Holomorphic Families Of Riemann Surfaces Research Articles

Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces

Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$. We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of $\overline{\mathcal{M}_g}$ in such a way that these measures form a continuous family of measures on the universal curve over $\overline{\mathcal{M}_g}$.

𝜆-lemma for families of Riemann surfaces and the critical loci of complex Hénon maps

We prove a version of the classical λ \lambda -lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex Hénon maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.

Veech surfaces and their periodic points

We give inequalities comparing widths or heights of cylinder decompositions of Veech surfaces with the signatures of their Veech groups. As an application of these inequalities, we estimate the numbers of periodic points of non-arithmetic Veech surfaces. The upper bounds depend only on the topological types of Veech surfaces and the signatures of Veech groups as Fuchsian groups. The upper bounds also estimate the numbers of holomorphic sections of holomorphic families of Riemann surfaces constructed from Veech groups of non-arithmetic Veech surfaces.

Kähler groups, $${\mathbb{R}}$$ R -trees, and holomorphic families of riemann surfaces

Let X be a compact Kahler manifold, and g a fixed genus. Due to the work of Parshin and Arakelov, it is known that there are only a finite number of non isotrivial holomorphic families of Riemann surfaces of genus \({g \geqslant 2}\) over X. We prove that this number only depends on the fundamental group of X. Our approach uses geometric group theory (limit groups, \({\mathbb{R}}\)-trees, the asymptotic geometry of the mapping class group), and Gromov-Shoen theory. We prove that in many important cases limit groups (in the sense of Sela) associated to infinite sequences of actions of a Kahler group on a Gromov-hyperbolic space are surface groups and we apply this result to monodromy groups acting on complexes of curves.

A criterion for holomorphic families of Riemann surfaces to be virtually isomorphic

In this paper, we give a rigidity theorem for holomorphic disks associated to holomorphic families of Riemann surfaces, aiming for studying the geometry (the asymptotic boundary behaviors) of such holomorphic disks. Indeed, our rigidity theorem is described with distributions of associated holomorphic disks and orbits of monodromies at infinity.

On holomorphic sections of Veech holomorphic families of Riemann surfaces

We give upper bounds of the numbers of holomorphic sections of Veech holomorphic families of Riemann surfaces. The numbers depend only on the topological types of the base Riemann surfaces and fibers. We also show a relation between signatures of Veech groups and moduli of cylinder decompositions of flat surfaces.

Veech holomorphic families of Riemann surfaces, holomorphic sections, and Diophantine problems

In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their holomorphic sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine equations over function fields. Moreover, we give upper bounds of the number of holomorphic sections of certain holomorphic families of Riemann surfaces.

A remark on universal coverings of holomorphic families of Riemann surfaces

We study the universal covering space $\tilde M$ of a holomorphic family (M, π, R) of Riemann surfaces over a Riemann surface R. The main result is that (1) $\tilde M$ is topologically equivalent to a two-dimensional cell, (2) $\tilde M$ is analytically equivalent to a bounded domain in C2, (3) $\tilde M$ is not analytically equivalent to the two-dimensional unit ball B2 under a certain condition, and (4) $\tilde M$ is analytically equivalent to the two-dimensional polydisc Δ2 if and only if the homotopic monodoromy group of (M, π, R) is finite.

On monodromies of holomorphic families of Riemann surfaces and modular transformations

A locally non-trivial holomorphic family of Riemann surfaces over a hyperbolic Riemann surface B determines a homomorphism (monodromy) of the fundamental group of B to the modular group. We give a necessary condition for a homomorphism to be a mondromy of a holomorphic family of Riemann surfaces. And we estimate the number of holomorphic families in terms of topological information for holomorphic families over Riemann surfaces of genus 0.

Remarks on holomorphic families of Riemann surfaces

Remarks on holomorphic families of Riemann surfaces

Holomorphic families of Riemann surfaces and Teichmüller spaces, III. Bimeromorphic embedding of algebraic surfaces into projective spaces by automorphic forms

Holomorphic families of Riemann surfaces and Teichmüller spaces, III. Bimeromorphic embedding of algebraic surfaces into projective spaces by automorphic forms

Holomorphic families of Riemann surfaces and Teichmüller spaces, II

Holomorphic families of Riemann surfaces and Teichmüller spaces, II