My research has been recently focused on the following four
subjects:
Here is list of my collaborators.
Potential
theory on Berkovich spaces
Berkovich theory was introduced in the
late 80's by Berkovich.
It gives a natural way to study analytic geometry
over non-archimedean fields. In this theory, analytic spaces (usually
referred to as Berkovich spaces) have the great advantage of being both
arcwise connected and locally compact. We refer to Ducros' webpage for a good
introduction to the subject and for an extensive bibliography on its
developements and applications.
What is potential theory
on Berkovich spaces? Berkovich spaces being the
analogs of complex analytic spaces over non-archimedean fields, it is
natural to trasnport complex tools to the non-archimedean setting.
Among these tools, the analysis
revolving around plurisubharmonic functions, positive currents, and the
ddc
operator plays
a particularly important role (see Demailly's webpage
where accessible surveys on the subject can be freely downloaded). It
thus seems natural to extend
this pluripotential analysis to the non-archimedean context. Beside
this fundamental aspect, pluripotential theory on Berkovich spaces has
potentially numerous applications: in arithmetics (see Chambert-Loir's
preprints on his webpage); in the study of singularities of psh function;
in holomorphic and non-archimedean dynamics.
The one-dimensional case,
i.e. the case of Berkovich curve, is now well understood, thanks to the
joint efforts of Baker-Rumely, Favre-Jonsson, and Thuillier. We refer to the
PhD thesis of the last named author for a general
construction of a Laplace operator on Berkovich curves; and to our
joint book with M. Jonsson The
Valuative Tree for a construction of a Laplace operator on a
general metrized tree. The main reason why the one-dimensional case is
easier to understand lies in the fact that locally any Berkovich curve
looks like a
tree. Below is an
artistic view (done by M. Jonsson) of a neighborhood of a point in a
Berkovich curve. Typically the boundary of such a neighborhood consists
of finitely many points.
The Laplace operator is then a kind of mixture of the standard
laplacian - d2/dx2 on the real line - which takes
into account the "continuous" part of the tree -, and the laplacian on
a
finite graph defined in a combinatorial way - which takes into account
the behaviour at the branching points- .
In
higher dimension, one should think of a local
neighborhood of a Berkovich space as a union of (affine)
simplices patched together in such a way that
the resulting space remains contractible. A more mathematical
formulation of this image is given by Kontesevich-Soibelman,
and also in a recent paper
of Thuiller. Although the affine structure on each simplex allows
one to build a natural Monge-Ampere operator, it seems complicated to
patch all these together. To overcom this difficulty, one is lead to a
more geometric construction of the Laplace operator.
The key idea is to realize that the Laplace operator (and
more
generally the potential theory) is the incarnation of the
intersection theory of divisors on all reductions of the Berkovich
space. A construction of a Monge-Ampere operator on
a Berkovich-like space (with a view towards application in singularities) is explained
in my joint paper Valuations
and plurisubharmonic singularities with S.
Boucksom and M. Jonsson.
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Singularities
of psh
function
A plurisubharmonic
function (or psh function in short) is a real-valued function
defined on a smooth complex space whose convexity properties are
adapted to the complex structure of the space. A singularity of
such a
function is a point where the value becomes negative infinite.
Singularities of psh function are particularly important as being a
source of construction of analytic objects, and as obstructions in the
intersection theory of positive closed current. We refer to the survey Plurisubharmonic
functions and potential theory in several
complex variables by Kiselman for more
informations on the history of the subject.
Various numbers have been introduced to measure the singularity of
a psh function by Lelong and subsequently by Demailly and Kiselman. For
instance, the Lelong number at 0 of the psh function log|f| with f
holomorphic is equal to the multiplicity of the hypersurface {f=0} at
0. In the aforementioned paper Valuations
and plurisubharmonic singularities written with S.
Boucksom and M.
Jonsson, we have shown that many of the fine properties of a psh
function near a singular point 0 can be understood through the data of
all its Lelong number at all infinitely near points (that is points
lying on some exceptional divisor of a sequence of blowups above 0).
Using potential theory
on Berkovich curves, we were also able with M. Jonsson to settle
positively the
following two questions in dimension
2:
- The openness conjecture (of
Demailly-Kollar). Pick u any psh function. Then the set of
t>0 such that e-tu is locally integrable is an open segment.
- Attenuation
of singularities. For
any ε>0, and any psh function u with an isolated
singularity, there
exists a
sequence of blowups such that the pull-back of the positive closed
current ddc u is a sum of a current of integration over the
exceptional divisors, and a current whose Lelong numbers are all ≤
ε.
We refer to our paper Valuations
and plurisubharmonic singularities for an approach to
these questions in higher dimension.
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Holomorphic dynamics
Holomorphic dynamics concerns the iteration of holomorphic
maps. There is a huge literature on the subject in one dimension (a
series of
surveys on the subject can be downloaded from Stony Brook). In
higher dimension,
the theory is much more recent and uses quite different techniques from
its one-dimensional analog. I am working on the more algebraic aspects
of the theory and more precisely on the problem of
description of the growth of
degrees. For a
fixed rational map F: Cn
→
Cn, define the
degree deg(F) as the
maximum of the degrees of its coordinates.
Problem. Describe the
behaviour of the sequence deg(Fk) when k tends to
infinity.
One can make this problem more precise as follows. Note first that
deg(Fk+l) ≤ deg(Fk) deg(Fl) hence
λ(F) = limk → ∞
deg(Fk)1/k is well-defined. It is called
the
asymptotic degree of F (or first dynamical degree), and
is invariant under conjugacy by a birational map.
Question 1: for any
rational map, the asymptotic degree is an
algebraic integer.
Question 2: deg(Fk)
= c. λ(F)k + O(hk) with h
<λ(F)
and c>0, except if F preserves an algebraic fibration.
With J. Diller, we have settled positively these two questions for
birational maps in dimension 2.
What we did was finding a good birational model in which the action of
F on the cohomology became compatible with the iteration.
In the case of polynomial maps of C2,
we have also settled these
questions with S. Boucksom and M. Jonsson in Eigenvaluation
and Degree
growth for meromorphic surface maps, and have made substantial
progresses for general rational surface maps. The main difference
with the invertible case is that in general no good birational model
exists where the action of F on the cohomology is analyzable. We thus
have to deal with all birational
models at the same time, and define and study a good cohomology theory
on the space one obtains after blowing up all points over P2.
We also refer to S. Cantat's paper Groupes
de
transformations birationnels du plan for another
applications of this kind of ideas.
In higher dimension though the situation is completely open, and the
two questions above remain
a great challenge. Viet-Anh Nguyen has introduced the class of quasi-algebraically
stable maps for which the analysis of growth of degrees is easy.
And there are series of interesting examples of
birational maps acting on the space of complex square matrices which
were worked out numerically
by a group of physicists leaded by J-M. Maillard, and theoretically
(see also)
by E. Bedford and K. Kim. But a general theoretical approach
remains to be built.
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Nonarchimedean dynamics
The basic problem is to study the dynamics of rational
maps R ∈ K(T) where K is a field endowed with a non-archimedean norm.
We usually assume K to be complete and algebraically closed but this is
not the only interesting case, see Benedetto-Briend-Perdry's
paper
for instance.
Benedetto and J. Rivera-Letelier have extensively developped the analog
of the complex Fatou-Julia theory in this context whose feature is as
follows.
Because the
standard projective line over K is both totally discontinuous and (at
least when the residue field of K is uncoutable) not locally compact,
one is naturally led to look at the action of R on the Berkovich
projective line P1ber(K).
This space is a compact tree (see the
picture above). One can then split the
Berkovich line into two totally invariant sets on which the dynamics of
R exhibits completely different behaviours.
On the Fatou set
(which is open), the dynamics is regular. This means that there is a
good understanding
of the dynamics on the preperiodic connected components of the Fatou
set (see Rivera-Letelier's
papers) ; and
there are reasonable conjectures concerning the appearance of wandering
components, as the following paper
of Benedetto indicates. Note that
the situation is here very different from the complex one, where
Sullivan's non-wandering theorem states that all connected components
of the Fatou set are necessarily preperiodic.
On the Julia set
(which is compact), the dynamics is essentially chaotic. In order to
describe it, it is thus natural to use statistichal tools and ergodic
theory. As a first step we have constructed with Rivera-Letelier a
mixing
measure whose support coincides with the Julia set, see . This measure
is not of maximal entropy as in the complex case in general, and it is
a very delicate task to compute explicitely the entropy of it. However
one can give bounds on the metric (and topological) entropy, and completely
characterize
those rational maps with zero topological entropy: these are rational
maps with good reduction (possibly after a conjugation by a Mobius
transformation). In this case, their Julia set is reduced to a point.
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Collaborators
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