On a theorem of Peters on automorphisms of Kahler surfaces

Abstract

For any Kahler surface which admits no nonzero holomorphic vectorfields, we consider the group of holomorphic automorphisms which induce identity on the second rational cohomology. Assuming the canonical linear system is without base points and fixed components, C.A.M. Peters showed that this group is trivial except when the Kahler surface is of general type and either $c_1^2=2c_2$ or $c_1^2=3c_2$ holds. Moreover, this group is a 2-group in the former case, and is a 3-group in the latter. The purpose of this note is to give further information about this group. In particular, we show that $c_1^2$ is divisible by the order of the group. Our argument is based on the results of C.H. Taubes on symplectic 4-manifolds, which are applied here in an equivariant setting.