Related papers: Singularities of local Ahlfors currents

We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or irreducible.

A smooth real curve is called separating in case the complement of the real locus inside the complex locus is disconnected. This is the case if there exists a morphism to the projective line whose inverse image of the real locus of the…

Consider the Loewner equation associated to the upper-half plane. Normally this equation is driven by a real-valued function. In this paper, we show that when the driving function is complex-valued with small $1/2$-H\"older norm, the…

If on a smooth bounded domain $\Omega\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $\Omega$ be a disk or an annulus? This question can be understood as a weaker analog…

In this paper, we generalize Ahlfors' lemma on logarithmic derivative to holomorphic tangent curves of directed projective manifolds intersecting closed subschemes. As a consequence, we obtain Algebro-Geometric Ahlfors' Lemma on Logarithmic…

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

Given a 2-sheeted torus over the circle with winding number 1, we prove that its polynomial hull is a union of 2-sheeted holomorphic discs. Moreover when the hull is non degenerate its boundary is a Levi-flat solid torus foliated by such…

In this note we give an overview of some applications of the Calabi-Yau theorem to the construction of singular positive (1,1) currents on compact complex manifolds. We show how recent developments allow us to give streamlined proofs of…

We classify convex ancient curve shortening flows in the disc with free boundary on the circle.

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities.

We study the interplay between the classical theory of linear series on curves, and the recent theory of linear series on graphs. We prove that every d-gonal (weighted) graph of Hurwitz type is the dual graph of a d-gonal curve. Conversely…

For a given singularity of a plane curve we consider the locus of nodal deformations of the singularity with the given number of nodes and describe possible components of the locus. As applications, we solve the local symplectic isotopy for…

An existence theorem for stationary discs of strongly pseudoconvex domains in almost complex manifolds is proved. More precisely, it is shown that, for all points of a suitable neighborhood of the boundary and for any vector belonging to…

We show that every finite type polarised curve in the conformal $2$-sphere with a polynomial conserved quantity admits a resonance point, under a non-orthogonality assumption on the conserved quantity. Using this fact, we deduce that every…

We show that the locally free sheaf of locally exact differentials on a smooth projective curve of genus at least two over an algebraically closed field k of characteristic p is a stable vector bundle. This answers a question of Raynaud.

We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local…

This short note is some obvious mathematical addendum to our papers on Wilson loops on polygon-like contours with circular edges \cite{Dorn:2020meb,Dorn:2020vzj}. Using the technique of osculating spheres and circles we identify the…

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to…

The stable pairs theory of local curves in 3-folds (equivariant with respect to the scaling 2-torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization…

We describe a generalization of the notion of a Hilbert space model of a function $\varphi$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $\varphi$ at a mild…