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.