we present a proof of the fact that given a hodge isometry psi between the rational second cohomology of two kahler k3 surfaces $s_1$ and $s_2$, we can find a finite sequence of k3 surfaces and analytic (2, 2)-classes supported on successive products, such that the isometry psi is the convolution of these classes. the proof of this fact implies that for projective k3 surfaces $s_1$, $s_2$ the class of psi is algebraic. this proves a conjecture of i. shafarevich.