Related papers: Singularities of local Ahlfors currents

We prove that an algebraic curve charged by a current coming from an entire curve is rational or elliptic. This answers a question by M. Paun.

We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be nonunique. Indeed, we will construct one exotic entire curve $f: \mathbb{C}\rightarrow X$ which produces infinitely many…

Let \(X\) be a compact complex manifold possessing the \emph{Runge approximation property on discs}, meaning that every holomorphic map from a closed disc into \(X\) is approximable by a global holomorphic map from \(\mathbb{C}\). We…

Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood. We perform a systematic study based on their associated Ahlfors currents. In particular, we generalize (by a…

We show that any simple holomorphic disc admits the annulus property, i.e., each interior point is surrounded by an arbitrary small annulus consisting entirely of injective points. As an application we show that interior singularities of…

First, inspired by a question of Sibony, we show that in every compact complex manifold $Y$ with certain Oka property, there exists some entire curve $f: \mathbb{C}\rightarrow Y$ generating all Nevanlinna/Ahlfors currents on $Y$, by…

We study the relation between the boundary of a simply connected domain being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk onto the domain. As an…

We give a constructive proof of the Carath\'eodory Theorem by means of the concept of a modulus of local connectivity and the extremal distance of the separating curves of an annulus.

We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.

Given a generic totally real torus unknotted in the unit sphere of the complex plane, we prove the following alternative : either there exists a filling of the torus by holomorphic discs and the torus is rationally convex, or its rational…

We investigate the existence, and lack of unicity, of a holomorphic fibration by discs transversal to a rational curve in a complex surface.

In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs. In this paper we consider the…

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X…

A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F) consisting of…

First, we show that every complex torus $\mathbb{T}$ contains some entire curve $g: \mathbb{C}\rightarrow \mathbb{T}$ such that the concentric holomorphic discs $\{g\restriction_{\overline{\mathbb D}_{r}}\}_{r>0}$ can generate all the…

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without…

A rigid current on a compact complex manifold is a closed positive current whose cohomology class contains only one closed positive current. Rigid currents occur in complex dynamics, algebraic and differential geometry. The goals of the…

We prove that 2 dimensional Integral currents (i.e. integer multiplicity 2 dimensional rectifiable currents) which are almost complex cycles in an almost complex manifold admitting locally a compatible symplectic form are smooth surfaces…

Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.

We show that $\mathbb{C}^2$ contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of $\mathbb{C}^2$. The construction is based on realizing corks as branched covers of…