The topology of conjugate varieties (Q1923230)

\textit{J.-P. Serre} and \textit{H. Abelson} [Topology 13, 161-176 (1974; Zbl 0247.14001)] have produced examples of conjugate algebraic varieties which are not homeomorphic. The work reported upon here aims to provide conditions sufficient to ensure that conjugate projective varieties are indeed homeomorphic. Two conditions are involved, one relating to whether the field of definition of the variety coincides with its field of moduli and the second a condition relating to stratifications. We show that the first condition is satisfied by complete intersections (whether in projective space or more generally in homogeneous varieties) and that the second condition is satisfied (at least) when the varieties are smooth. There do not appear to be any known examples of non-homeomorphic conjugates which satisfy the first but not the second one of these conditions.