Related papers: Specialization to the tangent cone and Whitney Equ…

We prove the existence of non-positively curved K\"ahler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact K\"ahler manifold, under a technical condition on the cone angles, and we also…

We consider a corank $1$, finitely determined, quasi-homogeneous map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by…

This work, together with \cite{KrumWica} and \cite{KrumWicc}, forms a series of articles devoted to an analysis of interior singularities of locally area minimizing $n$-dimensional rectifiable currents $T$ of codimension $\geq 2$. In the…

Tangent cones are preserved under ambient bilipschitz equivalence, but the behavior of the Nash cone is more delicate. This paper explores the behavior of the Nash cone and of exceptional rays under ambient bilipschitz equivalence for real…

In this paper we show that, if $T$ is an area-minimizing $2$-dimensional integral current with $\partial T = Q [\![ \Gamma ]\!]$, where $\Gamma$ is a $C^{1,\alpha}$ curve for $\alpha>0$ and $Q$ an arbitrary integer, then $T$ has a unique…

In this paper we give for all $n \geq 2$, d>0, $g \geq 0$ necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in $\matbf{P}^{n+1}$ and C is a smooth (reduced and irreducible) curve of…

In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace…

We discuss a particular class of rational Gorenstein singularities, which we call symplectic. A normal variety V has symplectic singularities if its smooth part carries a closed symplectic 2-form whose pull-back in any resolution X --> V…

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T.…

We prove a general theorem that gives a non trivial relation in the group of derived autoequivalences of a variety (or stack) X, under the assumption that there exists a suitable functor from the derived category of another variety Y…

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus.…

We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case. This version of the paper includes a substantial amount of new material (76% larger). The new material introduces the idea…

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular…

A germ of normal complex analytical surface is called a Hirzebruch-Jung singularity if it is analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if…

Let $X$ be a regular tame stack. If $X$ is locally of finite type over a field, we prove that the essential dimension of $X$ is equal to its generic essential dimension, this generalizes a previous result of P. Brosnan, Z. Reichstein and…

We associate to any irreducible germ S of complex quasi-ordinary hypersurface an analytically invariant semigroup. We deduce a direct proof (without passing through their embedded topological invariance) of the analytical invariance of the…

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points…

Let $k$ be a field, $X$ a variety with tame quotient singularities and $\tilde{X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde{X}$ by the Lang-Nishimura theorem, but if $x$ is singular this…

We provide a positive answer to Zariski's conjecture for families of singular surfaces in $\mathbb C^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization…

In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with…