Metrically thin singularities of integrable CR functions

Integrable deformations of local analytic fibrations with singularities

We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of in…

Wedge extendability of CR-meromorphic functions: the minimal case

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C^{2,a}-smooth embedded Cauchy-Riemann generic manifold M (CR functions on M - E) and more generally, we consider holomorphic functions defined in wedgelike domains attached to M - E. Our main result establishes the wedge- and the L^1-removability of E under the hypothesis that the (\dim M-2)-dimensional Hausdorff volume of E is zero and that M and M\backslash E are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everyw…

On non-integrable complex distributions: an approach by transversality with real domains

We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.

Note on the singular Hermitian metrics with analytic singularities

We study the sheaf of locally square integrable holomorphic section of vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.

Riemannian流形上满足逆Poletsky不等式映射的可去奇点

Removable singularities of mappings with inverse Poletksy inequality on Rie\-man\-nian manifolds

本文研究Riemannian流形上满足逆Poletsky型模不等式的开离散映射，证明在遗漏至少两点且控制函数几乎处处球面可积时，边界孤立点为可去奇点。

孤立canonical Gorenstein奇点处的canonical sheaves

Canonical sheaves at isolated canonical Gorenstein singularities

研究孤立canonical Gorenstein奇点处的canonical sheaf行为，探讨其在L²理论中的应用。

Local monodromy formula of Hadamard products

We find an explicit general formula for the iterated local monodromy of singularities of the Hadamard product of functions with integrable singularities. The formula implies the invariance by Hadamard product of the class of functions with integrable singularities with recurrent monodromies. In particular, it implies the recurrence of the local monodromy of functions with finite Hadamard grade as defined by Allouche and Mendès-France. We give other examples of natural classes of functions with recurrent monodromies, functions with algebro-logarithmic singularities, and more generally with poly…

$L^2$-sheaves of holomorphic functions and $n$-forms on complex spaces with isolated singularities

Let $X$ be a Hermitian complex space of pure dimension $n$ with isolated singularities. In the present paper, we give a natural resolution for the canonical sheaf of square-integrable holomorphic $n$-forms with Dirichlet boundary condition on $X$. As application, we obtain an explicit smooth model for the $L^2$-$\bar{\partial}$-cohomology, including natural resolutions for sheaves of $\bar{\partial}$-closed (holomorphic) $L^2$-functions.