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.