arXiv subjects

Guillaume Tahar

Publications and source records attributed to Guillaume Tahar.

At least 19 recordsLinked to original sources

arXiv2026

Hybrid connections on Hessian manifolds

We introduce a new class of affine connections on Hessian manifolds, called hybrid connections, characterized by the compatibility between their projective geometry with the underlying affine structure on the one hand and their infinitesimal holonomy with the Hessian metric on the other hand. In this paper, we investigate the properties of hybrid connections and prove that, on a given Hessian manifold, they are completely determined by the choice of a Hessian potential for the metric. In the special case of pseudo Euclidean manifolds, we identify canonical models and construct, in particular, a natural connection on the open unit ball that combines features of the Cayley Klein and Poincaré models of hyperbolic geometry. We also prove the existence and uniqueness (up to scaling) of a pseudo-Riemannian metric $h$ such that the geodesics of $\nabla$ admit parameterizations of constant speed with respect to $h$, which we call the isochrone metric.

math.DG↗

arXiv2026

Complete subvarieties in the projectivized strata of meromorphic differentials

Here we give an explicit construction of a globally defined strictly plurisubharmonic function on projectivized strata of strictly meromorphic differentials with prescribed orders of zeros and poles. In particular, this yields a flat-geometric proof that these strata do not contain positive-dimensional complete subvarieties.

math.AG↗

arXiv2026

On boundary points of minimal continuously Hutchinson invariant sets

A linear differential operator $T=Q(z)\frac{d}{dz}+P(z)$ with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, $T$ has a unique minimal Hutchinson-invariant set $M_{CH}^{T}$ in the complex plane. Using a geometric interpretation of its boundary in terms of envelopes of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field $\frac{Q(z)}{P(z)}\partial_{z}$), and singular points of different types which we classify below. The latter decomposition of the boundary of $M_{CH}^{T}$ is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the field $\frac{Q(z)}{P(z)}\partial_{z}$. We provide an upper bound for the number of local arcs in terms of degrees of $P$ and $Q$. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.

math.DS↗

arXiv2026

Inflection curves of rational vector fields

We initiate the study of inflection curves of rational vector fields on the Riemann sphere. For a rational vector field $v_R=-R(z)\frac{\partial}{\partial z}, \qquad R(z)=\frac{Q(z)}{P(z)} $ we define its affine regular inflection locus by $ \{z\in \mathbb{C}: R(z)\ne0,\ P(z)\neq 0,\ \text{Im} R'(z)=0\} $ and its algebraic inflection curve by the closure of this locus, equivalently by $ \mathfrak{I}_R=(R')^{-1}(\mathbb{RP}^1). $ We prove an explicit defining equation, degree bounds, local normal forms near poles, the asymptotic directions at infinity, and a maximum-principle obstruction excluding compact components without poles. We also explain that these curves are precisely the real dessins associated with exact rational differentials, i.e. rational differentials with zero residues. Finally, we give a reducibility criterion for the complexification, prove a generic irreducibility statement in the usual separated-variable sense, and classify the exact dessins of degree at most two. The degree-three problem is reduced to three explicit normal forms.

math.DS↗

arXiv2026

A lower bound for the genus of a knot using the Links-Gould invariant

The Links-Gould invariant of links $LG^{2,1}$ is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we prove that the degree of the Links-Gould polynomial provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander polynomial. As an example, unlike some classical tools such as the Alexander polynomial and Levine-Tristram signature, this new genus bound detects the fact that the Kinoshita-Terasaka and Conway knots have genus greater or equal to 2.

math.GT↗

arXiv2026

Extending Knot Polynomials of Braided Hopf Algebras to Links

Recently, a plethora of multivariable knot polynomials were introduced by Kashaev and one of the authors, by applying the Reshetikhin-Turaev functor to rigid $R$-matrices that come from braided Hopf algebras with automorphisms. We study the extension of these knot invariants to links, and use this to identify some of them with known link invariants, as conjectured in that same recent work.

math.QA↗

arXiv2026

Plane rectifiable curves: old and new

In this note we recall the classical notion of an algebraically rectifiable plane curve going back to J. A. Serret, E. Laguerre and G. Humbert. We provide new criteria of algebraic rectifiability, relate this notion to quadratic differentials, and generalize it to differentials of higher order.

math.AG↗

arXiv2026

Isoresidual curves

Given a partition $μ$ of $-2$, the stratum $\mathcal{H}(μ)$ parametrizes meromorphic differential one-forms on the Riemann sphere $\mathbb{CP}^{1}$ with~$n$ zeros and $p$ poles of orders prescribed by $μ$. The isoresidual fibration is defined by assigning to each differential in $\mathcal{H}(μ)$ its configuration of residues at the poles. In the case of differentials with $n=2$ zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe extensively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves, including the orders of the singularities and a period central charge encapsulating the linear dependence of periods on the underlying configuration of residues, provide rich discrete invariants for these fibers. We also determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, relying on the multi-scale compactification of strata of differentials. In particular, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition $μ$. Additionally, we classify the connected components of generic isoresidual fibers for strata in genus zero with an arbitrary number of zeros.

math.AG↗

arXiv2026

Bounds on saddle connections for flat spheres

We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to $2π$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of saddle connections with at most $k$ self-intersections. Additionally, we establish an upper bound on their lengths for a surface with a normalized area. Finally, we apply these bounds to the counting of singular trajectories in irrational polygonal billiards.

math.GT↗

arXiv2026

Skein theory for the Links-Gould polynomial

Building further on work of Marin and Wagner, we give a cubic braid-type skein theory of the Links--Gould polynomial invariant of oriented links and prove that it can be used to evaluate any oriented link, adding this polynomial to the list of polynomial invariants that can be computed by skein theory. As a consequence, we prove that this skein theory is also shared by the $V_1$-polynomial defined by two of the authors, deducing the equality of the two link polynomials. This implies specialization properties of the $V_1$-polynomial to the Alexander polynomial and to the $\mathrm{ADO}_3$-invariant, the fact that it is a Vassiliev power series invariant, as well as a Seifert genus bound for knots.

math.GT↗

arXiv2025

Isoperiodic foliation of the stratum $Ω\mathcal{M}_1(1,1,-2)$

This paper describes the geometry and topology of leaves of the isoperiodic foliation of the stratum $Ω\mathcal{M}_1(1,1,-2)$. We prove that each leaf is a surface of infinite genus homeomorphic to the Loch Ness monster surface and supports a singular Euclidean structure whose singularities correspond to isoperiodic forms in the lower stratum $Ω\mathcal{M}_1(2,-2)$. Along the way we also characterize the possible groups arising as the Veech group of a leaf and give a description of the large-scale conformal geometry of the wall-and-chamber decomposition of the leaves.

math.GT↗

arXiv2025

Simplicial arrangements with few double points

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Grünbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.

math.CO↗

arXiv2025

The translation geometry of Pólya's shires

In his shire theorem, G. Pólya proves that the zeros of iterated derivatives of a meromorphic function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. By recasting the local arguments of Pólya into the language of translation surfaces, we prove its generalisation describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator $T_ω: f \mapsto \frac{df}ω$ where $ω$ is a given meromorphic $1$-form. The accumulation set of these zeros is the union of edges of a generalised Voronoi diagram defined by the initial function $f$ together with the singular flat metric on the Riemann surface induced by $ω$. This result provides the ground for a novel approach to the problem of finding a flat geometric presentation of a translation surface initially defined in terms of algebraic or complex-analytic data.

math.GT↗

arXiv2025

The affine geometry of meromorphic connections with irregular singularities

A meromorphic connection on the tangent bundle of a Riemann surface induces a complex affine structure on the complement of the poles. Local models for Fuchsian singularities are already known. In this paper, we introduce a complete set of local invariants for a meromorphic connection and provide local models for a complex affine structure in a punctured neighborhood of an irregular singularity. Generalizing a construction attributed to Veech, we introduce the Delaunay decomposition of a compact Riemann surface endowed with a meromorphic connection with irregular singularities. In particular, we give upper bounds on the complexity of the decomposition.

math.CV↗

arXiv2025

Finite isoresidual covers in strata of $k$-differentials

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of $k$-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy.

math.AG↗

arXiv2025

On some log-concavity properties of the Alexander-Conway and Links-Gould invariants

The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $Δ_L(t)$ and has been shown to share some of its most geometric features in several recent works. Here we suggest that $\mathrm{LG}$ likely shares another of the Alexander polynomial's most distinctive - and mysterious - properties: for alternating links, the coefficients of the Links-Gould polynomial alternate and appear to form a log-concave two-indexed sequence with no internal zeros. The former was observed by Ishii for knots with up to 10 crossings. We further conjecture that they satisfy a bidimensional property of unimodality, thereby replicating a long-standing conjecture of Fox (1962) regarding the Alexander polynomial, and a subsequent refinement by Stoimenow. We also point out that the Stoimenow conjecture reflects a more structural phenomenon: after a suitable normalization, the Alexander polynomial of an alternating link appears to be a Lorentzian polynomial. We give compelling experimental and computational evidence for these different properties.

math.QA↗

arXiv2025

Conjugacies of geodesic flows in affine cylinders and tori

Affine cylinders (genus zero surfaces with two singularities) and affine tori (genus one surfaces without singularities) are among the simplest examples of surfaces endowed with a complex affine structure. Their geodesic flows are particularly tractable. In this article, we provide explicit necessary and sufficient conditions under which the geodesic flows on such surfaces are conjugate, in the topological and in the holomorphic category.

math.DS↗

arXiv2025

K-differentials with prescribed singularities

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g \geq 0$ can have for $k \geq 3$. These local invariants include the orders of zeros and poles, as well as the $k$-residues at the poles. We show that for a given pattern of orders of zeros, there exists, with a few exceptions, a primitive holomorphic $k$-differential having zeros of these orders. In the meromorphic case, for genus $g \geq 1$, every expected tuple appears as a configuration of $k$-residues. On the other hand, for certain strata in genus zero, finitely many tuples (up to simultaneous scaling) do not occur as configurations of $k$-residues for a $k$-differential.

math.CV↗