Article

Endomorphisms of Fano 3-folds and log Bott vanishing

January 2025
Mathematical Research Letters 31(6):1879-1902

DOI:10.4310/MRL.250211003919

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Article

Full-text available

Bott vanishing for Fano threefolds

April 2024 · Mathematische Zeitschrift

[...]
Burt Totaro

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that Hj(X,ΩXi⊗L)=0

H^j(X,\Omega ^i_X\otimes L)=0

for ... [Show full abstract] j>0

j>0

, i≥0

i\ge 0

, and L ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected.

View full-text

Article

Full-text available

Bott vanishing for elliptic surfaces

February 2022 · Mathematische Zeitschrift

[...]
Chengxi Wang

We explore Bott Vanishing for elliptic surfaces over P1

{\mathbb {P}}^1

. We show that Bott Vanishing is significantly affected by the geometric properties ... [Show full abstract] of fibers. For example, whether there exist certain types of singular fibers on the elliptic fibration such as cuspidal fibers. For an ample line bundle on the surface with large self-intersection, these geometric properties give criteria for Bott Vanishing.

View full-text

Preprint

Endomorphisms of Fano 3-folds and log Bott vanishing

May 2023

[...]
Burt Totaro

Kawakami and the author showed that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. That was a new way to analyze which varieties have nontrivial endomorphisms. In this paper, we extend that result to a logarithmic version of Bott vanishing for an endomorphism with a totally invariant divisor. We apply this to Fano 3-folds. ... [Show full abstract] Meng-Zhang-Zhong showed that the only smooth complex Fano 3-folds that admit an int-amplified endomorphism are the toric ones. Also, Achinger-Witaszek-Zdanowicz showed that the only smooth complex Fano 3-folds that are images of toric varieties are the toric ones. Using log Bott vanishing, we reprove both results and extend them to characteristic p, for morphisms of degree prime to p.

Preprint

Bott vanishing for Fano 3-folds

February 2023

[...]
Burt Totaro

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that

H^j(X,\Omega^i_X\otimes L)=0

for every

j>0

i\geq 0

, and L ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected. Along the way, we conjecture that for every projective birational ... [Show full abstract] morphism

\pi\colon X\to Y

of smooth varieties, and every line bundle A on X that is ample over Y, the higher direct image sheaf

R^j\pi_*(\Omega^i_X\otimes A)

is zero for every

j>0

and

i\geq 0

Last Updated: 12 Feb 2025

Endomorphisms of Fano 3-folds and log Bott vanishing

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