Appears in collection : Conférence à la mémoire de Jean-Pierre Demailly

The volume of a line bundle on a smooth projective variety is a rough measure for the asymptotic growth of the dimension of the space of sections of its high tensor powers, and line bundles with positive volume are called big. The volume extends naturally to a continuous function on the real Neron-Severi group, which vanishes outside the big cone and is C1 differentiable inside of it, by work of Boucksom-Favre-Jonsson and Lazarsfeld-Mustata. An interesting question is then to determine what the optimal regularity is, up to the boundary of the big cone. I will discuss joint work with Simion Filip and John Lesieutre where we construct examples where this volume function is C1 but not better (at the boundary), and use this to answer negatively a number of questions by Lazarsfeld and others.