As part of the structure of a projective variety, one remembers not only the topological subspace cut out in projective space by the vanishing of the defining homogeneous polynomials, but also a sheaf of rings on that subspace. One may wonder to what extent the topological space alone determines the variety. In spite of counterexamples in low dimension, such determination turns out to hold in sufficiently high dimension for normal, projective, geometrically irreducible varieties in characteristic 0. The latter is a recent result of Kollár (that builds on earlier work of Lieblich and Olsson) and it will be the subject of this talk.