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

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121 pages
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


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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
´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
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