Higher index Fano varieties with finitely many birational automorphisms

TL;DR

This paper introduces a new approach to studying the finiteness of birational automorphism groups of Fano varieties, demonstrating the existence of high-index Fano varieties with finite or trivial automorphism groups in all positive characteristics.

Contribution

It provides a novel method using differential forms on p-cyclic covers to show the existence of high-index Fano varieties with finite automorphism groups in positive characteristic.

Findings

Existence of high-index Fano varieties with finite automorphism groups in positive characteristic.

Application of differential forms on p-cyclic covers in the study of automorphisms.

New approach extends results beyond index one and two Fano varieties.

Abstract

Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying this problem for Fano varieties of index one and two. The purpose of this paper is to give a new approach to this problem, and to show that in every positive characteristic there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphisms. To do this we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Koll\'ar produces on p-cyclic covers in characteristic p>0.