Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$. Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d=2$ or $3$, we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$); for $d=2$, we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s ≥ 3. Also let ${\Omega }_{\pi }$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f:\partial {\Omega }_{\pi }\to ℝ$ that can be extended to a real-valued pluriharmonic function in ${\Omega }_{\pi }$.

Let $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^k$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda }^{-n}{\left(f^n\right)}^{*}S$ converge to the Green current $T_f$. For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$-invariant currents.

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

Given a coarse space $(X,ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X\times X$ to the natural coarse structure on $(ℬ\setminus \{\varnothing \left\}\right)\times (ℬ\setminus \{\varnothing \left\}\right)$ and say that a macro-uniform mapping $f:(ℬ\setminus \{\varnothing \left\}\right)\to X$ (or $f:{\left[X\right]}^2\to X$) is a selector (or 2-selector) of $(X,ℰ)$ if $f\left(A\right)\in A$ for each $A\in ℬ\setminus \left\{\varnothing \right\}$ ($A\in {\left[X\right]}^2$, respectively). We prove that a discrete coarse space $(X,ℰ)$ admits a selector if and only if $(X,ℰ)$ admits a 2-selector if and only if there exists a linear order “$\le$" on $X$ such that the family of intervals $\left\{\right[a,b]:a,b\in X,a\le b\}$ is a base for the bornology $ℬ$.

Let ${\Phi }_1,...,{\Phi }_{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U{\subset }^{N+m}$, let $𝕄$ be a $N$-dimensional $C^k$ submanifold of $U$ and define $$𝕋:=\{z\in 𝕄:{\Phi }_1\left(z\right),...,{\Phi }_{k+1}\left(z\right)\in T_z𝕄\}$$ where $T_z𝕄$ is the tangent space to $𝕄$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $𝕄$) of $𝕋$: If $z_0\in 𝕄$ is a $(N+k)$-density point (relative to $𝕄$) of $𝕋$ then all the iterated Lie brackets of order less or equal to $k$ $${\Phi }_{i_1}\left(z_0\right),[{\Phi }_{i_1},{\Phi }_{i_2}]\left(z_0\right),[[{\Phi }_{i_1},{\Phi }_{i_2}],{\Phi }_{i_3}]\left(z_0\right),...(h,i_h\le k+1)$$ belong to $T_{z_0}𝕄$. Such a property has been proved in [9] for $k=1$ and its proof in the...