The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_0\left(t\right)·x\left(t\right)+\mathrm{d}{f}_0\left(t\right)$, $x\left(t_0\right)=c_0$ $(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_k\left(t\right)·x\left(t\right)+\mathrm{d}{f}_k\left(t\right)$, $x\left(t_k\right)=c_k$ $(k=1,2,\cdots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k\left(t\right)\to x_0\left(t\right)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

We study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f\left(x_0\right))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\varsigma ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that...

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 ${\Omega }_{\pi }$.

The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices ${𝒮}^n$ that leave invariant the closed convex cones of copositive and completely positive matrices (${\mathrm{COP}}_n$ and ${\mathrm{CP}}_n$). A description of an invertible linear map on ${𝒮}^n$ such that $L\left({\mathrm{CP}}_n\right)\subset C{P}_n$ is obtained in terms of semipositive maps over the positive semidefinite cone ${𝒮}_{+}^n$ and the cone of symmetric nonnegative matrices ${𝒩}_{+}^n$ for $n\le 4$, with specific calculations for $n=2$. Preserver properties of the Lyapunov...

CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^1\text{'}*({\Delta }_{a,b};E)$ AND $L_1^{1,*}({\Delta }_{a,b};E)$......................... 71. Basic function classes ......................................................................................................................................................................................

Let $X$ denote either ${\mathrm{ℂ\mathbb{P}}}^m$ or ${ℂ}^m$. We study certain analytic properties of the space ${ℰ}^n(X,gp)$ of ordered geometrically generic $n$-point configurations in $X$. This space consists of all $q=(q_1,\cdots ,q_n)\in X^n$ such that no $m+1$ of the points $q_1,\cdots ,q_n$ belong to a hyperplane in $X$. In particular, we show that for a big enough $n$ any holomorphic map $f:{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ commuting with the natural action of the symmetric group $𝐒\left(n\right)$ in ${ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ is of the form $f\left(q\right)=\tau \left(q\right)q=(\tau \left(q\right)q_1,\cdots ,\tau \left(q\right)q_n)$, $q\in {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$, where $\tau :{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to \mathrm{𝐏𝐒𝐋}(m+1,ℂ)$ is an $𝐒\left(n\right)$-invariant holomorphic map. A similar result holds true for mappings of the configuration...

We give in ${ℝ}^6$ a real analytic almost complex structure $J$, a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$, such that there is no germ of $J$-holomorphic disc $\gamma$ included in $M$ with $\gamma \left(0\right)=0$ and $\frac{\partial \gamma }{\partial x}\left(0\right)=v$, although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$, we give sufficient conditions under which there exists such a germ of disc.

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

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 $(X,0)$ be a reduced, equidimensional germ of an analytic singularity with reduced tangent cone $(C_{X,0},0)$. We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part ${𝔛}^0$ of the specialization to the tangent cone $\varphi :𝔛\to ℂ$ to satisfy Whitney’s conditions along the parameter axis $Y$. This result is a first step in generalizing to higher dimensions Lê and Teissier’s result for hypersurfaces of ${ℂ}^3$ which establishes the Whitney equisingularity of $X$ and its tangent...

We consider a convexity notion for complex spaces $X$ with respect to a holomorphic line bundle $L$ over $X$. This definition has been introduced by Grauert and, when $L$ is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if $H^0(X,L)$ separates each point of $X$, then $X$ can be realized as a Riemann domain over the complex projective...

We study questions related to exceptional sets of pluri-Green potentials $V_{\mu }$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{\mu }$ are defined by $V_{\mu }\left(z\right)={\int }_{B}log(1/|{\varphi }_z\left(w\right)\left|\right)d\mu \left(w\right)$, where for a fixed z ∈ B, ${\varphi }_z$ denotes the holomorphic automorphism of B satisfying ${\varphi }_z\left(0\right)=z$, ${\varphi }_z\left(z\right)=0$ and $\left({\varphi }_z\circ {\varphi }_z\right)\left(w\right)=w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973). ...

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 $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^k(D,𝒪)=0$ for $2\le k\le n-1$. We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$.