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Valuations and plurisubharmonic singularities | | Sebastien Boucksom
; Charles Favre
; Mattias Jonsson
; | | Date: |
16 Feb 2007 | | Subject: | Complex Variables; Algebraic Geometry | | Abstract: | We extend to higher dimension the valuative analysis of singularities of psh functions developed by the last two authors in dimension 2. Following Kontsevich and Soibelman, we describe the geometry of the space V of all normalized valuations on C[z_1,...,z_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 over the origin, we construct a natural class of convex functions on V. For bounded convex functions on V, we define a mixed Monge-Ampere operator which reflects the intersection theory of divisors over the origin of C^n. This operator associates to any (n-1)-tuple g_i of such functions a positive measure of finite mass MA(g_i) 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 g_u on V. When F is a psh Holder weight in the sense of Demailly, the generalized Lelong number v_F(u) equals the integral of g_u w.r.t the positive measure MA(g_F). 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 F in the sense of Rashkovskii, in terms of g_u and g_F. | | Source: | arXiv, math/0702487 | | Services: | Forum | Review | PDF | Favorites | |
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