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Nordine Mir

Finite jet determination of CR maps of positive codimension

Abstract

We will present recent results about finite jet determination of CR maps of positive codimension from real-analytic CR manifolds into Nash manifolds in complex spaces of possibly different dimension. This is joint work with B. Lamel.

Lucas Kaufmann

Random walks on SL_2(C)

Abstract

Given a sequence of independent and identically distributed random 2 by 2 complex matrices, it is a classical problem to study the statistical properties of their products.

In this talk I will show how methods from complex analysis can be used to obtain several new limit theorems for these random processes, often in their optimal version. I will also discuss new results on the Fourier coefficients of the Furstenberg measure. This is based on joint works with T.-C. Dinh and H. Wu.

Keiji Oguiso

Some aspects of real form problem of a smooth complex projective variety

Abstract

My talk is a report of current joint works with Professors Tien-Cuong

Dinh and Xun Yu.

After Lesieutre, there made several works related to a long standing

open question since Kharlamov: "Is there a smooth complex projective

rational surface with infinitely many real forms up to isomorphisms, i.e.

, with infinitely many ways to describe by a system of equations with

real coefficients up to isomorphisms over the real number field?"

After a brief introduction on the real form problem with currently known

results and one way to reduce the problem to a problem of concrete

complex geometry, I would like to explain our affirmative answer to the

question above, with an outline of proof using a special Kummer K3

surface and its rich geometry.

I would like to close this talk by discussing relevant results in higher

dimensions and/or non-negative Kodaira dimensions, together with related open problems.

Min Ru

Nevanlinna hyperbolicity for complex manifolds

Abstract

Similar to the notion of the algebraic hyperbolicity introduced by J.P. Demailly and Xi Chen, we introduce the notion of Nevanlinna hyperbolicity for a pair (X, D), where X is a complex projective variety and D is an Cartier divisor on X which may be empty. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension property (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. This is a joint work with Yan He.

Fabrizio Bianchi

Higher bifurcations for polynomial skew products

Abstract

Given a holomorphic family of rational maps on the Riemann Sphere, one can decompose the parameter space into a stability locus and a bifurcation locus. The latter corresponds to maps whose global dynamics are very sensitive to a perturbation of the parameter and is characterized as the support of the so-called bifurcation current.

The changes in the global dynamics are dictated by changes in the dynamics of the critical set. When several critical points are present, it makes sense to define a stratification of the bifurcation locus, depending on how many critical points bifurcate independently. The good framework to do this is by means of the self-intersections of the bifurcation current, and one can prove that the resulting stratification is strict.

We consider in this talk dynamical systems on C^2 of the form f(z,w)= (p(z),q(z,w)), for suitable polynomials p,q. The stability/bifurcation dichotomy in higher dimensions was developed in previous joint work with Berteloot and Dupont. We prove here that, in contrast with the one-dimensional case, all the self-intersections of the bifurcation current have the same support.

Joint work with Matthieu Astorg, Orleans.

Ben Weinkove

The Chern-Ricci flow

Abstract

I will give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian

metrics on complex manifolds. I will emphasize open problems and new directions.

Qifeng Li

Unbendable rational curves of Goursat type and Cartan type

Abstract

In this talk we discuss the relation between the geometry of unbendable rational curves on a complex manifold and natural differential systems on the corresponding Douady space parameterizing these curves. We give a correspondence between the germs of unbendable rational curves of Goursat type with ordinary differential equations of order three, and give a correspondence between the germs of unbendable rational curves of Cartan type with germs of contact manifolds. This is a joint work with Jun-Muk Hwang.

Judith Brinkschulte

Dynamical aspects of holomorphic foliations with ample normal

bundle

Abstract

I will report on a joint paper with M. Adachi, where we proved a conjecture by Brunella: Let $X$ be a compact complex manifold of dimension $\geq 3$. Let $\mathcal{F}$ be a codimension one holomorphic foliaton with ample normal bundle. Then every leaf of $\mathcal{F}$ accumulates to the singular set of $\mathcal{F}$. I will also discuss related results concerning the nonexistence problem for Levi-flat real hypersurfaces.

Ngoc-Cuong Nguyen

The complex Sobolev space and H\"older continuous solutions to Monge-Amp\`ere equations

Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c u+\omega)^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $\omega$-plurisubharmonic function. We show that the equation admits a H\"older continuous solution {\it if and only if} the measure $\mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is H\"older continuous. A similar result is also obtained for the complex Monge-Amp\`ere equations on domains of $\bC^n$.

Viet-Anh Nguyen

Positive plurisubharmonic currents: Generalized Lelong numbers and Tangent theorems

Dinh--Sibony theory of tangent and density currents is a recent but powerful tool to study positive closed currents. Over twenty years ago, Alessandrini--Bassanelli initiated the theory of the Lelong number of a positive plurisubharmonic current in $\C^k$ along a linear subspace. Although the latter theory is intriguing, it has not yet been explored in-depth since then.

Introducing the concept of the generalized Lelong numbers and studying these new numerical values, we extend both theories to a more general class of positive plurisubharmonic currents and in a more general context of ambient manifolds.

%and define new numerical values associated to such a current

%and a submanifold in its domain of definition.

More specifically, in the first part of our talk,

we consider a positive plurisubharmonic current $T$ of bidegree $(p,p)$ on a complex manifold $X$ of dimension $k,$

and let $V\subset X$ be a K\"ahler submanifold of dimension $l$ and $B$ a relatively compact piecewise $\mathcal{C}^2$-smooth open subset of $V.$

We impose a mild reasonable condition on $T$ and $B,$ namely, $T$ is weakly approximable by $T_n^+-T_n^-$ on a neighborhood $U$ of $\overline B$ in $X,$ where

$(T^\pm_n)_{n=1}^\infty$ are some positive plurisubharmonic $\mathcal{C}^3$-smooth forms of bidegree $(p,p)$ defined on $U$ such that the masses $\|T^\pm_n\|$ on $U$ are uniformly bounded and

that the $\mathcal{C}^3$-norms of $T^\pm_n$ are uniformly bounded near $\partial B$

if $\partial B\not=\varnothing.$ Note that if $X$ is K\"ahler and $T$ is of class $\mathcal{C}^3$ near $\partial B$

(for example, this $\mathcal{C}^3$-smoothness near $\partial B$ is automatically fulfilled if either $\partial B=\varnothing$ or $V\cap \mathrm{supp}( T)\subset B$), then the above mild condition is satisfied.

Then, we define the notion of the $j$-th Lelong number of $T$ along $B$ for every $j$ with $\max(0,l-p)\leq j\leq \min(l,k-p)$ and prove their existence as well as their basic properties.

We also show that $T$ admits tangent currents and that all tangent currents are not only positive plurisubharmonic, but also partially $V$-conic and partially pluriharmonic. When the current $T$ is moreover pluriharmonic (resp. closed), then every tangent current is

also $V$-conic pluriharmonic (resp. $V$-conic closed).

We also prove that the generalized Lelong numbers are intrinsic.

In fact, if we are only interested in the {\it top degree} Lelong number of $T$ along $B$ (that is, the $j$-th Lelong number

for the maximal value $j=\min(l,k-p)$), then under a suitable holomorphic context, the above condition on the uniform regularity of $T^\pm_n$ near $\partial B$ can be removed. Our method relies on some Lelong-Jensen formulas for

the normal bundle to $V$ in $X,$ which are of independent interest.

The second part of our talk is devoted to geometric characterizations of the generalized Lelong numbers.

As a consequence of this study, we show that the top degree Lelong number of $T$ along $B$ is strongly intrinsic.

%: it is independent of the choice of metrics on the normal bundle to $V$ in $X.$

This is a generalization of the fundamental

result of Siu (for positive closed currents) and of Alessandrini--Bassanelli (for positive plurisubharmonic currents)

on the independence of Lelong numbers at a single point on the choice of coordinates.