Abstracts (pdf)
Let X be a surface of general type and let f:C -> X be an
entire curve. We show that if f(C) is contained in a real analytic
hypersurface then f(C) is actually contained in a complex algebraic
curve.
(J'insiste sur le fait que ce resultat n'a presque rien a voir avec
l?hyperbolicite' (si ce n'est que pour l'usage de certains resultats bien
connus). En fait, c'est un resultat sur les singularites des hypersurfaces
Levi-plates analytiques reelles, de nature completement locale.)
For his proof of rigidity for curves over function fields with
given locus of bad reduction, Arakelov uses Weierstrass points to achieve
ampleness of the relative dualizing sheaf. Combining his isomorphism
involving Wronskian differentials with an isomorphism observed by Mumford
one arrives at a certain explicit ample line bundle on the moduli space of
stable curves. We study some of the properties of this line bundle.
New results on the algebraic structure of jet differential operators
on surfaces will be presented : calculation of generators, relative
base locus, Chern classes. The ultimate goal is be to calculate the
asymptotics of the number of sections on an arbitrary subvariety of
the jet spaces, leading potentially to hyperbolicity results
of a very general nature.
Let n=2,3,4,5 and let X be a smooth complex projective manifold of dimension n and degree d. We determine an effective lower bound for d in order to have existence of global invariant jet differentials on X. This is done by using the algebraic version of holomorphic Morse inequalities on a particular subbundle of the bundle of invariant jet differentials.
We present an approach of Wu and Zheng to prove a structure
theorem for complex manifolds with non-positive holomorphic bisectional
curvature. In particular we point out the essential role of the maximal
foliation tangent to the null directions of the curvature on such
manifolds and suggest an algebraic method to construct such foliations
using the Harder-Narasimhan filtration of the tangent bundle.
Jahwer El Goul: Almost ampleness of the cotangent bundle of some surfaces of general type
We give an intrinsic characterization of the unit polydisk
in the complex euclidean n-space from the viewpoint of the holomorphic
automorphismn groups. (slides)
The Zalcman renormalization method is used
to prove renormalization of non normal sequences of harmonic functions (or maps)
to nonconstant affine functions. We give some applications
to hyperbolicity criteria for tube domains.
(reference: the paper: Applications harmoniques et hyperbolicite de domaines tubes
accepted in Enseignement Mathematique. A preliminary version is on arxiv).
We will review and strengthen some
inequalities of Miyaoka-Yau type in order to
obtain effective bounds on codimension-one
images of Calabi-Yau varieties
in varieties of general type.
Soit W -> X une variété projective non singulière réelle de dimension 3 fibrée en courbes rationnelles. On suppose que W(R) est orientable. Soit N une composante connexe de W(R). D'après Kollár, N est alors essentiellement une variété de Seifert ou une somme connexe d'espaces lenticulaires. Soit k un entier défini de la façon suivante : Si g : N -> F est une fibration de Seifert, on note k le nombre de fibres multiples de g. Si N est une somme connexe d'espaces lenticulaires, on note k le nombre d'espaces lenticulaires différents de P^3(R).
Théorème 1
Si X est une surface géometriquement rationnelle, alors k est majoré par 4.
Théorème 2
Si de plus $F$ est difféomorphe au tore S^1xS^1, alors k = 0.
Ces résultats répondent par l'affirmative à deux questions de Kollár qui avait montré en 1999 que k était majoré par 6. On déduit ce théorème d'une analyse fine de certaines surface de Del Pezzo singulières avec singularités Du Val.
(slides)
We will construct a bounded holomorphic map
whose radial cluster set at every point in
arbitrarily given sequence in the boundary
is a direct product of balls.
(slides)
Les resultats comprennent : etude des jets d'ordre 4 et 5 en dimension
2 ainsi que des jets d'ordre 4 en dimension 3 ; etude d'un sous-fibre
dont la caracteristique d'Euler croit plus rapidement que
celle du fibre de Demailly-Semple.
(slides)
Using the harmonic theory developed by Takegoshi for representation of
relative cohomology, and the framework
of computation of curvature of direct images bundles by Berndtsson, we
prove that the higher direct images by a smooth morphism of the
relative canonical bundle twisted by a semi-positive vector bundle
are locally free and semi-positively curved, when endowed with a
suitable Hodge type metric.
An estimate of Nevanlinna's SMT-type which is once established
is very effective for the study of degeneracy of holomorphic curves and the
Kobayashi hyperbolicity problem.
Beginning with Cartan's SMT, I will survey the estimates of SMT-type
for holomorphic curves so far obtained, referring related results.
I will mention open problems in this direction, too.
(slides)
I will describe some results obtained around the Kobayashi conjecture which predicts the hyperbolicity of generic hypersurfaces of large degree in the projective space and the hyperbolicity of their complements.
(slides)
In the talk, I'll present the following theorem:
Main Theorem. Let X \subset {\Bbb P}^N({\Bbb C}) be a smooth
complex projective variety of dimension $n \ge 1$ and degree $d$. Let
$f: {\Bbb C} \rightarrow X $ be an algebraically non-degenerate
holomorphic map, and let ${\bf f}=(f_0, \dots, f_N)$ be the reduced
representation of $f$. Define, for every $z \in \Bbb C$,
$$c_j(z)=\log {\|f(z)\|\over |f_j(z)|}, 0 \leq j \leq N,$$ and let
${\bf c}(z)=(c_0(z), \dots, c_N(z))$. Denote by $e_X({\bf c})$ the
Chow weight of $X$ with respect to ${\bf c}$. Let $L$ be an ample line
bundle and let $c_1(L)$ be the Chern form of $L$. Then, for every
$\epsilon >0$, $${1\over d(n+1)}\int_{0}^{2\pi} e_X({\bf
c}(re^{i\theta})){d\theta\over 2\pi} \leq (1+\epsilon) \int_{r_0}^r
{dt\over t}\int_{|z| \leq t} f^*c_1(L),$$ where the inequality holds for
all $r \in (0, +\infty)$ except for a possible set $E$ with finite
Lebesgue measure.}
Various consequences of the theorem, including
the recent solution to the Shiffman conjecture by the author, will
also be discussed. (slides)
In this talk, we would like to present two problems:
1) Characterization of domains in C^n by their noncompact automorphism groups.
2) Concerning the Green-Krantz-Conjecture, we study the parabolicity of domains in C^2 with boundary points of infinite type by showing that, generally, there is no
parabolic orbit accumulation point of infinite type for these domains.
(slides)
This talk contains two parts. In the first part, we discuss the Second Main Theorem for algebraically nondegenerate meromorphic mappings into CPn with moving hypersurfaces and multiplicities are truncated. This part is a joint work with Gerd Dethloff . In the second part, unicity theorems of meromorphic mappings with few hyperplanes are given. This is a joint work with Si Duc Quang.
(slides)
I would like to explain how to construct a singular hermitian
metric $\hat{h}_{can}$on $k_{X}+\Delta$ for a KLT pair $(X,\Delta)$.such
that
(1) $\hat{h}_[can}$ is uniquely determined by $(X,\Delta)$.
(2) $\hat{h}_{can}$ is an AZD of $K_{X} +\Delta$.
(3) The curvature current of $\hat{h}_{can}$ is semipositive on a
projective
family of KLT pairs.
I would like to discuss application of the metrics
such as
(1) Deformation invariance of (log) plurigenera.
(2) Semipositivity of multirelative canonical bundles
which relates to the Iitaka conjecture.
(3) Moduli problem for polarized varieties.
The Kobayashi problem suggests that generic hypersurfaces of degree 2n-1 in P^n are Kobayashi hyperbolic. So far the optimal bound in this problem
has not been reached. Therefore, constructing small degree examples is of interest. We will introduce a deformation method for constructing small degree examples. This is joint work of the speaker with B.Shiffman. (slides)