Valuations and plurisubharmonic singularities

By S. Boucksom, C. Favre, Mattias Jonsson. 2007

183 citations recorded by the source corpus.

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[x1 ,...,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,

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1042f859fb52c22495cd749b7ca29d87446ba9b

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Growth of balls of holomorphic sections and energy at equilibrium (2008, background): “It bears a strong resemblance with the differentiability property of the volume of divisors [27], which is in some sense a non-archimedean analogue of the present result (compare [26]).”
Singular semipositive metrics in non-Archimedean geometry (2011, background): “en combinatorial decompositions of X, in the spirit of the toroidal constructions of [KKMS]. These techniques can be extended to other geometric contexts such as the space of valuations considered in [BFJ08]. In particular they yield a uniform version of Izumi’s theorem [Izu85]. However, we shall po”
Geodesic rays and stability in the cscK problem (2020, background): “As pointed out by Boucksom, this lemma is essentially a local result about psh functions and related questions have been extensively studied in [12, 44].”
Boundary complexes and weight filtrations (2011, background): “ complex is their valuative tree [FJ04], and for the complement of a line it is their tree of valuations at innity [FJ07, FJ11]. The analogue of the valuative tree in higher dimensions is studied in [BFJ08]. These boundary complexes can also be applied to study invariants of singularities and, more ”
A proof of Demailly's strong openness conjecture (2015, background): “The result was conjectured in [3], where the authors show that I+(tu) = I+(tv) for all t > 0 is equivalent to (1) and (3) and implies (4).”
Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton (2012, background): “It is closely related to the thinness function constructed in [BFJ08] and the log discrepancy function in [JM11].”
K-semistability is equivariant volume minimization (2015, background): “ showing that every valuation of a surface can be resolved. Recently valuations have also become important in studying singularities of plurisubharmonic functions in complex analysis (see for example [BFJ08]). The concept of K-stability was introduced by Tian [Tia97] in his study of Kahler-Einstein ”
Analytification is the limit of all tropicalizations (2008, background): “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].”
An algebraic approach to the openness conjecture of Demailly and Kollár (2012, methodology): “The idea of studying psh functions using valuations was systematically developed in [2, 12, 13] but appears already in the work of Lelong [24] and Kiselman [20, 21].”
Dynamics on Berkovich Spaces in Low Dimensions (2012, methodology): “[z1;:::zn]] whose restriction to (˘) is trivial and such that v(m˘) >0 (resp., v(m˘) = 1). This shows that when Kis algebraically closed, the set V˘ above is isomorphic to the space considered in [BFJ08b]. This space was rst introduced in dimension n= 2 in [FJ04] where it was called the valuative”
The volume of an isolated singularity (2010, methodology): “ormal singularity at a given point 0 ∈X, and m⊂OX denotes the maximal ideal of 0. We ﬁrst show how to extend to this setting the intersection theory of nef b-divisors introduced in the smooth case in [BFJ08]. The main ingredient to do so is the approximation theorem from the previous section. We nex”