanishing

This paper demonstrates that among all currently known smooth Fano fourfolds of Picard rank 1, only the projective space satisfies Bott vanishing, implying it is the unique such variety admitting an endomorphism of degree greater than 1, while also introducing new Schubert2 functions for symmetric and skew-symmetric degeneracy loci and weighted projective spaces.

Original authors: Jiahe Wang

Published 2026-06-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiahe Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the mathematical world of shapes as a vast, infinite library. Inside this library, there is a special section dedicated to "Fano varieties." You can think of these as particularly beautiful, perfectly balanced geometric shapes that mathematicians love to study.

This paper focuses on a specific, rare breed of these shapes: smooth Fano fourfolds.

"Fourfolds" means they exist in four dimensions (hard to visualize, like trying to imagine a 3D object inside a 4D room).
"Smooth" means they have no sharp corners or tears; they are perfectly polished.
"Picard Rank 1" is a fancy way of saying they have a very simple, unified structure, like a single, solid building block rather than a complex Lego castle made of many different pieces.

The Big Mystery: The "Degree 1" Rule

For a long time, mathematicians have been hunting for a specific rule about these shapes. The rule is a conjecture (a guess that is widely believed but not yet proven) that goes like this:

"If you have one of these special shapes, and you can stretch or shrink it in a specific way (an 'endomorphism') that makes it bigger without tearing it, then it must be a standard 4D space (like a perfect, empty 4D room)."

Think of it like this: Imagine you have a magical balloon. If you can blow it up to be twice as big, three times as big, or any size larger than its original state without it popping or changing its fundamental shape, the conjecture says the balloon must have been a perfect sphere to begin with. If it was a weird, lumpy shape, you wouldn't be able to stretch it that way.

The Tool: The "Bott Vanishing" Test

To prove this, the authors use a mathematical "test" called Bott vanishing.

The Metaphor: Imagine trying to fill a bucket with water. If the bucket has a hole in the bottom, the water "vanishes." In math, "vanishing" means certain complex numbers (called cohomology groups) become zero.
The Logic: The paper relies on a previous discovery by Kawakami and Totaro, which says: If a shape can be stretched (has that special endomorphism), it must pass the Bott vanishing test (the water must vanish).
The Reverse: If a shape fails the test (the water doesn't vanish), then it cannot be stretched. Therefore, it cannot be the answer to our mystery unless it is the standard 4D space.

The Investigation: Checking the "Known" Shapes

The authors looked at 32 specific families of these 4D shapes that were already known to mathematicians. These shapes were built in three different ways:

Weighted Intersections: Like carving a shape out of a block of stone where the stone has different densities in different directions.
Grassmannian Zero Loci: Shapes found where specific mathematical "forces" cancel each other out inside a giant, complex space.
Pfaffian Subvarieties: Shapes defined by a special kind of symmetry in a matrix (a grid of numbers), often called "skew-symmetric" (where flipping the grid changes the sign of the numbers).

For each of these 32 families, the authors ran a complex calculation to check the "Bott vanishing" test. They calculated a specific number called $\chi(X, TX)$ .

The Analogy: Think of $\chi$ $χ$ as a "balance score."
- If the score is positive or zero, the shape might pass the test (the water might vanish).
- If the score is negative, the shape fails the test (the water definitely does not vanish).

The Results: A Clean Sweep

The authors found that for all 32 families of these specific shapes, the balance score was negative.

Translation: None of these 32 families pass the Bott vanishing test.
Conclusion: Because they fail the test, they cannot be stretched. Therefore, they are not the "perfect sphere" (the standard 4D space).

When you combine this with previous work that already checked the other types of these shapes (those with higher "indices"), the result is a complete picture: Among all the smooth Fano fourfolds of Picard rank 1 that we currently know, the only one that can be stretched is the standard 4D space ( $P^4$ ).

The New Tools: A Better Calculator

To do this, the authors (Jiahe Wang and Burt Totaro) had to build new tools. They wrote new functions for a computer program called Macaulay2 (specifically for a package named Schubert2).

The Metaphor: Imagine trying to calculate the weight of a complex, floating sculpture. The old tools could only weigh simple cubes. Wang and Totaro built new "scales" that can weigh these weird, floating, skew-symmetric shapes and shapes in "weighted" spaces (where some parts of the space count more than others).
These new tools allow mathematicians to compute the "balance scores" for these complex shapes much faster and more accurately than before.

Summary

In simple terms:

Mathematicians have a rule: "Only the perfect 4D space can be stretched."
They tested 32 known "imperfect" 4D shapes to see if they could be stretched.
They used a new, custom-built calculator to prove that none of these 32 shapes can be stretched.
Therefore, the rule holds true for every single known example: If you find a shape like this that can be stretched, it is definitely the perfect 4D space.

Technical Summary: Families of Smooth Fano Fourfolds of Picard Rank 1 Without Bott Vanishing

Problem Statement
The paper addresses a specific case of a well-known conjecture regarding endomorphisms of smooth Fano varieties. Conjecture 1.1 posits that if a smooth Fano variety $X$ of Picard number $\rho(X)=1$ over an algebraically closed field of characteristic zero admits an endomorphism of degree greater than 1, then $X$ must be isomorphic to projective space $\mathbb{P}^n$ .

Recent work by Kawakami and Totaro [16] established a general implication: the existence of an int-amplified endomorphism (which is equivalent to an endomorphism of degree $>1$ when $\rho(X)=1$ ) implies that $X$ satisfies "Bott vanishing." Bott vanishing requires that $H^i(X, \Omega^j_X(A)) = 0$ for all $i > 0$ , $j \ge 0$ , and every ample Weil divisor $A$ .

The specific problem tackled in this paper is the verification of this conjecture for the currently known families of smooth Fano fourfolds with Picard rank 1 and Fano index 1. While the conjecture was already known for index $>1$ cases and other specific dimensions or structures, the index 1 case in dimension 4 remained open for the known families. The authors aim to determine if any of these known families satisfy Bott vanishing; if they do not, they cannot admit endomorphisms of degree $>1$ .

Methodology
The authors employ a computational approach using the Macaulay2 package Schubert2. The core strategy relies on the relationship between Bott vanishing and the Euler characteristic of the tangent bundle, $\chi(X, T_X)$ .

Logical Reduction: The authors observe that if Bott vanishing holds for a Fano variety $X$ , then $H^i(X, T_X) = 0$ for all $i > 0$ (since $T_X \cong \Omega^3_X \otimes (-K_X)$ and $-K_X$ is ample). Consequently, $\chi(X, T_X) = h^0(X, T_X) \ge 0$ . Therefore, if the authors can demonstrate that $\chi(X, T_X) < 0$ for a given family, Bott vanishing fails, and the variety cannot admit an endomorphism of degree $>1$ .
Computational Tools: To compute $\chi(X, T_X)$ $χ (X, T_{X})$ for complex geometric constructions, the authors (jointly with Burt Totaro) developed new functions for the Schubert2 package. These functions handle:
- Symmetric and Skew-Symmetric Degeneracy Loci: Functions skewDegeneracyLocus, symmetricDegeneracyLocus, and their variants (...2, ...KernelBundle) compute characteristic classes and Euler characteristics for loci defined by rank conditions on symmetric or skew-symmetric morphisms. These utilize resolutions via relative Grassmannians and Chern class computations based on theorems by Fulton, Harris–Tu, and Józefiak–Lascoux–Pragacz.
- Weighted Projective Spaces: Functions weightedProjectiveBundle and weightedProjectiveSpace enable intersection-theoretic computations in weighted projective settings, modeling the intersection ring and pushforwards appropriately.
Scope of Calculation: The authors apply these tools to 32 specific families of smooth Fano fourfolds of Picard rank 1 and index 1:
- 13 families realized as complete intersections in weighted projective spaces.
- 13 families realized as zero loci of homogeneous vector bundles on Grassmannians.
- 6 additional families realized as skew-symmetric degeneracy loci (Pfaffian subvarieties), including constructions by Qureshi and Küchle.

Key Contributions

Algorithmic Development: The paper introduces and documents new computational functions for Schubert2 that allow for the systematic calculation of characteristic classes and Euler characteristics for degeneracy loci (both symmetric and skew-symmetric) and weighted projective varieties.
Systematic Verification: The authors perform a comprehensive calculation of $\chi(X, T_X)$ for all 32 currently known families of smooth Fano fourfolds with Picard rank 1 and index 1.

Results
The computations yield the following definitive results:

Theorem 1.3: For every one of the 32 known families of smooth Fano fourfolds of Picard rank 1 and index 1, the Euler characteristic $\chi(X, T_X)$ is strictly negative.
Failure of Bott Vanishing: Since $\chi(X, T_X) < 0$ , the condition $H^i(X, T_X) = 0$ for $i > 0$ cannot hold. Thus, none of these 32 families satisfy Bott vanishing.
Corollary 1.4: Combining these results with the previously known cases (Picard rank 1, index $>1$ ), the authors conclude that among all currently known smooth Fano fourfolds of Picard rank 1, the only variety that can admit an endomorphism of degree greater than 1 is the projective space $\mathbb{P}^4$ .

Significance
The paper claims to verify the endomorphism conjecture for the complete set of currently known smooth Fano fourfolds of Picard rank 1. By demonstrating that the negative Euler characteristic of the tangent bundle precludes Bott vanishing in all non-projective known cases, the authors rule out the existence of non-trivial endomorphisms for these varieties. The work relies on the implication established by Kawakami–Totaro [16] and extends the verification of Conjecture 1.1 to the index 1 case in dimension 4, filling a gap in the classification of Fano varieties with respect to endomorphism properties. The paper does not claim to prove the conjecture for all possible Fano fourfolds (as the classification of index 1 families is not yet complete), but strictly limits its conclusion to the "currently known" families.