arXiv subjects

Charles Favre

Publications and source records attributed to Charles Favre.

43 records · Page 3Linked to original sources

arXiv2007

Eigenvaluations

We study the dynamics in C^2 of superattracting fixed point germs and of polynomial maps near infinity. In both cases we show that the asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the adequate invariance property. This is done by finding an infinitely near point at which the map becomes rigid: the critical set is contained in a totally invariant set with normal crossings. We locate this infinitely near point through the induced action of the dynamics on a space of valuations. This space carries an real-tree structure and conveniently encodes local data: an infinitely near point corresponds to a open subset of the tree. The action respects the tree structure and admits a fixed point--or eigenvaluation--which is attracting in a certain sense. A suitable basin of attraction corresponds to the desired infinitely near point.

math.DS↗

arXiv2006

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over C_p, for each prime p.

math.NT↗

arXiv2005

Valuations and multiplier ideals

We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollar. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on the realization of ideals as multiplier ideals.

math.CV↗

arXiv2005

Valuative analysis of planar plurisubharmonic functions

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 V of valuations on R and--by way of a natural Laplace operator defined in terms of the tree structure on V--a positive measure on 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-Ampere 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.

math.CV↗

arXiv2004

Theoreme d'equidistribution de Brolin en dynamique p-adique

We prove an analog of the famous equidistribution theorem of Brolin for rational mappings in one variable defined over the p-adic field C_p. We construct a mixing invariant probability measure which describes the asymptotic distribution of iterated preimages of a given point. This measure is supported on the Berkovich space associated to the projective line over C_p. We show that its support is precisely the Julia set as defined by Rivera-Letelier. Our results are based on the construction of a Laplace operator on real trees with arbitrary number of branching as done by Favre-Jonsson.

math.DS↗

arXiv2003

The valuative tree

We describe the set V of all real valued valuations v on the ring C[[x,y]] normalized by min{v(x),v(y)}=1. It has a natural structure of an R-tree, induced by the order relation v is less than v' iff v(f) is less than v'(f) for all f. It can also be metrized, endowing it with a metric tree structure. From the algebraic point of view, these structures are obtained by taking a suitable quotient of the Riemann-Zariski variety of C[[x,y]], in order to force it to be a Hausdorff topological space. The tree structure on V also provides an identification of valuations with balls of irreducible curves in a natural ultrametric. We show that the dual graphs of all sequences of blow-ups patch together, yielding an R-tree naturally isomorphic to V. Altogether, this gives many different approaches to the valuative tree V. We then describe a natural Laplace operator on V. It associates to (special) functions of V a complex Borel measure. Using this operator, we show how measures on the valuative tree can be used to encode naturally both integrally closed ideals in R and cohomology classes of the local analog of voute etoilee over the complex plane.

math.AC↗

arXiv2001

Brolin's Theorem for curves in two complex dimensions

Given a holomorphic selfmap f of the complex projective plane of algebraic degree at least 2, we give sufficient conditions on a positive closed (1,1) current S of unit mass under which the normalized pullbacks of S under iterates of f converge to the Green current T. In particular, we completely characterize the plane algebraic curves whose normalized pullbacks converge weakly to T.

math.DS↗