arXiv research

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

Abstract framework for two meromorphic forms on punctured surfaces.

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.

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}^{…

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.

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

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.

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

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

Study geodesics of meromorphic connections on Riemann surfaces.

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

This paper classifies components of meromorphic differential strata.

Classifies meromorphic affine connections on complex surfaces.

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…

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 paper studies complex affine structures near irregular singularities.

The isoresidual fibration maps Riemann sphere strata to resonance arrangements.

We show that the Atiyah-Patodi-Singer reduced $η$-invariant of the twisted Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight $2m$ up to an integral $q$-series. We prove this resu…

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

In this paper we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-Δ)^{-α/2}$, and we develop basic analogous prop…

A new projective structure on Riemann surfaces differs from the uniformization theorem's structure.

Using the locally compact abelian group $\BT \times \BZ$, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2--3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a …

Researchers compute the index of meromorphic functions on tori.

We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.

A meromorphic projective structure on a punctured Riemann surface $X\setminus P$ is determined, after fixing a standard projective structure on $X$, by a meromorphic quadratic differential with poles of order three or more at each puncture in $P$. In this article we prove the analogue of Thurston's grafting theorem for…

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

The paper explores anti-hyperbolicity for hyperkähler varieties.

Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.

We extend Vasy's results on semiclassical high energy estimates for the meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds to metrics that are not necessarily even. Vasy's method gives the meromorphic continuation of the resolvent and high energy estimates in strips, assuming that the geo…

Geometric approach to meromorphic differentials' periods and their holonomy representations.

Study meromorphic k-differentials with prescribed singularities on Riemann surfaces.