We prove that for a real analytic generic submanifold $M$ of ${\mathbb{C}}^n$ whose Levi-form has constant rank, the tangential ${\overline{\partial }}_M$-system is non-solvable in degrees equal to the numbers of positive and $M$ negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with $M$ non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.

CONTENTSIntroduction...................................................................................................................................................................51. Notation and some definitions...................................................................................................................................6  (a) Analytic families.....................................................................................................................................................6  (b)...

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

Let $M$ be a two dimensional totally real submanifold of class $C^2$ in ${\mathbf{C}}^2$. A continuous map $F:\bar{\Delta }\to {\mathbf{C}}^2$ of the closed unit disk $\bar{\Delta }\subset \mathbf{C}$ into ${\mathbf{C}}^2$ that is holomorphic on the open disk $\Delta$ and maps its boundary $b\Delta$ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk $F^0$ with boundary in $M$, we describe the existence and behavior of analytic disks near $F^0$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve $F^0\left(b\Delta \right)$ in $M$. We also prove a regularity...

We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map $f_{*}{|}_Y:Y\subseteq J^r(D,M)\to J^r(D,N)$ is generically (for $f:M\to N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\to P$ to the preimage ${\left(j^{s}f\right)}^{-1}\left(A\right)$ of a submanifold $A\subseteq J^s(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction ${g|}_{{\left(j^{s}f\right)}^{-1}\left(A\right)}$ is also generic. We also present an example...

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...