A simple remark on fields of definition

Let K< L be an extension of fields, in characteristic zero, with L algebraically closed and let ˉK < L be the algebraic closure of K in L. Let X and Y be irreducible projective algebraic varieties, X defined over ˉK and Y defined over L, and let π : X →Y be a non-constant morphism, defined over L. If we assume that ˉK ≠ L, then one may wonder if Y is definable over ˉK. In the case that K = Q, L = C and that X and Y are smooth curves, a positive answer was obtained by Gonzalez-Diez. In this short note we provide simple conditions to have a positive answer to the above question. We also state a conjecture for a class of varieties of general type.