Rational equivariant holomorphic maps of symmetric domains (Q1606077)

Let \(G\) be the semisimple Lie group of real points of a semisimple algebraic group defined over \(\mathbb{Q}\) and let \(K\) be a maximal compact subgroup of \(G\) such that the symmetric domain \(G/K\) has a \(G\)-invariant complex structure. The author studies equivariant holomorphic maps between symmetric domains of this type which are induced by homomorphisms of semisimple algebraic groups. A special family of polarized abelian varieties, called Kuga fiber variety, parametrized by a complex quasi-projective arithmetic quotient of a symmetric domain, is associated to any such map. The main result is as follows: Let \(\varphi\) be an equivariant holomorphic map as above such that the associated Kuga fiber variety is rigid. Then \(\varphi\) is rational.