Bounding automorphism groups
It is an old problem to find an upper bound on the size of the group of birational automorphisms of a complex variety \(X\). It is known that this group is finite when \(X\) is of general type. In dimension 1 and 2 there is an upper bound which is linear in terms of Chern numbers of the minimal model of \(X\), and there is some (much larger) upper bound in all dimensions. Here we try to get a polynomial bound for \(n\)-dimensional \(X\). We have to assume that \(X\) has a minimal model \(Y\) and besides \(K_Y\) our bound will depend on the index \(r\) of \(Y\) as well. The bound is a polynomial of \((rK^n_Y)\). In the course of the proof it turns out to be useful to get a bound on the number of linear automorphisms of an arbitrary projective variety whenever this number is finite. This bound is a polynomial of the degree of the variety. We get some (much weaker) results in characteristic \(p > 0\). In the projective setup our bound is a polynomial of the degree and the size of the \(p\)-Sylow subgroup, this can be used to get some bounds for surfaces of general type (involving the size of the \(p\)-Sylow subgroup).
scientific article; zbMATH DE number 892575