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

Jiahe Wang

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arXiv · 2606.13466

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

Jiahe Wang

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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.

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Jiahe Wang. 2026-06-11. Families of smooth Fano fourfolds of Picard rank 1 without Bott vanishing. https://arxiv.org/abs/2606.13466

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