Related papers: Holomorphic curves with shift-invariant hyperplane…

We consider unbounded curves without endpoints. Isomorphism is equivalence up to translation. Self-avoiding plane-filling curves cannot be periodic, but they can satisfy the local isomorphism property: We obtain a set $\Omega $ of coverings…

The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must satisfy certain global algebraic or…

A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study…

Let $X$ be a smooth complete intersection over $\mathbb{C}$ of dimension $n-k$ in the projective space $\mathbf{P}^n_{\mathbb{C}}$, for given positive integers $n$ and $k$. For a given integral homology cycle $[\gamma] \in…

We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic…

Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and…

We study the problem of existence of a periodic point in the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carath\'eordory's prime…

In 1941, L. Ahlfors gave another proof of a 1933 theorem of H. Cartan on approximation to hyperplanes of holomorphic curves in P^n. Ahlfors' proof built on earlier work of H. and J. Weyl (1938), and proved Cartan's theorem by studying the…

In this note, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of…

Let $A_1, A_2\in \mathbb C(z)$ be rational functions of degree at least two that are neither Latt\`es maps nor conjugate to $z^{\pm n}$ or $\pm T_n.$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of…

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the…

We study the compactness problem for moduli spaces of holomorphic supercurves which, being motivated by supergeometry, are perturbed such as to allow for transversality. We give an explicit construction of limiting objects for sequences of…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…

We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…

We study the preperiodic dynatomic curves $\mathcal{X}\_{n,p}$, the closure of set of $(c,z)\in \C^2$ such that $z$ is a preperiodic point of $f\_c$ with preperiod $n$ and period $p$ ($n,p\geq1$). We prove that each $\mathcal{X}\_{n,p}$ has…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H.…

In this paper, we give some results on the number of meromorphic mappings of C^m into P^n under a condition on the inverse images of hyperplanes in P^n. At the same time, we give an answer for an open question by H.Fujimoto.

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and…

In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hypersurfaces with truncated counting functions. A uniqueness theorem for these mappings sharing…