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 V of all normalized valuations on 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 C n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we define a mixed Monge-Ampère operator which reflects the intersection theory of divisors above the origin of C n . This operator associates to any (n − 1)-tuple of formal psh functions a positive measure of finite mass on 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 functionû on V. When ϕ is a psh Hölder weight in the sense of Demailly, the generalized Lelong number ν ϕ (u) equals the integral of u against the Monge-Ampère measure ofφ. 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 ϕ in the sense of Rashkovskii, in terms ofû andφ.

Analytification is the limit of all tropicalizations

“…Kontsevich and Soibelman have identified the analytification with an inverse limit of "Clemens polytopes" of simple normal crossing models over the valuation ring, using theorems on existence of semistable reductions [20]. Similar inverse limit constructions with simple normal crossing resolutions appear in work of Boucksom, Favre, and Jonsson on valuations and singularities in several complex variables [10]. Arguments close to the spirit of this paper also appear in Gubler's elegant study of tropicalization of nonarchimedean analytic spaces [15].…”

Effectiveness of Demailly’s strong openness conjecture and related problems

“…In [19,20], the authors solved the strong openness conjecture posed by Demailly in [6] and [7] (see also [9], [11], [3], [29], [24], [22], [4], [25], [44], [30], etc. ):…”

Algebraic bounds on analytic multiplier ideals

“…-Denote the Siu decomposition of T as There are deeper results relating valuations to analytic multiplier ideals in [15], [16], and in [3], but we do not need them.…”