UNIVERSITE NICE SOPHIA ANTIPOLIS UFR SCIENCES Ecole Doctorale Sciences Fondamentales et Appliquees

Tony Pantev

Description

Niveau: Supérieur, Doctorat, Bac+8
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

Sujets

• Simpson
• Rapporteur
• Mónaco
• Moussa

Informations

• Publié par: profil-zyak-2012
• Publié le: 01 mai 2011
• Nombre de lectures: 160
• Langue: English

Extrait

´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
´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 cr