Valuations and Plurisubharmonic Singularities

  • Sébastien Boucksom

    Université Paris 6, France
  • Charles Favre

    École Polytechnique, Palaiseau, France
  • Mattias Jonsson

    University of Michigan, Ann Arbor, United States


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 of all normalized valuations on 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 above the origin, we define formal psh functions on , designed to be analogues of the usual psh functions. For bounded formal psh functions on , we define a mixed Monge–Ampère operator which reflects the intersection theory of divisors above the origin of . This operator associates to any -tuple of formal psh functions a positive measure of finite mass on . Next, we show that the collection of Lelong numbers of a given germ of a psh function at all infinitely near points induces a formal psh function on . When is a psh Hölder weight in the sense of Demailly, the generalized Lelong number equals the integral of against the Monge–Ampère measure of . 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 with respect to in the sense of Rashkovskii, in terms of and .

Cite this article

Sébastien Boucksom, Charles Favre, Mattias Jonsson, Valuations and Plurisubharmonic Singularities. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, pp. 449–494

DOI 10.2977/PRIMS/1210167334