arXiv research

The paper proves a grafting theorem for meromorphic projective structures and shows the monodromy map is a local homeomorphism.

Proves theorem about meromorphic projective structures with complex poles.

Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the puncture…

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

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

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

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

The study introduces a new equivalence for Poisson modules on complex projective varieties.

This paper generalizes monodromy maps for projective structures with poles.

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in…

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve …

For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least…

The paper studies deformations of Lagrangian fibrations on symplectic manifolds.

The $\mathcal D$-groupoid of symmetries is minimal under specific conditions.

The article extends Thurston's Grafting Theorem to signed spaces and defines a framed monodromy map.

Researchers find a method to construct projective structures on a specific surface.

Let F be a codimension one singular holomorphic foliation on a compact complex manifold M. Assume that there exists a meromorphic vector field X on M generically transversal to F. Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N, or F is transversely projective out…

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

Classifies meromorphic affine connections on complex surfaces.

Characterizes monodromy groups for projective structures on surfaces with specified poles.

The paper studies complex affine structures near irregular singularities.

The paper explores anti-hyperbolicity for hyperkähler varieties.

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.

Constructs metrics on Riemann surfaces with singularities.

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

We study meromorphic actions of unipotent complex Lie groups on compact Kähler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable Kähler structures obtained by symplectic reduct…

Study geodesics of meromorphic connections on Riemann surfaces.

Paper defines quasi-Strebel structures for meromorphic k-differentials and proves their existence.

The paper explores meromorphic connections over Frobenius manifolds.

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

Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the …

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

The paper explores Kodaira dimension on almost complex manifolds.

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan, which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K=T(M). Given a conformal immersion of M into \bbR^3, the un…

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 …

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 …

Study non-commutative function algebras using contact geometry.

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…

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 …

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic …

The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.

For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disc, and they generalise the conjugacy class example o…

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…

The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M de…

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}^{…

The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natu…

Study shows non-polyhedral structure in moduli spaces for n≥8.

Study symplectic structures in moduli spaces of meromorphic connections.