Surfaces via Almost - Primes

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following facts: (1) For any given positive integer N , there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard groups are not isomorphic but their transcendental lattices are Hodge isometric, or equivalently, their derived categories are mutually equivalent. (2) For any given projective K3 surface there are at most finitely many nonisomorphic projective K3 surfaces having Hodge isometric transcendental lattices, i.e. having equivalent derived categories. We also give an explicit formula for the cardinality of the isomorphism classes of projective K3 surfaces having derived categories equivalent to the one of X with Picard number 1 in terms of the degree of X.