Summary: Let G be a finite p-group acting on a complex projective variety \(V^ n\) and suppose the degree of V is prime to p. Does G have a fixed point on V? We will always assume G preserves the hyperplane class in \(H^ 2(V)\), or even that it acts projectively on the ambient projective space. In Topology 27, No.4, 459-472 (1988; Zbl 0669.57023), we showed that if \(G\cong \prod {\mathbb{Z}}/p\) and in addition \(n\not\equiv -1(mod p)\) then G does have a fixed point, while for non-abelian G this is not true as shown by the author in Invent. Math. 87, 331-342 (1987; Zbl 0629.57026). Both of the above papers used algebraic topology exclusively and proved fixed point theorems under certain topological assumptions. In this paper, we combine some of these methods with simple geometrical arguments in projective space to get more delicate results.