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

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.