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UNIVERSITE NICE-SOPHIA ANTIPOLIS - UFR SCIENCES Ecole Doctorale Sciences Fondamentales et Appliquees THESE Pour obtenir le titre de Docteur en Sciences de l'Universite Nice Sophia-Antipolis Specialite : MATHEMATIQUES Presente et soutenue publiquement par CHADI TAHER Titre de la these CALCULATING THE PARABOLIC CHERN CHARACTER OF A LOCALLY ABELIAN PARABOLIC BUNDLE - THE CHERN INVARIANTS FOR PARABOLIC BUNDLES AT MULTIPLE POINTS. These dirigee par Professeur CARLOS SIMPSON Soutenue le 16 Mai 2011 a la Faculte des Sciences de l'Universite de Nice Membre du jury: Mr.Tony PANTEV Professeur, Universite de Pennsylvania USA Rapporteur Mrs.Jaya IYER Professeur, Universite de Hyderabad India Rapporteur Mr.Alexandru DIMCA Professeur, Universite Nice Sophia-Antipolis France Examinateur Mr.Sorin DUMITRESCU Professeur, Universite Nice Sophia-Antipolis France Examinateur Mr.Bertrand Toen Directeur de Recherche, Universite Montpellier 2 France Examinateur Mr.Carlos SIMPSON DR1 CNRS, Universite Nice Sophia-Antipolis France Directeur 1

- nice sophia-antipolis
- riemann-roch theorem
- chern
- el-solh
- docteur en sciences de l'universite de nice
- parabolic invariant
- quasi-parabolic structures
- universite de nice

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´Ecole Doctorale Sciences Fondamentales et Appliquees´

`THESE

Pour obtenir le titre de

Docteur en Sciences

de l’Universite´ Nice Sophia-Antipolis

´Specialit´ e´ : MATHEMATIQUES

Present´ e´ et soutenue publiquement par

CHADI TAHER

Titre de la these`

CALCULATING THE PARABOLIC CHERN CHARACTER

OF A LOCALLY ABELIAN PARABOLIC BUNDLE -

THE CHERN INVARIANTS

FOR PARABOLIC BUNDLES AT MULTIPLE POINTS.

These` dirigee´ par Professeur CARLOS SIMPSON

Soutenue le 16 Mai 2011

a` la Faculte´ des Sciences de l’Universite´ de Nice

Membre du jury:

Mr.Tony PANTEV Professeur, Universite´ de Pennsylvania USA Rapporteur

Mrs.Jaya IYER, Universite´ de Hyderabad India

Mr.Alexandru DIMCA Professeur, Universite´ Nice Sophia-Antipolis France Examinateur

Mr.Sorin DUMITRESCU, Universite´ Nice

Mr.Bertrand Toen¨ Directeur de Recherche, Universite´ Montpellier 2 France

Mr.Carlos SIMPSON DR1 CNRS, Universite´ Nice Sophia-Antipolis Directeur

1Acknowledgment

First of all I would like to express my gratitude and my deepest thanks to professor Car-

los Simpson for his supervision, advice, and guidance from the very early stage of this

research as well as giving me extraordinary experiences through out the work. During

these years i have beneﬁted from his experience and his vast knowledge mathematics.

While working under his supervision, he always offered unlimited support to bring forth

the best possible work. He has a special method to simplify every thing in a way that

made me capable of achieving more and more. I am very grateful to the care and the

attention that he gave during this work. I feel unable to ﬁnd the appropriate words to

express my gratitude to him properly. I am indebted to him more than he knows. Thank

you very much.

I am very grateful to Professor Alexandre Dimca, who introduced me to Algebraic

Geometry and Singularity Theory. I thank him for agreeing to be the rapporteur and

to the jury. Also a lot of thanks go to the professors who accepted to judge this work.

I thank Professor.Tony Pantev, Professor.Jaya Iyer, Professor.Sorin Dumitrescu, Pro-

fessor.Bertrand Toen. I gratefully thank for the panel who gave me the opportunity to

represent this work and a big honor by their present.

Many thanks go in particular to Professor Nicole Simpson, for her valuable advice

in science discussion.

Words fail me to express my appreciation to my wife Pharmacy Doctor.Marwa

Awada Taher(My Love), whose dedication, love and persistent conﬁdence in me, has

taken the load off my shoulder. I owe her for being unselﬁshly let her intelligence, pas-

sions, and ambitions collide with mine. Therefore, I would also thank Made Awada’s

family for letting me take her hand in marriage, and accepting me as a member of

the family. I would like to thank My uncle Dr.Hassan Awada, Mrs.Joumana Baraket,

Mrs.Souna Awada, Mrs.Kawthar Awada, Mr.Ali Awada, and ﬁnally Mr.Moussa Hassan.

Where would I be without my family? My parents deserve special mention for their

inseparable support and prayers. My Father Hassan Taher in the ﬁrst place is the per-

son who put the fundament my learning character, showing me the joy of intellectual

pursuit ever since I was a child. My Mother Samia El-hajj, is the one who sincerely

raised me with her caring and gently love. A Lot of thanks to my sisters and brothers.

First I gratefully thank my sister Professor.Fadia Taher for her support me and advise,

guidance, from the ﬁrst year at the university I feel unable to ﬁnd the appropriate words

to express my gratitude to her properly. Thank you very much. Dr.Nachaat Mansour,

2Dr.Ismat Taher, Dr.Hanadi Taher(cuty nana), Dr.Isam Taher, Dr.Tania Taher, Dr.Adel

Taher, Mrs.Lama Akel, Dr.Fadi Taher and Mrs.Rola Kassem. Thanks for being support-

ive and caring siblings.

My special thanks to Mr.Mostapha El-Solh Honorary Consul of Lebanon in Monaco

for giving me the opportunity to work with him as assistant during the period of study. I

feel unable to ﬁnd the appropriate words to express my gratitude to him properly. Thank

you very much. I would also acknowledge Dr.Samih Beik El-Solh, and Mrs.Souad

Mikati El-Solh, Mr.Marek Sinno, Mrs.Maya El-Solh Sinno, Mr.Mohamad El-Solh, and

Mrs.Cecile EL-Solh.

It is a pleasure to express my gratitude wholeheartedly to Mikati’s family. Many

thanks to Mr.Taha Mikati, Mr.Najib Mikati the Prime minister of lebanon, Mrs.Nada

Miskawi Mikati, Mrs.May Mikati, Mr.Azmi Mikati, Mr.Maher Mikati, Mr.Fouad Mikati,

Mr.Malek Mr.Ali Bdeir, Mrs.Mira Azmi Mikati, Mrs.Mira Mikati Bdeir, and

Mrs.Dana Mikati.

I would like also to express my gratitude and my deepest thanks to Colonel.Ramez

Khamiss and Colonel. Pierre Neghawi.

I want to thank also My University Nice Sophia-Antipolis, from which I, as well as

thousands of students, have graduated and to my teachers in Master who gave me the

chance to get the proper and high level of education.

Collective and individual acknowledgments are also owed to my colleagues at the

University of Nice Sophia-Antipolis. Many thanks go in particular to Dr.Mohamed

Sarrage, Dr.Osman Khodor, Dr.Mouhamad Hanzal, Dr.Hayssam, Dr.Samer Alouch,

Dr.Hamad hazim, Dr.Brahim Benzeghli... Also i would like to thank Mr.Samir Chahine,

Dr.Kifah Yehya, Mrs.Rouba Yehya, Mr.Mazen Yehya, Mr.Jad Abou Khater, Mr.Maher

Raed, Dr.Ali Hamzi, Mr.Ali Mousawi, Mrs.Manar Akel, Mr.Youssef Hachouch, Dr.Rola

Abou-Taam, Dr.Hassan Kalakech, Mrs.Nancy Kalakech, Mr.Samer Shahine, Dr.Kamel

Kalakech, Mr.Abdalah Rammel, Mr.Ramzi Abi Haydar, Mr.Ziad Dagher, Mr.Dani Kan-

daleft, Captain.Abdalah Charkawi and Mr.Nouhad Kechle(Abou Taha)

Finally, I would like to thank everybody who was important to the successful real-

ization of thesis, as well as expressing my apology that I could not mention personally

one by one.

3Contents

1 Introduction 1

1.1 Algebraic geometry background . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Afﬁne varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.2 Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.3 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Divisors on curves . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.5 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.6 Vector bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Intersection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Chow group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Cartier divisor . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.4 Segre and Chern classes vector bundles . . . . . . . . . . . . . 18

1.2.5 Statement of the Hirzeburch-Rimann-Roch theorem . . . . . . 24

1.2.6 Parabolic bundles . . . . . . . . . . . . . . . . . . . . . . . . . 33

par1.2.7 Sections of the line bundleL . . . . . . . . . . . . . . . . . 36

1.3 Blowing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.3.1 Elementary transformation of algebraic bundles . . . . . . . . . 43

1.3.2 Generalization of elementary transformation . . . . . . . . . . 45

2 Calculating the parabolic Chern character of a locally abelian parabolic

bundle 47

2.1 Quasi-parabolic structures . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.1 Index sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.2 Two approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.3 Locally abelian condition . . . . . . . . . . . . . . . . . . . . . 53

2.2 Weighted parabolic structures . . . . . . . . . . . . . . . . . . . . . . . 58

2.2.1 Riemann-Roch theorem . . . . . . . . . . . . . . . . . . . . . 65

2.3 Computation of parabolic Chern characters of a locally abelian parabolic

Par ParbundleE in codimension one and twoch (E); ch (E) . . . . . . . 691 22.3.1 The characteristic numbers for parabolic bundles in codimen-

sion 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Parabolic Chern character of a locally abelian parabolic bundleE in codi-

Parmension 3, ch (E) 783

3.1 The characteristic number for a parabolic bundle in codimension 3 . . . 80

4 Chern invariants for parabolic bundles at multiple points 82

4.1 Calculating the invariant of a locally abelian parabolic bundle . . . . 83

4.2 Parabolic bundles with full ﬂags . . . . . . . . . . . . . . . . . . . . . 86

4.3 Resolution of singular divisors . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Local Bogomolov-Gieseker inequality . . . . . . . . . . . . . . . . . . 93

4.5 Modiﬁcation of ﬁltrations due to elementary transformations . . . . . . 98

4.6 The local parabolic invariant . . . . . . . . . . . . . . . . . . . . . . . 100

4.7 Normalization via standard elementary transformations . . . . . . . . . 104

4.8 The rank two case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.8.1 Panov differentiation . . . . . . . . . . . . . . . . . . . . . . . 109

4.8.2 The Bogomolov-Gieseker inequality . . . . . . . . . . . . . . . 111

5Abstract

In this thesis we calculate the parabolic Chern character of a bundle with locally abelian

parabolic structure on a smooth strict normal crossings divisor, using the deﬁnition in

terms of Deligne-Mumford stacks. We obtain explicit formulas for ch , ch and ch ,1 2 3

and verify that these correspond to the formulas given by Borne forch and Mochizuki1

forch .2

The second part of the thesis we take D X is a curve with multiple points in

a surface, a parabolic bundle deﬁned on (X;D) away from the singularities can be

extended in several ways to a parabolic bundle on a resolution of singularities. We

investigate the possible parabolic Chern classes for these extensions.Chapter 1

Introduction

The ﬁrst part of this thesis we supposeX be a smooth projective variety with a strict

normal crossings divisorD =D +::: +D X. The aim of this paper is to give an1 n

explicit formula for the parabolic Chern character of a locally abelian parabolic bundle

on (X;D) in terms of:

—the Chern character of the underlying usual vector bundle,

—the divisor componentsD in the rational Chow groups ofX,i

—the Chern characters of the associated-graded pieces of the parabolic ﬁltration along

the multiple intersections of the divisor components, and

—the parabolic weights.

After giving a general formula, we compute explicitly the parabolic ﬁrst, second,

Par Par Parthird parabolic Chern charactersch (E);ch (E) andch (E).1 2 3

The basic idea is to use the formula given in [IS2]. However, their formula did

not make clear the contributions of the different elements listed above. In order to

adequately treat this question, we start with a somewhat more general framework of

unweighted quasi-parabolic sheaves [Se]. These are like parabolic sheaves except that

the real parabolic weights are not speciﬁed. Instead, we consider linearly ordered sets

0 indexing the parabolic ﬁltrations over the componentsD . Let denote the linearlyi i i

ordered set of links or adjacent pairs in . We also call these “risers” as can bei

thought of as a set of steps. The parabolic weights are then considered as functions

0 : ! ( 1; 0] R. This division allows us to consider separately some Cherni i

class calculations for the unweighted structures, and then the calculation of the parabolic

Chern character using the parabolic weights.

A further difﬁculty stems from the fact that there are classically two different ways

to give a parabolic structure: either as a collection of sheaves included in one another; or

by ﬁxing a bundleE (typically the zero-weight sheaf) plus a collection of ﬁltrations of

Ej . The formula of [IS2] is expressed in terms of the collection of sheaves, whereasDi

we look for a formula involving the ﬁltrations. Thus, our ﬁrst task is to investigate the

relationship between these two points of view.An important axiom concerning the parabolic structures considered here, is that

they should be locally abelian. This means that they should locally be direct sums of

parabolic line bundles. It is a condition on the simultaneous intersection of three or more

ﬁltrations; up to points where only two divisor components intersect, the condition is

automatic. This condition has been considered by a number of authors (Borne [Bo1]

[Bo2], Mochizuki [Mo2], Iyer-Simpson [IS1], Steer-Wren [Sr-Wr] and others) and is

necessary for applying the formula of [IS2].

A quasi-parabolic sheaf consists then of a collection of sheavesE with 2 ;:::; i1 n

onX, whereas a quasi-parabolic structure given by ﬁltrations consists of a bundleEi

ionX together with ﬁltrationsF Ej of the restrictions to the divisor components.D ii

In the locally abelian case, these may be related by a long exact sequence (2.1.5):

nM M

i ij0! E !E! ( ) (L )! ( ) (L )!:::!L ! 0: ;:::; i ? i ? ;:::; n n1 j ; 1i i j

i=1 i<j

i ;:::;iq1WhereL denote the quotient sheaves supported on intersections of the divisors ;:::; i i1 q

D \:::\D .i i1 q

Using this long exact sequence we get a formula (2.1.6) for the Chern characters

ofE in terms of the Chern character of sheaves supported on intersection of the ;:::; 1 n

divisorsD \:::\D of the form:i i1 q

n X X

Vb Vb q Ich (E ) =ch (E) + ( 1) ch (L ) : ;::: I;?1 n ;:::; i iq1

q=1 i <i <:::<i1 2 q

The notion of parabolic weight function is then introduced, and the main work of

this paper begins: we obtain the Chern characters for theE for any 2 ( 1; 0]; ;:::; i1 n

these are then put into the formula of [IS2], and the result is computed. This computation

requires some combinatorial manipulations with the linearly ordered sets notably thei

0associated sets of risers in the ordering. It yields the following formula (2.2.2) ofi

Theorem 2.2.4:

Par Vb Dch (E) =ch (E)e +

" #

n q D ( ( )+1)D i i i iX X X Y j j j je (1 e )i ;:::;iD q 1 qe : ( 1) ch (Gr ) :I;? ;:::; Di i i1 q je 10q=1 i <i <:::<i j=11 2 q 2i ij j

In this formula, the associated-graded sheaves corresponding to the multiple ﬁltrations

i ;:::;iq1on intersections of divisor componentsD =D \\D are denoted byGr .I i iq1 ;:::; i iq1

These are sheaves on D but are then considered as sheaves on X by the inclusionI

i ;:::;i1 q : D ,! X. The Chern characterch (Gr ) is the Chern character ofI;? I I;? ;:::; i iq1

the coherent sheaf onX. This is not satisfactory, since we want a formula involving the

2i ;:::;i1 qChern characters of theGr onD . Therefore inx2.2.1 we use the Grothendieck-I ;:::; i i1 q

Riemann-Roch theorem to interchangech and , leading to the introduction of ToddI;?

Dclasses of the normal bundles of the D . Another difﬁculty is the factor of e mul-I

Vbtiplying the term ch (E); we would like to consider the parabolic Chern class as a

Vbperturbation of the Chern class of the usual vector bundle ch (E). Using the same

Vbformula for the case of trivial parabolic weights, which must give backch (E) as an

Vb Vb Danswer, allows us to rewrite the difference betweench (E) andch (E)e in a way

compatible with the rest of the formula. After these manipulations the formula becomes

(2.2.5) of 2.2.14:

IfX be a smooth projective variety with a strict normal crossings divisorD = D +1

::: +D X. Then the explicit formula for the parabolic Chern character of a locallyn

abelian parabolic bundle on (X;D) in terms of:

—the Chern character of the underlying usual vector bundle,

—the divisor componentsD in the rational Chow groups ofX,i

—the Chern characters of the associated-graded pieces of the parabolic ﬁltration along

the multiple intersections of the divisor components, and

—the parabolic weights, is deﬁned as follows:

Par Vbch (E) =ch (E)

!

n q D iX X X Y j1 e i ;:::;iD q 1 qe : ( 1) : ch(Gr ) +I;? ;:::; i i1 qDij0q=1 i <i <:::<i j=11 2 q 2i ij j

" !#

qn ( )+1 D ( i i ) iX X X Y j j j1 e i ;:::;iqD q 1e : ( 1) : ch(Gr ) :I;? ;:::; i iq1Dij0q=1 i <i <:::<i 2 j=11 2 q ij ij

Par ParFinally, we would like to compute explicitly the terms ch (E), ch (E) and1 2

Parch (E). For these, we expand the different terms3

" !# !

q q ( )+1 D D( )i i i iY Yj j j j1 e 1 eDe ; ;

D Di ij jj=1 j=1

in low-degree monomials ofD , and then expand the whole formula dividing the termsij

up according to codimension. Denoting byS :=f1;:::;ng the set of indices for divisor

components, we get the following formulae:

Parch (E) :=rank(E):[X]0

3X X

iPar Vb 1ch (E) :=ch (E) ( ):rank(Gr ):[D ]i i i1 1 1 1 1i1P

0i2S1 2i1 i1

X X

DiPar Vb i1 1ch (E) := ch (E) ( ):( ) c (Gr )i i i ?2 2 1 1 1 1 i1P0i2S1 2i i1 1

X X1 2 i 21+ ( ):rank(Gr ):[D ]i ii 1 11 i12 P0i2S1 2i i1 1

X X X

i ;i1 2+ ( ): ( ):rank (Gr ):[D ]:i i i i p p1 1 2 2 ; i i1 2

0i <i 2 p2Irr(D \D )1 2 i i i1 i 1 21

0 2i i2 2

ParForch (E), see Chapter 3.3

ParThe formula forch (E) is well-known (Seshadri et al) and, in terms of the deﬁnition1

of Chern classes using Deligne-Mumford stacks, it was shown by Borne in [Bo1]. The

Parformula forch (E) was given by Mochizuki in [Mo2], and also stated as a deﬁnition2

by Panov [Pa]. In both cases these coincide with our result (see the discussion on page

Par76). As far as we know, no similar formula forch (E) has appeared in the literature.3

There are some of the motivations for the present work. In heterotic string theory

[OPP] physicists look for a vector bundle with speciﬁc Chern classes. The third Chern

class corresponds to numbers of families of quarks and leptons on the observable brane.

In future works where these vector bundles might be replaced by parabolic or orbifold

Parbundles, it would be important to have the formula for ch (E). The Bogomolov3

Gieseker inequality says thatch (E) 0 whereE is a stable bundle withch (E) = 0.2 1

Donaldson’s theorem says that in case of equality one gets a ﬂat unitary connection.

These facts have been extended to the parabolic case notably in work of Li, Panov and

Mochizuki ([Mo2], [Pa], [L]). Our calculations conﬁrm their formulas for ch (E)—2

getting the right formula is essential for applying the Bogomolov-Gieseker inequality.

ParThe formula forch (E) will be useful in the Donagi-Pantev approach to the geometric2

Langlands program [DP]. Iyer and Simpson have pointed out that Reznikov’s theorem

of vanishing of certain regulations of ﬂat bundles, extends to the parabolic case, and for

applications it would be important to know the explicit formulas.

Mochizuki deﬁnes the Chern classes using the curvature of an adapted metric and

obtains his formula as a result of a difﬁcult curvature calculation. It should be noted

that our formula concerns the classes deﬁned via Deligne-Mumford stacks in the ra-

tional Chow groups ofX whereas Mochizuki’s deﬁnition involving curvature can only

4