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...

Dans cet article on étudie les $𝒟$-modules dont le support singulier est un croisement normal dans ${\mathbf{C}}^n$, par l’intermédiaire de la catégorie équivalente de faisceaux pervers. On montre qu’ils sont caractérisés, à isomorphisme près, par la donnée suivante : un hypercube constitué par des espaces vectoriels de dimension finie $F_I$ indexés par les parties de $\{1,...,n\}$, et des applications linéaires $F_I\rightleftarrows F_{I\cup \left\{i\right\}}$ soumises à certaines conditions de commutativité et d’inversibilité. Ce résultat est exprimé sous forme d’une équivalence...

Dans un article sur la transformation de Radon-Penrose, A. D’Agnolo et P. Schapira ont montré qu’au-dessus d’une variété complexe $X$ de dimension $\ge 3$, tout $\widehat{ℰ}$- module localement libre de rang $1$ est de la forme $\widehat{ℰ}{\otimes }_{{\pi }^{-1}𝒪}{\pi }^{-1}ℒ$ pour un fibré inversible $ℒ$ sur $X$. Ce résultat est faux en dimension $2$, et le but de ce travail est de déterminer la structure des $𝒟$- modules micro-localement libres de rang $1$ dans ce cas. Un des principaux résultat est la description des $𝒟$-modules micro-localement libres de rang un en termes...

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

In 1958, H. Grauert proved: If D is a strongly pseudoconvex domain in a complex manifold, then D is holomorphically convex. In contrast, various cases occur if the Levi form of the boundary of D is everywhere zero, i.e. if ∂D is Levi flat. A review is given of the results on the domains with Levi flat boundaries in recent decades. Related results on the domains with divisorial boundaries and generically strongly pseudoconvex domains are also presented. As for the methods, it is explained how Hartogs...

A mapping $F:ℝⁿ\to {ℝ}^m$ is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping $F:ℝⁿ\to {ℝ}^m$ can be reduced to the case m = n.

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber $f^{-1}\left(t₀\right)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}\left(t₀\right)$. We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...

For every polynomial F in two complex variables we define the Łojasiewicz exponents ${ℒ}_{p,t}\left(F\right)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents ${ℒ}_{p,t}\left(F\right)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.

A compact set $K\subset {ℂ}^N$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that $V_K\left(z\right)\ge B{\left(dist(z,K)\right)}^{\beta }$ if dist(z,K) ≤ 1. (LS) Here $V_K$ denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

Let $a=(a_1,...,a_n)$ be a tuple of commuting operators on a Banach space $X$. We discuss various conditions equivalent to that the holomorphic (Taylor) functional calculus has an extension to the real-analytic functions or various ultradifferentiable classes. In particular, we discuss the possible existence of a functional calculus for smooth functions. We relate the existence of a possible extension to existence of a certain (ultra)current extension of the resolvent mapping over the (Taylor) spectrum of $a$. If $a$...