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.