In this short paper, we show that the only proper holomorphic self-maps of bounded domains in ${ℂ}^k$ whose iterates approach a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type theorem for a sequence of maps whose degrees are a priori unbounded.

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some...

Let us consider a projective manifold $M^n$ and a smooth volume form $\Omega$ on $M$. We define the gradient flow associated to the problem of $\Omega$-balanced metrics in the quantum formalism, the $\Omega$-balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the $\Omega$-balancing flow converges towards a natural flow in Kähler geometry, the $\Omega$-Kähler flow. We also prove the long time existence of the $\Omega$-Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...

In 1996, Braaksma and Faber established the multi-summability, on suitable multi-intervals, of formal power series solutions of locally analytic, nonlinear difference equations, in the absence of “level $1^{+}$”. Combining their approach, which is based on the study of corresponding convolution equations, with recent results on the existence of flat (quasi-function) solutions in a particular type of domains, we prove that, under very general conditions, the formal solution is accelero-summable. Its sum...

Soit $G_{\mathbf{C}}$ un groupe de Lie complexe et $G_{\mathbf{R}}$ une forme réelle fermée de $G_{\mathbf{C}}$. Un couple $(G_{\mathbf{C}},G_{\mathbf{R}})$ est dit pseudo-convexe, s’il existe sur $G_{\mathbf{C}}$ une fonction régulière, strictement p.s.h., invariante par l’action de $G_{\mathbf{R}}$ et d’exhaustion sur $G_{\mathbf{C}}/G_{\mathbf{R}}$. On dit que $G_{\mathbf{R}}$ est à spectre imaginaire pur, si pour tout $X$ de Lie$\left(G_{\mathbf{R}}\right)$, les valeurs propres de ad$X$ sont imaginaires pures. Pour $G_{\mathbf{C}}$ à radical simplement connexe, cette dernière propriété équivaut à la pseudo-convexité de $(G_{\mathbf{C}},G_{\mathbf{R}})$. Pour $(G_{\mathbf{C}},G_{\mathbf{R}})$ pseudo-convexe et sous une hypothèse de sous-groupe discret,...

In this paper we establish the action of the Grothendieck-Teichmüller group $\widehat{GT}$ on the prime order torsion elements of the profinite fundamental group ${\pi }_1^{geom}\left({ℳ}_{0,\left[n\right]}\right)$. As an intermediate result, we prove that the conjugacy classes of prime order torsion of ${\widehat{\pi }}_1\left({ℳ}_{0,\left[n\right]}\right)$ are exactly the discrete prime order ones of the ${\pi }_1\left({ℳ}_{0,\left[n\right]}\right)$.

We prove that every compact, normal Riemannian homogeneous manifold admits an adapted complex structure on its entire tangent bundle.

This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem...