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arxiv: 2606.13466 · v1 · pith:ZVV6OB2Enew · submitted 2026-06-11 · 🧮 math.AG

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

Jiahe Wang This is my paper

Pith reviewed 2026-06-27 05:30 UTC · model grok-4.3

classification 🧮 math.AG
keywords Fano fourfoldsPicard rank 1Bott vanishingEuler characteristicSchubert calculusdegeneracy lociweighted projective spaces
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The pith

Among all currently known smooth Fano fourfolds of Picard rank 1, only the projective space satisfies Bott vanishing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the Euler characteristic of the tangent bundle for all currently known families of smooth Fano fourfolds with Picard rank one and index one, showing the value is negative in each case. This negativity implies that none of these varieties can satisfy Bott vanishing. When added to existing results for the higher-index cases, the conclusion is that projective four-space is the only known smooth Fano fourfold of Picard rank one that does satisfy the condition. The same negativity also forces any endomorphism of degree greater than one on these varieties to occur only on projective space. The authors introduce new computational functions to handle the required Schubert calculus on symmetric, skew-symmetric, and weighted degeneracy loci.

Core claim

We show that χ(X,T_X)<0 for the currently known families of smooth Fano fourfolds of Picard rank 1 and index 1. Combining this with the known Picard rank 1 index > 1 cases, we show that among all currently known smooth Fano fourfolds of Picard rank 1, the only variety satisfying Bott vanishing is the projective space.

What carries the argument

The Euler characteristic χ(X, T_X) of the tangent bundle, whose negativity for each listed family rules out Bott vanishing.

If this is right

  • Only the projective space satisfies Bott vanishing among all currently known smooth Fano fourfolds of Picard rank 1.
  • Any endomorphism of degree greater than 1 on one of these varieties must occur on the projective space.
  • The new Schubert2 functions enable explicit verification of the Euler characteristic for symmetric, skew-symmetric, and weighted degeneracy loci.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A new family with non-negative Euler characteristic would be the first candidate beyond projective space for satisfying Bott vanishing.
  • The result isolates projective space as the only known member of the class that can carry non-constant endomorphisms of degree greater than one.
  • Further work could test whether the negativity persists for any hypothetical new families constructed by different geometric methods.

Load-bearing premise

The computation that χ(X, T_X) < 0 holds for every listed family of smooth Fano fourfolds of Picard rank 1 and index 1.

What would settle it

A calculation showing χ(X, T_X) ≥ 0 for one of the listed families, or the discovery of a new smooth Fano fourfold of Picard rank 1 that satisfies Bott vanishing.

read the original abstract

We show that $\chi(X,T_X)<0$ for the currently known families of smooth Fano fourfolds of Picard rank $1$ and index $1$. Combining this with the known Picard rank $1$ index $> 1$ cases, we show that among all currently known smooth Fano fourfolds of Picard rank $1$, the only variety satisfying Bott vanishing is the projective space. By a result of Kawakami--Totaro, the existence of an endomorphism of degree greater than 1 implies Bott vanishing. Therefore, among the currently known smooth Fano fourfolds of Picard rank $1$, any variety admitting an endomorphism of degree greater than 1 must be $\mathbb P^4$. Together with Burt Totaro, we develop new Schubert2 functions for symmetric and skew-symmetric degeneracy loci, and weighted projective spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript computes χ(X, T_X) < 0 for all currently known families of smooth Fano fourfolds of Picard rank 1 and index 1 by realizing each as a symmetric or skew-symmetric degeneracy locus (or in weighted projective space) and applying newly developed Schubert2 functions. Combined with prior results for index >1 cases, this implies that among all known smooth Fano fourfolds of Picard rank 1, only ℙ⁴ satisfies Bott vanishing. By the Kawakami–Totaro theorem, this further implies that no such variety except ℙ⁴ admits an endomorphism of degree >1. The paper also introduces the new Schubert2 functions developed jointly with Burt Totaro.

Significance. If the computations are correct, the result gives a complete accounting of Bott vanishing (and its implication for endomorphisms) across the known list of Picard-rank-1 Fano fourfolds. The development of new Schubert2 functions for symmetric/skew-symmetric degeneracy loci and weighted projective spaces is a concrete computational contribution that may be reusable in related problems.

major comments (1)
  1. [Computations of χ(X, T_X) via new degeneracy-locus functions] The central claim that χ(X, T_X) < 0 holds for every listed index-1 family rests entirely on the correctness of the newly implemented degeneracy-locus formulas and the subsequent Chern-class arithmetic. No independent cross-check (e.g., Macaulay2 resolution or direct Hirzebruch–Riemann–Roch computation) is supplied for any family; an error in even one case would falsify the “only ℙ⁴” statement for the known list.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for verification of the central computations. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim that χ(X, T_X) < 0 holds for every listed index-1 family rests entirely on the correctness of the newly implemented degeneracy-locus formulas and the subsequent Chern-class arithmetic. No independent cross-check (e.g., Macaulay2 resolution or direct Hirzebruch–Riemann–Roch computation) is supplied for any family; an error in even one case would falsify the “only ℙ⁴” statement for the known list.

    Authors: We agree that the absence of independent cross-checks is a limitation of the current manuscript. The new Schubert2 functions for symmetric/skew-symmetric degeneracy loci and weighted projective spaces were developed jointly with Burt Totaro using standard Schubert calculus, and the explicit formulas are given in the paper. However, to strengthen the result, we will revise the manuscript to include a direct Hirzebruch–Riemann–Roch computation (via the standard formula on the ambient space) as a cross-check for at least two representative families. We will also add implementation details and sample code snippets for the new functions to improve reproducibility. A full Macaulay2 resolution for every family is not feasible within the scope of this work, but the added checks address the core concern. revision: partial

Circularity Check

0 steps flagged

No significant circularity; central computations are independent of the target claim

full rationale

The derivation enumerates known families from prior literature, realizes them as degeneracy loci, and applies newly developed Schubert2 functions (joint with Totaro) to compute Chern classes of T_X and hence χ(X, T_X) < 0. These steps are presented as explicit calculations rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The Kawakami-Totaro theorem is an external citation whose content is independent of the present computations. No equation or step reduces the claim that only P^4 satisfies Bott vanishing among the listed families to a tautology or to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the completeness of the list of known families and on the correctness of the Euler characteristic computations for each family.

axioms (1)
  • domain assumption The families enumerated in the paper exhaust all currently known smooth Fano fourfolds of Picard rank 1 and index 1.
    The statement is explicitly limited to 'currently known' families.

pith-pipeline@v0.9.1-grok · 5669 in / 1132 out tokens · 17895 ms · 2026-06-27T05:30:13.431660+00:00 · methodology

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Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    Amerik, M

    E. Amerik, M. Rovinsky, and A. Van de Ven. A boundedness theorem for morphisms between threefolds.Ann. Inst. Fourier (Grenoble), 49(2):405–415, 1999

    1999

  2. [2]

    Maps onto certain Fano threefolds.Doc

    Ekaterina Amerik. Maps onto certain Fano threefolds.Doc. Math., 2:195–211, 1997

    1997

  3. [3]

    V. V. Batyrev. On the classification of toric Fano4-folds. volume 94, pages 1021–1050. 1999. Algebraic geometry, 9

    1999

  4. [4]

    Endomorphisms of hypersurfaces and other manifolds.Internat

    Arnaud Beauville. Endomorphisms of hypersurfaces and other manifolds.Internat. Math. Res. Notices, (1):53–58, 2001

    2001

  5. [5]

    Fano fourfolds database.https://fano-fourfolds.apps.math.cnrs.fr/

    Marcello Bernardara, Enrico Fatighenti, Laurent Manivel, and Fabio Tanturri. Fano fourfolds database.https://fano-fourfolds.apps.math.cnrs.fr/. Accessed May 21, 2026

    2026

  6. [6]

    Fano fourfolds of K3 type.arXiv:2506.18014v1, 2025

    Marcello Bernardara, Enrico Fatighenti, Laurent Manivel, and Fabio Tanturri. Fano fourfolds of K3 type.arXiv:2506.18014v1, 2025

    arXiv 2025

  7. [7]

    Coates, A

    T. Coates, A. Kasprzyk, and T. Prince. Four-dimensional Fano toric complete intersections. Proc. A, 471(2175):20140704, 14, 2015

    2015

  8. [8]

    On the geometry of some quiver zero loci Fano fourfolds.arXiv:2406.04389, 2024

    Enrico Fatighenti, Fabio Tanturri, and Federico Tufo. On the geometry of some quiver zero loci Fano fourfolds.arXiv:2406.04389, 2024

    arXiv 2024

  9. [9]

    William Fulton.Intersection theory, volume 2 ofErgebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition, 1998

    1998

  10. [10]

    Grayson and Michael E

    Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available athttp://www2.macaulay2.com

  11. [11]

    https://macaulay2.com/

    Daniel R. Grayson, Michael E. Stillman, Stein A. Strømme, David Eisenbud, and Charley Crissman. Schubert2: Characteristic classes for varieties without equations. Version 0.7. A Macaulay2package available at "https://macaulay2.com/"

  12. [12]

    Joe Harris and Loring W. Tu. On symmetric and skew-symmetric determinantal varieties. Topology, 23(1):71–84, 1984. 10

    1984

  13. [13]

    Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles.J

    Jun-Muk Hwang and Ngaiming Mok. Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles.J. Algebraic Geom., 12(4):627–651, 2003

    2003

  14. [14]

    Józefiak, A

    T. Józefiak, A. Lascoux, and P. Pragacz. Classes of determinantal varieties associated with symmetric and skew-symmetric matrices.Izv. Akad. Nauk SSSR Ser. Mat., 45(3):662–673, 1981

    1981

  15. [15]

    Four-dimensional Fano quiver flag zero loci.Proc

    Elana Kalashnikov. Four-dimensional Fano quiver flag zero loci.Proc. A, 475(2225):20180791, 23, 2019

    2019

  16. [16]

    Endomorphisms of varieties and Bott vanishing.J

    Tatsuro Kawakami and Burt Totaro. Endomorphisms of varieties and Bott vanishing.J. Al- gebraic Geom., 34(2):381–405, 2025

    2025

  17. [17]

    Rational connectedness and boundedness of Fano manifolds.J

    János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rational connectedness and boundedness of Fano manifolds.J. Differential Geom., 36(3):765–779, 1992

    1992

  18. [18]

    On Fano4-fold of index1and homogeneous vector bundles over Grassmanni- ans.Math

    Oliver Küchle. On Fano4-fold of index1and homogeneous vector bundles over Grassmanni- ans.Math. Z., 218(4):563–575, 1995

    1995

  19. [19]

    Some remarks and problems concerning the geography of Fano4-folds of index and Picard number one.Quaestiones Math., 20(1):45–60, 1997

    Oliver Küchle. Some remarks and problems concerning the geography of Fano4-folds of index and Picard number one.Quaestiones Math., 20(1):45–60, 1997

    1997

  20. [20]

    A. G. Kuznetsov and Yu. G. Prokhorov. On higher-dimensional del Pezzo varieties.Izv. Ross. Akad. Nauk Ser. Mat., 87(3):75–148, 2023

    2023

  21. [21]

    Biregular classification of Fano3-folds and Fano manifolds of coindex3.Proc

    Shigeru Mukai. Biregular classification of Fano3-folds and Fano manifolds of coindex3.Proc. Nat. Acad. Sci. U.S.A., 86(9):3000–3002, 1989

    1989

  22. [22]

    K. H. Paranjape and V. Srinivas. Self-maps of homogeneous spaces.Invent. Math., 98(2):425– 444, 1989

    1989

  23. [23]

    V. V. Przyjalkowski and C. A. Shramov. Smooth prime Fano complete intersections in toric varieties.Mat. Zametki, 109(4):590–596, 2021

    2021

  24. [24]

    Bounds for smooth Fano weighted complete intersections.Commun

    Victor Przyjalkowski and Constantin Shramov. Bounds for smooth Fano weighted complete intersections.Commun. Number Theory Phys., 14(3):511–553, 2020

    2020

  25. [25]

    Smooth Fano 4-folds in Gorenstein formats.Bull

    Muhammad Imran Qureshi. Smooth Fano 4-folds in Gorenstein formats.Bull. Aust. Math. Soc., 104(3):424–433, 2021

    2021

  26. [26]

    Boundedness of finite morphisms onto Fano manifolds with large Fano index.J

    Feng Shao and Guolei Zhong. Boundedness of finite morphisms onto Fano manifolds with large Fano index.J. Algebra, 639:678–707, 2024

    2024

  27. [27]

    P. M. H. Wilson. Fano fourfolds of index greater than one.J. Reine Angew. Math., 379:172– 181, 1987. UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA Email address:jiahewang@math.ucla.edu 11

    1987