Hacen ZELACI - Academia.edu (original) (raw)

Papers by Hacen ZELACI

arXiv (Cornell University), Apr 13, 2018

Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [... more Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is ... more Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E *. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X .

arXiv (Cornell University), Nov 22, 2017

We prove a canonical identifications between the spaces of generalized theta functions on the mod... more We prove a canonical identifications between the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras. We also show a strange duality on level one in the unramiffied case, this gives the dimensions of the spaces of generalized theta functions of level one.

HAL (Le Centre pour la Communication Scientifique Directe), Nov 15, 2017

In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν :... more In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν : C → X such that the class of C is m times the minimal class. In [11], Welters introduced the formalism of complementary pairs to handle this problem in the case m = 2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any m and compute the dimension of the locus of these abelian varieties.

arXiv (Cornell University), Nov 15, 2017

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is ... more Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E *. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X. Contents 1. Introduction 1 2. Bruhat-Tits parahoric G−torsors 2 3. σ−quadratic and σ−alternating modules 5 4. Moduli spaces of anti-invariant vector bundles 9 4.1. Semistability of anti-invariant vector bundles 9 4.2. Construction of the moduli space 11 4.3. Some properties of M σ,+ X (r) and M σ,− X (r) 15 References 16

arXiv (Cornell University), Dec 20, 2016

Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for ... more Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over X under σ. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.

arXiv (Cornell University), Nov 9, 2016

In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν :... more In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν : C → X such that the class of C is m times the minimal class. In [9], Welters introduced the formalism of complementary pairs to handle this problem in the case m = 2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any m and compute the dimension of the locus of these abelian varieties. Contents 1 Introduction 1 2 Quotient of Jacobians 4 3 Quotient of pull-back of Jacobians 5 4 Quotient of Prym varieties 8

Proceedings of the American Mathematical Society, Jul 10, 2019

In this paper, we consider the conformal embedding of so(r) into sl(r) and study relations betwee... more In this paper, we consider the conformal embedding of so(r) into sl(r) and study relations between level one SO(r)-theta functions and twisted SL(r)-theta functions coming from parahoric moduli spaces. In particular, we give another proof of a theorem by Pauly-Ramanan [PR01].

Manuscripta Mathematica, Jul 3, 2018

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is ... more Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E *. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X. Contents 1. Introduction 1 2. Bruhat-Tits parahoric G−torsors 2 3. σ−quadratic and σ−alternating modules 5 4. Moduli spaces of anti-invariant vector bundles 9 4.1. Semistability of anti-invariant vector bundles 9 4.2. Construction of the moduli space 11 4.3. Some properties of M σ,+ X (r) and M σ,− X (r) 15 References 16

Soit X une courbe projective lisse et irreductible munie d'une involution σ. Dans cette these... more Soit X une courbe projective lisse et irreductible munie d'une involution σ. Dans cette these, nous etudions les fibres vectoriels invariants and anti-invariants sur X sous l'action induite par σ. On introduit la notion de modules σ-quadratiques et on l'utilise, avec GIT, pour construire ces espaces de modules, puis on en etudie certaines proprietes. Ces espaces de modules correspondent aux espaces de modules de G-torseurs parahoriques sur la courbe X/σ , pour certains schemas en groupes parahoriques G de type Bruhat-Tits, qui sont twistes dans le cas des anti-invariants. Nous developpons les systemes de Hitchin sur ces espaces de modules et on les utilise pour deriver une classification de leurs composantes connexes en les dominant par des varietes de Prym. On etudie aussi le fibre determinant sur les espaces de modules des fibres vectoriels anti-invariants. Dans certains cas, ce fibre en droites admet certaines racines carrees appelees fibres Pfaffiens. On montre que l...

In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν :... more In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν : C → X such that the class of C is m times the minimal class. In [11], Welters introduced the formalism of complementary pairs to handle this problem in the case m = 2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any m and compute the dimension of the locus of these abelian varieties.

arXiv: Algebraic Geometry, 2016

In this paper, we study principally polarized abelian varieties XXX of dimension ggg that contain... more In this paper, we study principally polarized abelian varieties XXX of dimension ggg that contain a curve nu:CtoX\nu:C\to Xnu:CtoX such that the class of CCC is mmm times the minimal class. Welters introduced the formalism of stable pairs to handle this problem in the case m=2m=2m=2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any mmm and compute the dimension of the locus of these abelian varieties.

Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for ... more Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over X under σ. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.

Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [... more Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.

Mathematical Research Letters

We prove a canonical identifications between the spaces of generalized theta functions on the mod... more We prove a canonical identifications between the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras. We also show a strange duality on level one in the unramiffied case, this gives the dimensions of the spaces of generalized theta functions of level one.

Proceedings of the American Mathematical Society

In this paper, we consider the conformal embedding of s o ( r ) \mathfrak {so}(r) into s l ( r ) ... more In this paper, we consider the conformal embedding of s o ( r ) \mathfrak {so}(r) into s l ( r ) \mathfrak {sl}(r) and study relations between level one SO ⁡ ( r ) \operatorname {SO}(r) -theta functions and twisted SL ⁡ ( r ) \operatorname {SL}(r) -theta functions coming from parahoric moduli spaces. In particular, we give another proof of a theorem by Pauly-Ramanan [J. London Math. Soc. (2) 63 (2001), pp. 513–532].

arXiv (Cornell University), Apr 13, 2018

Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [... more Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is ... more Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E *. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X .

arXiv (Cornell University), Nov 22, 2017

We prove a canonical identifications between the spaces of generalized theta functions on the mod... more We prove a canonical identifications between the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras. We also show a strange duality on level one in the unramiffied case, this gives the dimensions of the spaces of generalized theta functions of level one.

HAL (Le Centre pour la Communication Scientifique Directe), Nov 15, 2017

In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν :... more In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν : C → X such that the class of C is m times the minimal class. In [11], Welters introduced the formalism of complementary pairs to handle this problem in the case m = 2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any m and compute the dimension of the locus of these abelian varieties.

arXiv (Cornell University), Nov 15, 2017

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is ... more Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E *. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X. Contents 1. Introduction 1 2. Bruhat-Tits parahoric G−torsors 2 3. σ−quadratic and σ−alternating modules 5 4. Moduli spaces of anti-invariant vector bundles 9 4.1. Semistability of anti-invariant vector bundles 9 4.2. Construction of the moduli space 11 4.3. Some properties of M σ,+ X (r) and M σ,− X (r) 15 References 16

arXiv (Cornell University), Dec 20, 2016

Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for ... more Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over X under σ. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.

arXiv (Cornell University), Nov 9, 2016

In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν :... more In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν : C → X such that the class of C is m times the minimal class. In [9], Welters introduced the formalism of complementary pairs to handle this problem in the case m = 2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any m and compute the dimension of the locus of these abelian varieties. Contents 1 Introduction 1 2 Quotient of Jacobians 4 3 Quotient of pull-back of Jacobians 5 4 Quotient of Prym varieties 8

Proceedings of the American Mathematical Society, Jul 10, 2019

In this paper, we consider the conformal embedding of so(r) into sl(r) and study relations betwee... more In this paper, we consider the conformal embedding of so(r) into sl(r) and study relations between level one SO(r)-theta functions and twisted SL(r)-theta functions coming from parahoric moduli spaces. In particular, we give another proof of a theorem by Pauly-Ramanan [PR01].

Manuscripta Mathematica, Jul 3, 2018

Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is ... more Let X be a smooth irreducible projective curve with an involution σ. A vector bundle E over X is called anti-invariant if there exists an isomorphism σ * E → E *. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over X. Contents 1. Introduction 1 2. Bruhat-Tits parahoric G−torsors 2 3. σ−quadratic and σ−alternating modules 5 4. Moduli spaces of anti-invariant vector bundles 9 4.1. Semistability of anti-invariant vector bundles 9 4.2. Construction of the moduli space 11 4.3. Some properties of M σ,+ X (r) and M σ,− X (r) 15 References 16

Soit X une courbe projective lisse et irreductible munie d'une involution σ. Dans cette these... more Soit X une courbe projective lisse et irreductible munie d'une involution σ. Dans cette these, nous etudions les fibres vectoriels invariants and anti-invariants sur X sous l'action induite par σ. On introduit la notion de modules σ-quadratiques et on l'utilise, avec GIT, pour construire ces espaces de modules, puis on en etudie certaines proprietes. Ces espaces de modules correspondent aux espaces de modules de G-torseurs parahoriques sur la courbe X/σ , pour certains schemas en groupes parahoriques G de type Bruhat-Tits, qui sont twistes dans le cas des anti-invariants. Nous developpons les systemes de Hitchin sur ces espaces de modules et on les utilise pour deriver une classification de leurs composantes connexes en les dominant par des varietes de Prym. On etudie aussi le fibre determinant sur les espaces de modules des fibres vectoriels anti-invariants. Dans certains cas, ce fibre en droites admet certaines racines carrees appelees fibres Pfaffiens. On montre que l...

In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν :... more In this paper, we study principally polarized abelian varieties X of dimension g with a curve ν : C → X such that the class of C is m times the minimal class. In [11], Welters introduced the formalism of complementary pairs to handle this problem in the case m = 2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any m and compute the dimension of the locus of these abelian varieties.

arXiv: Algebraic Geometry, 2016

In this paper, we study principally polarized abelian varieties XXX of dimension ggg that contain... more In this paper, we study principally polarized abelian varieties XXX of dimension ggg that contain a curve nu:CtoX\nu:C\to Xnu:CtoX such that the class of CCC is mmm times the minimal class. Welters introduced the formalism of stable pairs to handle this problem in the case m=2m=2m=2. We generalize the results of Welters and construct families of principally polarized abelian varieties for any mmm and compute the dimension of the locus of these abelian varieties.

Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for ... more Given a smooth projective complex curve X with an involution σ, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over X under σ. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.

Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [... more Let X be a smooth irreducible projective curve. In this notes, we generalize the main result of [PPN18] to principal G−bundles for any semisimple linear algebraic group G. After defining very stability of principal G−bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of SL 2 −bundles.

Mathematical Research Letters

We prove a canonical identifications between the spaces of generalized theta functions on the mod... more We prove a canonical identifications between the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras. We also show a strange duality on level one in the unramiffied case, this gives the dimensions of the spaces of generalized theta functions of level one.

Proceedings of the American Mathematical Society

In this paper, we consider the conformal embedding of s o ( r ) \mathfrak {so}(r) into s l ( r ) ... more In this paper, we consider the conformal embedding of s o ( r ) \mathfrak {so}(r) into s l ( r ) \mathfrak {sl}(r) and study relations between level one SO ⁡ ( r ) \operatorname {SO}(r) -theta functions and twisted SL ⁡ ( r ) \operatorname {SL}(r) -theta functions coming from parahoric moduli spaces. In particular, we give another proof of a theorem by Pauly-Ramanan [J. London Math. Soc. (2) 63 (2001), pp. 513–532].