Related papers: Lipschitz geometry of complex surfaces: analytic i…

Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component. They are also natural Poincar\'e duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed cohomology…

We prove that the topological type of a normal surface singularity $(X,0)$ provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$, as well as for the combinatorics of the families of generic hyperplane sections and of…

We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in ${\bf C}^3$. We study irreducible (i.e. $gcd (m,k,l) = 1$) non-isolated (i.e. $1 \leq k < l$)…

This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve.…

Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to…

In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of isolated hypersurface singularities is Whitney equisingular if and only if the…

Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S). We use…

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the $n$-dimensional Heisenberg group into $CAT(0)$ spaces. Our main theorem establishes that these maps have the…

The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…

Given a normal surface singularity (X,0), its link, M is a closed differentiable three dimensional manifold which carries much analytic information. It is an interesting question to ask whether, under suitable analytic and topological…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic…

Let $F(z)=\mathcal{R}e(P(z)) + h.o.t$ be such that $M=(F=0)$ defines a germ of real analytic Levi-flat at $0\in\mathbb{C}^{n}$, $n\geq{2}$, where $P(z)$ is a homogeneous polynomial of degree $k$ with an isolated singularity at…

We define normalized versions of Berkovich spaces over a trivially valued field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by rescaling semivaluations. We associate such a normalized space to any special formal…

Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an…

Let $(S, 0) \subset (\mathbb{C}^{d+1},0)$ be an irreducible germ of hypersurface. The germ $(S,0)$ is quasi-ordinary if $(S,0)$ has a finite projection to $(\mathbb{C}^d,0)$ which is unramified outside the coordinate hyperplanes. This…

Let X be an irreducible hypersurface in $\mathbb{P}^n$ of degree $d\geq 3$ with only isolated semi-weighted homogeneous singularities, such that $exp(\frac{2\pi i}{k})$ is a zero of the Alexander polynomial. Then we show that the…

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions,…

We give an elementary combinatorial proof of the following fact: Every real or complex analytic complete intersection germ X is equisingular -- in the sense of the Hilbert-Samuel function -- with a germ of an algebraic set defined by…

We show that the knot type of the link of a real analytic map germ with isolated singularity $f\colon(\mathbb{R}^2,0)\to(\mathbb{R}^4,0)$ is a complete invariant for $C^0$-$\mathscr A$-equivalence. Moreover, we also prove that isolated…

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all…