Related papers: Classifying singularities up to analytic extension…

We give a classification of superattracting germs in dimension one over a complete normed algebraically closed field of positive characteristic up to conjugacy. In particular we show that formal and analytic classifications coincide for…

In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one.…

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper…

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all…

In this paper we study the problem of analytic extension of germs of holonomy of algebraic foliations. More precisely we prove that for a Riccati foliation associated to a branched projective structure over a finite type surface which is…

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…

We define an Isometry germ at any given event $x$ of space-time as a vector field $\xi$ defined in a neighborhood of $x$ such that the Lie derivative of both the metric and the Riemannian connection are zero at this event. Two isometry…

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$…

The thesis deals with holomorphic germs $ \Phi: (\mathbb{C}^2, 0) \to (\mathbb{C}^3,0) $ singular only at the origin, with a special emphasis on the distinguished class of finitely determined germs. The results are published in two articles…

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic…

We show that any holomorphic germ $f \colon (X,x_0) \to (Y,y_0)$ of topological degree $1$ between normal surface singularities can be written as $f=\pi \circ \sigma$, where $\pi \colon Y' \to (Y,y_0)$ is a modification and $\sigma \colon…

C points, that is isolated points of circular polarization in transverse fields of varying polarization, are classified morphologically into three distinct types, known as lemons, stars and monstars. These morphologies are interpreted here…

Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…

For a germ $(X,0) \subset (\mathbb{C}^n,0)$ of reduced, equidimensional complex analytic singularity its Nash modification can be constructed as an analytic subvariety $ Z \subset \mathbb{C}^n \times G(k,n)$. We give a characterization of…

We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric…

T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic…

We prove that separable extensions of noetherian rings and finite \'etale morphisms of noetherian schemes give rise to separable extensions of singularity categories.

We determine the singularity category of an arbitrary finite dimensional gentle algebra $\Lambda$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of…

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related $L^2$ extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded…

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…