Let Ω be a domain of finite type in ℂ² and let f be a function holomorphic in Ω and belonging to ${}^{k,\alpha }\left(\Omega ̅\right)$. We prove the existence of boundary values for some suitable derivatives of f of order greater than k. The gain of derivatives holds in the complex-tangential direction and it is precisely related to the geometry of ∂Ω. Then we prove a property of non-isotropic Hölder regularity for these boundary values. This generalizes some results given by J. Bruna and J. M. Ortega for the unit ball.

For $z\in \partial B^n$, the boundary of the unit ball in ${ℂ}^n$, let $\Lambda \left(z\right)=\left\{\lambda \right|\lambda |\le 1\}$. If $f\in 𝕆\left(B^n\right)$ then we call $E\left(f\right)=\{z\in \partial B^n{\int }_{\Lambda \left(z\right)}|f\left(z\right){|}^2\mathrm{d}\Lambda \left(z\right)=\infty \}$ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_{\delta }$ and $F_{\sigma }$ subset of the projective $(n-1)$-dimensional space ${\mathbb{P}}^{n-1}=\mathbb{P}\left({ℂ}^n\right)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E\left(f\right)=E$.

Let $D$ be a domain in ${ℂ}^2$. For $w\in ℂ$, let $D_w=\{z\in ℂ\mid (z,w)\in D\}$. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E(D,f)$ of all $w$ such that $f(.,w)$ is not square-integrable in $D_w$ is of measure zero. We call this set the exceptional set for $f$. In this note we prove that for every $0<r<1$,and every $G_{\delta }$-subset $E$ of the circle $C(0,r)=\{z\in ℂ\mid |z|=r\}$,there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in ${ℂ}^2$ such that $E(B,f)=E.$

The moduli space of stable vector bundles over a moving curve is constructed, and on this a generalized Weil-Petersson form is constructed. Using the local Riemann-Roch formula of Bismut-Gillet-Soulé it is shown that the generalized Weil-Petersson form is the curvature of the determinant line bundle, equipped with the Quillen metric, for a vector bundle on the fiber product of the universal moduli space with the universal curve.

Let $A$ be a closed polar subset of a domain $D$ in $ℂ$. We give a complete description of the pluripolar hull ${\Gamma }_{D\times ℂ}^{*}$ of the graph $\Gamma$ of a holomorphic function defined on $D\setminus A$. To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.

On étudie la dimension moyenne de l’espace de courbes $1$-Brody à valeurs dans deux surfaces complexes  : d’abord pour des surfaces de Hopf, et ensuite pour ${\mathbf{P}}^2$ privé d’une droite. On montre dans le premier cas que la dimension moyenne est nulle via une borne sur la croissance des fonctions holomorphes faisant apparaître le lemme de la dérivée logarithmique. Pour montrer la positivité dans le deuxième exemple, on relève de la droite à son complémentaire un espace de courbes de Brody de dimension moyenne...