Related papers: Caratheodory completeness on the complex plane

We show that the hypercomplete $\infty$-topos associated with any replete topos is Postnikov complete, positively answering a question of Bhatt and Scholze; this will be deduced from the Milnor sequences for sheaves of spaces on replete…

We study the continuity of half-plane capacity as a function of boundary hulls with respect to the Carath\'eodory convergence. In particular, our interest lies in the case that hulls are unbounded. Under the assumption that every hull is…

We prove that if two plane curve singularities are equisingular, then they are topologically equivalent. The method we will use is P.~Fortuny~Ayuso's who proved this result for irreducible plane curve singularities.

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…

Under a mild technical assumption, we prove a necessary and sufficient condition for a totally real compacdt set in $\mathbb{C}^n$ to be rationally convex. This generalizes a classical result of Duval-Sibony

We investigate the $\mathcal F$-Borel complexity of topological spaces in their different compactifcations. We provide a simple proof of the fact that a space can have arbitrarily many different complexities in different compactifications.…

We clarify and discuss a misunderstanding between uniform completeness and metric completeness, that has appeared in the literature in a study on the Alexandrov topology for a spacetime.

We give a new proof of the fact that the complement of the complexification of a real hyperplane arrangement is homotopy equivalent to the Salvetti complex of the associated oriented matroid. Our proof involves no choices, is relatively…

We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a new proof for…

We prove a Caratheodory-Fejer type interpolation theorem for certain matrix convex sets in $\C^d$ using the Blecher-Ruan-Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S.…

The idea of pairwise paracompactness was studied by many authors in a bitopological space. Here we study the same in the setting of more general structure of a bispace using the thoughts of the same given by Bose et al[2].

Noguchi proved that the set of dominant maps from a fixed variety to a fixed hyperbolic variety is finite. We extend this result to the setting of orbifold pairs, as introduced by Campana, under suitable assumptions. Certain compactness…

We show that C^2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.

We prove the formula C(a,b) = K(a|C(a,b)) + C(b|a,C(a,b)) + O(1) that expresses the plain complexity of a pair in terms of prefix and plain conditional complexities of its components.

These notes were written for a presentation given at the university Paris VII in January 2012. The goal was to explain a proof of a famous theorem by P. Deligne about coherent topoi (coherent topoi have enough points) and to show how this…

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by…

A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some…

We prove the finiteness of leaps of modules of $m$-integrable derivations for algebras essentially of finite type and, more generally, for schemes essentially of finite type over an algebraically closed field of positive characteristic.…

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…

We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic…