Hodge similarities, algebraic classes, and Kuga–Satake varieties

Abstract

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperkähler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove the Hodge conjecture for the square of the general member of the first four-dimensional families of K3 surfaces with totally real multiplication of degree two. We then show the functoriality of the Kuga–Satake construction with respect to Hodge similarities. This implies that, if the Kuga–Satake Hodge conjecture holds for two hyperkähler manifolds, then every Hodge similarity between their transcendental lattices is algebraic after composing it with the Lefschetz isomorphism. In particular, we deduce that Hodge similarities of transcendental lattices of hyperkähler manifolds of generalized Kummer deformation type are algebraic.