Free actions of finite groups on varieties. II
[Part I appeared in ''Current Trends in Algebraic Topology'', CMS Conf. Proc. 2, part 2, 23-33 (1982; Zbl 0551.57022).] Suppose a finite group G acts freely on a complex projective variety V such that \(H^ 1(V)=0\) and \(g^*h=h\in H^ 2(V)\) for all \(g\in G\), where h is the hyperplane section class. Then exponent G divides degree V and {\#}(G)\(=order\) G divides (degree V)\({}^ 2\). The theorem is actually proved in more generality for polarized 2n near manifolds X which generalize projective varieties in a topological context, and for a generalization of degree involving classes \(h_ 1,...,h_ k\in H^ 2(X)\). Part of the proof is the theorem that if G is a p-group and \(h_ 1,....,h_ k\in H^ 3(G)\) then there exists a subgroup \(i_ 0: G_ 0\subset G\) such that \(i^*_ 0(h_ i)=0\), \(i=1,...,k\), and {\#}(G) divides {\#}(G\({}_ 0)^{k+1}\).
scientific article; zbMATH DE number 3877350