Valuations and Plurisubharmonic Singularities

Abstract

We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the first two authors. Following Kontsevich and Soibelman we describe the geometry of the space $\mathcal{V}$ of all normalized valuations on $\mathbf{C}[x_1,...,x_n]$ 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 ${\mathbf{C}}^n$ above the origin, we define formal psh functions on $\mathcal{V}$, designed to be analogues of the usual psh functions. For bounded formal psh functions on $\mathcal{V}$, we define a mixed Monge–Ampère operator which reflects the intersection theory of divisors above the origin of ${\mathbf{C}}^n$. This operator associates to any $(n-1)$-tuple of formal psh functions a positive measure of finite mass on $\mathcal{V}$. Next, we show that the collection of Lelong numbers of a given germ $u$ of a psh function at all infinitely near points induces a formal psh function $\hat{u}$ on $\mathcal{V}$. When $\phi$ is a psh Hölder weight in the sense of Demailly, the generalized Lelong number ${\nu }_{\phi }(u)$ equals the integral of $\hat{u}$ against the Monge–Ampère measure of $\hat{\phi }$. In particular, any generalized Lelong 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 $\phi$ in the sense of Rashkovskii, in terms of $\hat{u}$ and $\hat{\phi }$.