Bound of automorphisms of projective varieties of general type

TL;DR

This paper establishes a universal bound on the size of automorphism groups of smooth projective varieties of general type, linking it to the volume of the variety with respect to its canonical bundle.

Contribution

It proves a new uniform bound on automorphism groups of varieties of general type, depending only on the dimension, and relates it to the volume of the canonical bundle.

Findings

Existence of a constant C_n depending only on dimension n.

Automorphism group size is bounded by C_n times the volume of the canonical bundle.

The bound applies uniformly to all smooth projective varieties of general type in dimension n.

Abstract

We prove that there exists a positive number $C_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, the automorphism group $Aut(X)$ satisfies the inequality $\sharp{Aut}(X)\leq C_{n}\cdot\mu (X,K_{X})$, where $\mu (X,K_{X})$ is the volume of $X$ with respect to $K_{X}$.