Symmetries of Fano varieties

TL;DR

This paper investigates the symmetry properties of Fano varieties, establishing bounds on symmetric group actions, analyzing special classes like toric and weighted complete intersections, and exploring implications for the boundedness of Fano varieties.

Contribution

It provides new bounds on symmetric group actions on Fano varieties, characterizes maximally symmetric cases, and links symmetry to boundedness in the classification of Fano varieties.

Findings

Bound on symmetric group actions: $k \,\leq\, m(n)$ for n-dimensional Fano varieties.

Exact bounds for toric varieties: $m(n)=n+2$ for $n\geq 4$.

Connection between symmetry and boundedness of Fano varieties.

Abstract

We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded.