Algebraic K3 surfaces with finite automorphism groups

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The purpose of this paper is to give a proof to the result announced in [3]. Let X be an algebraic surface defined over C. X is called a K3 surface if its canonical line bundle Kx is trivial and dim H1(X, ϕX) = 0. It is known that the automorphism group Aut (X) of X is isomorphic, up to a finite group, to the factor group O(Sx)/Wx, where O(Sx) is the automorphism group of the Picard lattice of X (i.e. Sx is the Picard group of X together with the intersection form) and Wx is its subgroup generated by all reflections associated with elements with square (–2) of Sx ([11]). Recently Nikulin [8], [10] has completely classified the Picard lattices of algebraic K3 surfaces with finite automorphism groups.