Valuations and multiplier ideals

By C. Favre, Mattias Jonsson. 2004

154 citations recorded by the source corpus.

This article is the third of a series of work on a new approach to the study of singularities of various objects in a local, two-dimensional setting. Our focus in the present paper is on multiplier ideals and singularity exponents. In the discussion below, we fix an equicharacteristic zero, two-dimensional regular local ring (R, m) with algebraically closed residue field. An important example is the ring R = (Do o? holomorphic germs at the origin in C2. In [FJ1], we introduced the space V consisting of all R+U{+00}-valued valuations on R centered at m, and normalized by ^(m) = 1. This space is naturally a tree: it is a union of (uncountably many) real segments patched together in such a way that V remains homotopic to a point. It is also an R-tree in the classical sense for a natural metric. We hence call V the valuative tree. It encodes in a natural way all possible blowups of R centered at m and therefore gives a way of measuring quite precisely singularities of different kinds of objects. The points in V that are not ends form the subtree Vqm of quasimonomial valuations. These valuations, which can alternatively be characterized as Abhyankar valuations of rank 1 or as

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1217af4f4efbdefe0d251d15e2a8dc783a893b5f

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Citations in the corpus

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]).”
Toric plurisubharmonic functions and analytic adjoint ideal sheaves (2010, background): “This approach has been widely developed in [8] in the two-variable case, and in [1] in higher dimensions.”
A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “Using the characterization of integrability exponents of psh functions in terms of Lelong numbers [FJ05b, BFJ08] (see Appendix B) we prove that the L θ part of the Ding functional D where the right-hand side is deﬁned by the same formula as above.”
Dynamics on Berkovich Spaces in Low Dimensions (2012, methodology): “ above proof, which uses tree arguments together with Lipman’s result in Proposition 7.4, does not generalize. Instead, one can construct the ideals an as valuative multiplier ideals. This is done in [FJ05b] in dimension two, and in [BFJ08b] in higher dimensions. 7.9. Intrinsic description of the tr”
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].”
Valuations and asymptotic invariants for sequences of ideals (2010, result): “. If Conjecture C holds for all n≤d, then Conjecture B holds for all Xwith dim(X) ≤d. We give a proof of the strong version of Conjecture C in dimension ≤2. The argument is similar to the one used in [FJ3], where a version of Conjecture B is proved. However, as opposed to [FJ3], the proof given here”
The openness conjecture for plurisubharmonic functions (2013, background): “This conjecture has attracted a good deal of attention; in pa rticular it has been completely proved in dimension 2 by Favre and Jonsson, [6].”
Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties (2011, background): “has analytic singularities, as one sees by spelling out the integrability condition on a log-resolution of the singularities of ψ−. The 2-dimensonal case of the openness conjecture was established in [FJ05]. Remark 1.7. We also recall that ∆ϕ∈ Lq loc for some q<+∞ implies that ϕlocally belongs to”
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]).”
Integral closure of ideals, rings, and modules (2006, background): “For more information about this correspondence and related topics see [78], [79], [50], [53], and [94].”
An algebraic approach to the openness conjecture of Demailly and Kollár (2012, methodology): “The strategy in [13] is to consider a subspace V of semivaluations v: Ox→ [0,+∞] satisfying v(mx) = 1, where mx is the maximal ideal.”
The openness conjecture and complex Brunn-Minkowski inequalities (2014, background): “The openness conjecture was first proved in dime nsion 2 by Favre and Jonsson (see [11]), and then for all dimensions in [2].”
The weighted log canonical threshold (2014, background): “In 2005, this conjecture was proved in dimension 2 by Favre and Jonsson [12,20,21].”
A proof of Demailly's strong openness conjecture (2015, background): “The dimension two case of the openness conjecture was proved by Favre and Jonsson in [12] (see also [13]).”