The Mukai conjecture via Cox rings for smooth canonical ambient embeddings

Abstract

We prove the Mukai conjecture on the characterisation of powers of projective spaces among smooth Fano varieties $n+\rho -i\rho \ge 0$ for a class of locally factorial Fano varieties naturally defined in terms of their Cox rings. Recall that all klt Fano varieties $X$ embed into a projective toric variety $Z$ via their Cox ring, where $Z$ is uniquely determined up to isomorphism by the isomorphism class of $X$. We prove the Mukai conjecture when the ambient $Z$ is smooth, where we prove the stronger result $n+\rho -i\rho \ge n+\rho -\sum a_i\ge 0$, where $\sum a_i{D}_i$ is a distinguished anticanonical divisor supported on the Cox ring divisors of $X$. Importantly, this replicates the same stronger lower bound as proved in the case of spherical varieties. Here we are able to interpret the terms in the Mukai conjecture: $i\rho$ in terms of the geometry of the moment polytope of $Z$, and $n+\rho$ as an upper bound on the sum of coefficients in a decomposition of the anticanonical divisor into the Cox ring divisors. This decomposition is obtained by passing to a positive characteristic reduction of the Cox ring, where we obtain the coefficients from strong $F$-regularity, and the upper bound on the coefficient sum from the $F$-pure threshold.

arXiv commentThis version greatly strengthens the first version, extending the scope of the main theorem to the full class of Fano varieties whose canonical ambient toric variety is smooth. Here we now also introduce this class of varieties as the "ambiently smooth Fano varieties''