arXiv research

Study shows how volume forms on degenerating varieties converge to a non-Archimedean measure.

Study of asymptotics of meromorphic 3D-index as q approaches 1.

The paper defines and computes volumes of meromorphic differentials with simple poles.

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $κ=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $Ω^d\mathcal{M}_{g,n}(κ)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $κ$. We…

Normal forms found for meromorphic connections over a specific F-manifold.

Abstract framework for two meromorphic forms on punctured surfaces.

The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quad…

The study predicts large genus behavior of quadratic differential volumes and constants.

The paper explores meromorphic connections over Frobenius manifolds.

This work is based on the approach developed by J.~Dorfmeister, F.~Pedit and H.~Wu [GANG and KITCS preprint, Report KITCS94-4-1] to construct maps $Φ:D\rightarrow R^3$, $D$ being the unit disk in $C$, whose images are surfaces of constant mean curvature. They start from certain meromorphic one forms, so called meromorp…

The paper counts ends of differential forms on surfaces.

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $|ω|^2$, where $ω$…

The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.

The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.

Constructs metrics on Riemann surfaces with singularities.

We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…

Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.

We prove in this article that given a linearly concave domain $D$ in the projective space $\Bbb{CP}^{n}$, a 1-dimensional comlex analytic set $V$ in $D$, and a meromorphic 1-form $φ$ on $V$, $V$ is a subset of an algebraic variety of $\Bbb{CP}^{n}$ and $φ$ is the restriction to $V$ of an algebraic 1-form on $\Bbb{CP}^{…

We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previous…

Pólya's theorem extended to meromorphic functions on Riemann surfaces.

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We …

The Bergman kernel and period map for curves are studied.

We consider a log-Riemann surface $\mathcal{S}$ with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism $π: \mathcal{S} \to \C$. We prove that $\mathcal{S}$ is biholomorphic to a compact Riemann surface with finit…

Constructs universal local deformations for curves and differential forms.

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their $GL^{+}(2,\mathbb{R})…

Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that $a(M)\leq k$, where $a(M)$ is the algebraic dimension $a(M)$ (i.e. the transcendence degre…

Generalizes Gauss-Bonnet to metrics with logarithmic singularities.

In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical …

The 3D-index connects to Turaev-Viro invariant and knot periods.

In this paper, we analyze the theory of meromorphic $(1,0)$-forms $ω\in\mathcal{M}Ω^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on a compact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ every non-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros as poles, where each is counted acc…

Study describes how to realize periods of meromorphic differentials with specific properties.

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singul…

The paper studies residues of manifolds and their applications in geometry.

The paper studies the distribution of random degeneracy sets on complex manifolds.

Study geodesics on flat tori, focusing on convex bodies.

Study of meromorphic connections and their spectral duals in $\mathfrak{gl}_3(\mathbb{C})$.

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of …

Study geodesics of meromorphic connections on Riemann surfaces.

Constructs a function to prove meromorphic differential strata don't have complete subvarieties.

We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stra…

This paper classifies components of meromorphic differential strata.

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on t…

Classifies meromorphic affine connections on complex surfaces.

Study rationality of meromorphic functions between real algebraic sets in the plane.

For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue …

Classifies connected components of meromorphic differentials with residue conditions.

The isoresidual fibration maps Riemann sphere strata to resonance arrangements.

The paper studies complex affine structures near irregular singularities.