Frank Loray
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Frank Loray has a total of 44 co-authors. Here are the top co-authors by number of shared publications:
- Frédéric Touzet
IRMAR
11 shared publications - Jorge Vitório Pereira
IMPA, Rio de Janeiro, Brazil
10 shared publications - Thiago Fassarella
Universidade Federal Fluminense -UFF Rua Alexandre Moura 8 Niterói RJ Brazil
5 shared publications - Maycol Falla Luza
Univ Fed Fluminense, UFF, Rua Mario Santos Braga S-N, Niteroi, RJ, Brazil
4 shared publications - Benoît Claudon
Univ Lorraine, IECL, BP 70239, F-54506 Vandoeuvre Les Nancy, France
3 shared publications - Masa-Hiko Saito3 shared publications
Latest Publications
- 21 June 2024
- Published by Springer Nature
- p. 267-305
The Riemann-Hilbert Correspondence for Rank 2 Meromorphic Connections on Curves
- 21 June 2024
- Published by Springer Nature
- p. 267-305
The Riemann-Hilbert Correspondence for Rank 2 Meromorphic Connections on Curves
- 6 May 2024
- Published by Elsevier in Journal of Geometry and Physics
- Vol. 201
Automorphisms of projective structures
- 6 May 2024
- Published by Elsevier in Journal of Geometry and Physics
- Vol. 201
Automorphisms of projective structures
- Open Access
Canonical coordinates for moduli spaces of rank two irregular connections on curves
In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a ``companion'' normal form. This allow us to naturally identify an open set of the moduli space of $\mathrm{GL}_2$-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for $\mathrm{SL}_2$-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree $2$.Open AccessCanonical coordinates for moduli spaces of rank two irregular connections on curves
In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a ``companion'' normal form. This allow us to naturally identify an open set of the moduli space of $\mathrm{GL}_2$-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for $\mathrm{SL}_2$-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree $2$. - Open Access
Canonical coordinates for moduli spaces of rank two irregular connections on curves.
In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a “companion” normal form. This allow us to naturally identify an open set of the moduli space of GL2-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for SL2-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree 2.
Open AccessCanonical coordinates for moduli spaces of rank two irregular connections on curves.
In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a “companion” normal form. This allow us to naturally identify an open set of the moduli space of GL2-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for SL2-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree 2.
- Open Access
Automorphisms of projective structures
We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of flat projective structures having positive dimensional Lie algebra of projective vector fields.
Open AccessAutomorphisms of projective structures
We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of flat projective structures having positive dimensional Lie algebra of projective vector fields.
- Open Access
Hitchin fibration under ramified coverings
We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $\pi$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to describe this map.Open AccessHitchin fibration under ramified coverings
We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $\pi$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to describe this map. - Open Access
Moduli of Higgs bundles over the five punctured sphere
We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $μ\in[0,1]^5$. The moduli space corresponding to the central weight $μ_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus $n$ points is false.Open AccessModuli of Higgs bundles over the five punctured sphere
We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $μ\in[0,1]^5$. The moduli space corresponding to the central weight $μ_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus $n$ points is false. - 1 February 2023
- Published by National Research University, Higher School of Economics (HSE) in Moscow Mathematical Journal
- Vol. 23 (1) , 59-95
Projective Structures, Neighborhoods of Rational Curves and Painlevé equations
- 1 February 2023
- Published by National Research University, Higher School of Economics (HSE) in Moscow Mathematical Journal
- Vol. 23 (1) , 59-95
Projective Structures, Neighborhoods of Rational Curves and Painlevé equations
- Open Access
- 7 December 2022
- Published by MathDoc/Centre Mersenne in Publications mathématiques de l'IHÉS
- Vol. 136 (1) , 149-224
Two dimensional neighborhoods of elliptic curves: analytic classification in the torsion case
We investigate the analytic classification of two dimensional neighborhoods of an elliptic curve with torsion normal bundle. We provide the complete analytic classification for those neighborhoods in the simplest formal class and we indicate how to generalize this construction to general torsion case.
- 7 December 2022
- Published by MathDoc/Centre Mersenne in Publications mathématiques de l'IHÉS
- Vol. 136 (1) , 149-224
Open AccessTwo dimensional neighborhoods of elliptic curves: analytic classification in the torsion case
We investigate the analytic classification of two dimensional neighborhoods of an elliptic curve with torsion normal bundle. We provide the complete analytic classification for those neighborhoods in the simplest formal class and we indicate how to generalize this construction to general torsion case.
- Open Access
Primitive Lie algebras of rational vector fields
A transitive Lie algebra g of rational vector fields on a projective manifold which do not preserve any foliation determines a rational map to an algebraic homogenous space G/H which maps g to lie(G).
Open AccessPrimitive Lie algebras of rational vector fields
A transitive Lie algebra g of rational vector fields on a projective manifold which do not preserve any foliation determines a rational map to an algebraic homogenous space G/H which maps g to lie(G).