Let $\tilde{X}$ be a desingularization of a normal surface $X$. The group Pic$\left(\tilde{X}\right)$ is provided with an order relation $L\underset{\_}{\gg }0$, defined by $L$. $V\le 0$ for any effective exceptional divisor $V$. Comparing to the usual order relation we define the ceiling of $L$ which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which...

We prove a uniqueness result for Coleff-Herrera currents which in particular means that if $f=(f_1,...,f_m)$ defines a complete intersection, then the classical Coleff-Herrera product associated to $f$ is the unique Coleff-Herrera current that is cohomologous to $1$ with respect to the operator ${\delta }_f-\overline{\partial }$, where ${\delta }_f$ is interior multiplication with $f$. From the uniqueness result we deduce that any Coleff-Herrera current on a variety $Z$ is a finite sum of products of residue currents with support on $Z$ and holomorphic...

Let $V$ be an algebraic variety in $C^n$ and when $k\ge 0$ is an integer then $\mathrm{Pol}(V,k)$ denotes all holomorphic functions $f\left(z\right)$ on $V$ satisfying $\left|f\left(z\right)\right|\le C_f(1+|z|{)}^k$ for all $z\in V$ and some constant $C_f$. We estimate the least integer $ϵ(V,k)$ such that every $f\in \mathrm{Pol}(V,k)$ admits an extension from $V$ into $C^n$ by a polynomial $P(z_1,...,z_n)$, of degree $k+ϵ(V,k)$ at most. In particular ${lim}_{k\>\infty }ϵ(V,k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$. The existence of the finite integer $ϵ(V,k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem. ...

Let $X$ be a Banach space with a countable unconditional basis (e.g., $X={\ell }_2$), $\Omega \subset X$ an open set and $f_1,...,f_k$ complex-valued holomorphic functions on $\Omega$, such that the Fréchet differentials $d{f}_1\left(x\right),...,d{f}_k\left(x\right)$ are linearly independant over $ℂ$ at each $x\in \Omega$. We suppose that $M=\{x\in \Omega :f_1\left(x\right)=...=f_k\left(x\right)=0\}$ is a complete intersection and we consider a holomorphic Banach vector bundle $E\to M$. If $I$ (resp.${𝒪}^E$) denote the ideal of germs of holomorphic functions on $\Omega$ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$), then the sheaf cohomology groups...

The webs have been studied mainly locally, near regular points (see a short list of references on the topic in the bibliography). Let d be an integer ≥ 1. A d-web on an open set U of ℂ² is a differential equation F(x,y,y’) = 0 with $F(x,y,y^{\text{'}})={\sum }_{i=0}^d{a}_i(x,y){\left(y^{\text{'}}\right)}^{d-i}$, where the coefficients $a_i$ are holomorphic functions, a₀ being not identically zero. A regular point is a point (x,y) where the d roots in y’ are distinct (near such a point, we have locally d foliations mutually transverse to each other, and caustics appear...

Let $\Omega$ an open set in ${\mathbf{C}}^4$ near $z_0\in \partial \Omega$, $\lambda$ a suitable holomorphic function near $z_0$. If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\bar{\partial }u=\lambda f$, ($f$ is a $(0,1)$ form, $\bar{\partial }$ closed in $U\left(z_0\right)$ in $U\left(z_0\right)$ with supp$\left(u\right)\subset \bar{\Omega }\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U\left(z_0\right)$, as holomorphic fonctions in $\Omega \cap V\left(z_0\right)$.