An equivariant holomorphic map of symmetric domains associated to a homomorphism of semisimple algebraic groups defined over Q is rational if it carries a point belonging to a set determined by an arithmetic subgroup to a point in a similar set. We prove that an equivariant holomorphic map of symmetric domains is rational if the associated Kuga fiber variety does not have a nontrivial deformation. Introduction.In this paper we study the relation between the rationality of equivariant holomorphic maps of symmetric domains and the rigidity of Kuga fiber varieties which are families of polarized abelian varieties parametrized by a locally symmetric arithmetic variety.Let G be a semisimple algebraic group defined over Q that does not contain factors defined over Q and compact over R, and let G = G(R) be the associated real semisimple Lie group. Let K be a maximal compact subgroup of G, and assume that the corresponding symmetric space D = G/K has a G-invariant complex structure. Let V be a vector space over Q of dimension 2m, and let β be a nondegenerate alternating bilinear form on V . We set G = Sp(V, β) and denote by D the symmetric domain associated to G = G (R). Let ρ : G → G be a homomorphism of Lie groups induced by a morphism ρ : G → G of algebraic groups defined over Q. We define the space D ρ to be the set of all holomorphic maps τ : D → D satisfying τ(gz) = ρ(g)τ(z) for all g ∈ G and z ∈ D.Let L be a lattice in V R , and let Γ be a torsion-free subgroup of Sp(L, β) of finite index. We set