Valuative analysis of planar plurisubharmonic functions
By C. Favre, Mattias Jonsson. 2004
81 citations recorded by the source corpus.
We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set $\mathcal{V}$ of valuations on R and – by way of a natural Laplace operator defined in terms of the tree structure on $\mathcal{V}$ – a positive measure on $\mathcal{V}$. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampère mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed (1,1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/49409932503c046767547d82549be0b4a837e694
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- 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].”
- A sharp effectiveness result of Demailly's strong openness conjecture (2017, background): “Depending on the truth of OC, the above conjecture was proved in [16] (the two-dimensional case was proved by Favre-Jonsson [10]).”
- Valuations and multiplier ideals (2004, methodology): “Equisingularity may be geometrically interpreted as follows: for any composition π of blowups, the Lelong numbers of the pull-backs π ∗ u and π ∗ v are the same at any point on the exceptional divisor π − 1 (0); see [FJ2, Proposition 6.2].”
- Effectiveness of Demailly’s strong openness conjecture and related problems (2014, background): “For n ≤ 2, the above conjecture was proved by Favre and Jonsson in [13] (see also [12]).”
- Valuations and plurisubharmonic singularities (2007, methodology): “position to relate the generalized Lelong number with respect to a tame weight to an average of valuations, largely by repeating the argument given in [ FJ2 ].”
- A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “For each v ∈ X div we can make sense of v ( θ ) = v ( ψ ) as a generic Lelong number on some blowup; see [FJ05a,BFJ08] and Appendix B.”
- Open problems in pluripotential theory (2015, unknown): “When n = dimC ≤ 2, it follows from the work of Blel and Mimouni [83], the second author [84] and Favre and Jonsson [85] that φ can be further decomposed as follows: for any ε > 0, there exists πε : Xε → X a finite composition of blow-ups such that π∗ ε (ω + ddcφ) = [Dε] + Rε ,”