We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.

In this paper, we study finite symplectic actions on K3 surfaces X, that is, actions of finite groups G on X which act on H2,0(X) trivially. We show that the action on the K3 lattice H2(X, ℤ) induced by a symplectic action of G on X depends only on G up to isomorphism, except for five groups.

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in ${{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296 ± .010.