Valuations and plurisubharmonic singularities

We extend to higher dimension our valuative analysis of singularities of psh functions started in [FJ2]. Following [KoSo], we describe the geometry of the space V of all normalized valuations on C[z1, . . . , zn] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C n over the origin, we construct a natural class of convex functions on V. For bounded convex functions on V, we define a mixed Monge-Ampère operator which reflects the intersection theory of divisors over the origin of C. This operator associates to any (n − 1)-tuple gi of such functions a positive measure of finite mass MA (gi) on V. Next, we show that the collection of Lelong numbers of a given germ of a psh function at all infinitely near points induces a convex function gu on V. When φ is a psh Hölder weight in the sense of Demailly [De], the generalized Lelong number νφ(u) equals R V gu MA(gφ). In particular, any such number is an average of valuations. We also show how to compute the multiplier ideal of u and the relative type of u with respect to φ in the sense of Rashkovskii [Ras], in terms of gu and gφ.