Annales de la Faculté des Sciences de Toulouse

Gerd Dethloff

Pham Hoang Ha

Ramification of the Gauss map of complete minimal surfaces in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ on annular ends

François Delgove

Nicolas Retailleau

Sur la classification des hexagones hyperboliques à angles droits en dimension 5

Matthieu Gendulphe

Yohei Komori

Polyhedral Realization of a Thurston Compactification

Corey Dunn

Matthew Miller

Max Wakefield

Sebastian Zwicknagl

Equivalence classes of Latin squares and nets in ${\mathbb{C}P}^2$

Sandro Manfredini

Simona Settepanella

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
SANDRO MANFREDINI, SIMONA SETTEPANELLA
On the Configuration Spaces of Grassmannian Manifolds
Tome XXIII, no 2 (2014), p. 353-359.
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© Université Paul Sabatier, Toulouse, 2014, tous droits réservés.
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Annales de la Faculté des Sciences de Toulouse Vol. XXIII, n◦ 2, 2014
pp. 353-359
On the Configuration Spaces
of Grassmannian Manifolds
Sandro Manfredini(1), Simona Settepanella(2)
ABSTRACT. — Let F ih(k, n) be the i-th ordered configuration space of all
distinct points H1, . . . , Hh in the Grassmannian Gr(k, n) of k-dimensional
subspaces of Cn, whose sum is a subspace of dimension i. We prove that
F ih(k, n) is (when non empty) a complex submanifold of Gr(k, n)h of
dimension i(n − i) + hk(i − k) and its fundamental group is trivial if
i = min(n, hk), hk = n and n > 2 and equal to the braid group of the
sphere CP 1 if n = 2. Eventually we compute the fundamental group in
the special case of hyperplane arrangements, i.e. k = n− 1.
RÉSUMÉ. — Soit F ih(k, n) le i-ème espace de configuration ordonnée de
tous les points distincts H1, . . . , Hh dans la Grassmannienne Gr(k, n) de
sous-espaces de dimension k de Cn, dont la somme est un sous-espace
de dimension i. Nous prouvons que F ih(k, n) est (si non vide) une sous-
variété complexe de Gr(k, n)h de dimension i(n − i) + hk(i − k) et que
son groupe fondamental est trivial si i = min(n, hk), hk = n et n > 2 et
égal au groupe de tresses de la sphère CP 1 si n = 2. Finalement, nous
calculons le groupe fondamental dans le cas particulier des arrangements
d’hyperplans, c’est-à-dire k = n− 1.
1. Introduction
Let M be a manifold. The ordered configuration space
Fh(M) = {(x1, . . . , xh) ∈Mh|xi = xj , i = j}
of h distinct points inM has been widely studied after it has been introduced
by Fadell and Neuwirth [5] and Fadell [3] in the sixties. It is well known that
(1) Department of Mathematics, University of Pisa
manfredi@dm.unipi.it
(2) LEM, Scuola Superiore Sant’Anna, Pisa
s.settepanella@sssup.it.
– 353 –
Sandro Manfredini, Simona Settepanella
for a simply connected manifold M of dimension greater or equal than 3,
the pure braid group π1(Fh(M)) on h strings of M is trivial. This is not
the case when the dimension of M is lower than 3 as, for example, the pure
braid group of the sphere S2 ≈ CP 1 with presentation:
π1(Fh(CP 1)) ∼=
〈
αij , 1  i < j  h− 1
∣∣(Y B 3)h−1, (Y B 4)h−1, D2h−1 = 1
〉
where Dk = α12(α13α23)(α14α24α34) · · · (α1kα2k · · ·αk−1 k) and (Y B 3)n
and (Y B 4)n are the Yang-Baxter relations (see [2] and [4]):
(Y B 3)n : αijαikαjk = αikαjkαij = αjkαijαik, 1  i < j < k  n,
(Y B 4)n : [αkl, αij ] = [αil, αjk] = [αjl, α
−1
jk αikαjk] = [αjl, αklαikα
−1
kl ] = 1,
1  i < j < k < l  n.
In a recent paper ([1]) Berceanu and Parveen introduced new configuration
spaces. They stratify the classical configuration spaces Fh(CPn) with com-
plex submanifolds F ih(CPn) defined as the ordered configuration spaces of
all h points in CPn generating a projective subspace of dimension i. They
prove that the fundamental groups π1(F ih(CPn)) of these submanifolds are
trivial except when i = 1 providing, in this last case, a presentation similar
to those of the pure braid group of the sphere.
In a subsequent paper ([6]), authors apply similar techniques to the
affine case, that is to the ordered configuration space F i,nh = F ih(Cn) of all
h points in Cn generating an affine subspace of dimension i. They prove
that the spaces F i,nh are simply connected except for i = 1 or i = n = h− 1
and, in the last cases, they provide a presentation of the fundamental groups
π1(F i,nh ).
In this paper we generalize the result in [1] to the Grassmannian manifold
Gr(k, n) parametrizing k-dimensional subspaces of Cn. We define the i-th
ordered configuration space F ih(k, n) as the ordered configuration space of
all distinct points H1, . . . , Hh in the Grassmannian Gr(k, n) such that the
sum (H1 + · · ·+Hh) is an i-dimensional space.
We prove that the i-th ordered configuration space F ih(k, n) is (when non
empty) a complex submanifold of Gr(k, n)h and we compute its dimension.
As a corollary, we prove that if n = hk and i =min(n, hk) then the i-th
ordered configuration space F ih(k, n) has trivial fundamental group except
when n = 2, that is:
π1(Fmin(n,hk)h (k, n)) = 0 if (k, n) = (1, 2) (1.1)
π1(F11 (1, 2)) = π1(F2(CP1)).
– 354 –
On the Configuration Spaces of Grassmannian Manifolds
As a consequence, the fundamental group of the i-th ordered configuration
space F ih(n − 1, n) of hyperplane arrangements of cardinality h vanishes
except when n = 2.
Using a dual argument, we also get that the fundamental group of the or-
dered configuration space of all distinct k-dimensional subspaces H1, . . . , Hh
in Cn such that the intersection (H1 ∩ · · · ∩ Hh) is an i-dimensional sub-
space is a simply connected manifolds when i = max(0, n−hk), except when
n = 2.
We conjecture that similar results to that obtained in [1] for projective
spaces holds also for Grassmannian manifolds and the fundamental group of
the i-th ordered configuration space F ih(k, n) vanishes except for low values
of i. This will be the object of forthcoming publications.
2. Main Section
LetGr(k, n) be the Grassmannian manifold parametrizing k-dimensional
subspaces of the n-dimensional complex space Cn, 0<k<n, and Fh(Gr(k, n))
be its ordered configuration spaces.
2.1. The spaces F ih(k, n)
Let’s define the i-th ordered configuration space F ih(k, n) as the space of
all distinct points H1, . . . , Hh in the Grassmannian Gr(k, n) whose sum is
an i-dimensional subspace of Cn, i.e.
F ih(k, n) = {(H1, . . . , Hh) ∈ Fh(Gr(k, n)) | dim(H1 + · · ·+Hh) = i}.
It is easy to see that the following results hold:
1. if h = 1 then F i1(k, n) is empty unless i = k, in which case Fk1 (k, n) =
Gr(k, n);
2. if i = 1 then F i1(k, n) is empty unless k = h = 1 and we get F11 (1, n) =
Gr(1, n) = CPn−1;
3. for h  2, F ih(k, n) = ∅ if and only if i  k + 1 and i  min(hk, n);
4. for i = hk  n, then the h subspaces giving a point of Fhkh (k, n) form
a direct sum;
5. for h  2, Fh(Gr(k, n)) =
n∐
i=2
F ih(k, n);
– 355 –
Sandro Manfredini, Simona Settepanella
6. for h  2, the adjacency of the strata is given by
F ih(k, n) = F ih(k, n)
∐
F i−1h (k, n)
∐
. . .
∐
F2h(k, n).
By above remarks, it follows that the case h = 1 is trivial, hence from
now on, we will consider h > 1 (and hence i > k).
We want to show that F ih(k, n) is (when non empty) a complex subma-
nifold of Gr(k, n)h and compute its dimension. We need to briefly recall few
easy facts and introduce some notations.
2.2. The determinantal variety
Let’s recall that the determinantal variety Dr(m,m
′) is the variety of
m × m′ matrices with complex entries of rank less than or equal to r 
min(m,m′). It is an analytic (algebraic, in fact) variety of dimension r(m+
m′ − r) whose set of singular points is given by those matrices of rank less
than r. From now on, Dr(m,m
′)∗ will denote the set of non-singular points
of the determinantal variety Dr(m,m
′), that is the set of m×m′ matrices
of rank equal to r.
2.3. A system of local coordinates for Gr(k, n)h
Let V0 ⊂ Cn be a subspace of dimension dimV0 = n− k, then the set
UV0 = {H ∈ Gr(k, n) | H ⊕ V0 = Cn}
is an open dense subset of Gr(k, n).
Let B = {w1, . . . , wk, v1, . . . , vn−k} be a basis of Cn such that
{v1, . . . , vn−k} is a basis of V0. We get a (complex) coordinate system on
UV0 as follows.
Let H be an element in UV0 , then the affine subspace V0 +wj intersects
H in one point uj for any j = 1, · · · , k and {u1, · · · , uk} form a basis of
H. Hence H is uniquely determined by a n × k matrix of the form
(
I
A
)
,
where I is the k × k identity matrix and A is the (n− k)× k matrix of the
coordinates of u1 − w1, . . . , uk − wk with respect to vectors {v1, . . . , vn−k}.
The coefficients of A give complex coordinates in UV0
∼= Ck(n−k).
Let (H1, . . . , Hh) be a point in Gr(k, n)
h, the open sets UH1 , . . . , UHh in
the Grassmannian manifold Gr(n− k, n) have non empty intersection, that
is there exists an element V0 ∈ Gr(n− k, n) such that V0 ⊕Hj = Cn for all
– 356 –
On the Configuration Spaces of Grassmannian Manifolds
j = 1, . . . , h. Thus, Gr(k, n)h is covered by the open sets UhV0 as V0 varies
in Gr(n − k, n). Taking a basis as defined above, each element in UhV0 is
uniquely determined by a n× hk matrix of the form
(
I I · · · I
A1 A2 · · · Ah
)
and the coefficients of (A1 A2 · · · Ah ) give complex coordinates in
UhV0
∼= Chk(n−k).
2.4. A system of local coordinates for F ih(k, n)
In terms of the above coordinates, (H1, . . . , Hh) ∈ UhV0 belongs to F ih(k, n)
if and only if Aj = Al when j = l and rank
(
I I · · · I
A1 A2 · · · Ah
)
= i. Let
us remark that
rank
(
I I · · · I
A1 A2 · · · Ah
)
= rank
(
I I · · · I
0 A2 −A1 · · · Ah −A1
)
= k + rank (A2 −A1 · · · Ah −A1 ) .
Then the coefficients of Bj = Aj − A1 are new coordinates, in which
the intersection UV0 ∩ F ih(k, n) corresponds, in Chk(n−k), to the product
Ck(n−k) ×Di−k(n − k, hk − k)∗ minus the closed sets given by Bj = 0 for
2  j  h and by Bj = Bl for 2  j, l  h, j = l. We get the following
theorem.
Theorem 2.1. — The i-th ordered configuration space F ih(k, n) is a com-
plex submanifold of the Grassmannian manifold Gr(k, n) of dimension
dih(k, n) = i(n− i) + hk(i− k). (2.2)
Equation (2.2) is an easy consequence of the equality:
k(n− k) + (i− k)(n− k + hk − k − (i− k)) = i(n− i) + hk(i− k).
Let us remark that the dimension dih(k, n) attains its maximum hk(n−k) if
and only if i = n or i = hk. Hence dih(k, n) is a strictly increasing function
of i when i min(n, hk).
2.5. The fundamental group of Fmin(n,hk)h (k, n)
The space Fmin(n,hk)h (k, n) is an open subset of the ordered configuration
space Fh(Gr(k, n)) and all other (non void) Fjh(k, n) have strictly lower di-
mension. Moreover, if i = n the difference of dimensions dih(k, n)−di−1h (k, n)
– 357 –
Sandro Manfredini, Simona Settepanella
equals 1 + hk − n and if i = hk it equals 1 + n − hk. Then if n = hk, all
(non void) Fjh(k, n) with j <min(n, hk) have real codimension at least 4 in
Fh(Gr(k, n)). Then, if n = hk and i =min(n, hk), the fundamental group
of F ih(k, n) = Fh(Gr(k, n)) \ F i−1h (k, n) is the same as the fundamental
group of Fh(Gr(k, n)) (since, by the adjacency of the strata, the closure
F i−1h (k, n) is the finite union of complex subvarieties of Fh(Gr(k, n)) of real
codimension at least 4).
Let us recall that the complex Grassmannian manifolds Gr(k, n) are
simply connected and have real dimension at least 4 except Gr(1, 2) = CP1
and that for a simply connected manifold of real dimension at least 3 the
pure braid groups vanish, i.e. π1(Fh(Gr(k, n))) = 0 if (k, n) = (1, 2). We
get the following corollary.
Corollary 2.2. — The fundamental group of the i-th ordered configu-
ration space F ih(k, n) vanishes if n = hk and i = min(n, hk) except when
n = 2 in which it is the pure braid group of the sphere.
2.6. The dual case
LetGr(k, n)∗ be the Grassmannian manifold parametrizing k-dimensional
subspaces in the dual space (Cn)∗. Then we can define the i-th dual ordered
configuration space F ih(k, n)∗ as
F ih(k, n)∗ = {(H1, . . . , Hh) ∈ Fh(Gr(k, n)∗) | dim(H1 ∩ · · · ∩Hh) = i}.
The spaces F ih(k, n)∗ stratify the ordered configuration space Fh(Gr(k, n)∗)
of the Grassmannian manifold Gr(k, n)∗.
The annihilators define homeomorphisms Ann: Gr(n − k, n) → Gr(k, n)∗
which induce homeomorphisms between the (n− i)th ordered configuration
space Fn−ih (n−k, n) and the i-th dual ordered configuration space F ih(k, n)∗.
As a consequence the spaces F
max(0,n−hk)
h (n − k, n)∗ are simply connected
manifolds except when n = 2. In this case the fundamental group is the
pure braid group of the sphere.
2.7. i-th ordered configuration spaces of hyperplane arrangements
If k = n− 1 points in the ordered configuration space Fh(Gr(n− 1, n))
are h-uple of hyperplanes in Cn, i.e. ordered arrangements of hyperplanes.
In this case, h = 1 implies i = n − 1 and the i-th ordered configuration
space is the Grassmannian manifold, i.e. Fn−11 (n − 1, n) = Gr(n − 1, n).
While h > 1 implies i = n, since the sum of two (different) hyperplanes is
– 358 –
On the Configuration Spaces of Grassmannian Manifolds
the whole space Cn, and the following equalities hold
Fnh (n− 1, n) = Fh(Gr(n− 1, n)) = Fh(CPn−1).
Hence, the fundamental group of the i-th ordered configuration space of
hyperplane arrangements F ih(n− 1, n) vanishes except when n = 2. In this
case it is the fundamental group of the sphere CP1.
In the dual case there are homeomorphisms F ih(n−1, n)∗ ∼= Fn−ih (1, n) and
fundamental groups of Fn−ih (1, n) are zero except if i = n − 1 (see [1]).
Hence the space of h-uples of distinct hyperplanes in Cn whose intersection
has dimension equal to i is simply connected except if i = n− 1.
Acknowledges. — The second author gratefully acknowledge the sup-
port given to this research by the European Commission, within the 6th FP
Network of Excellence “DIME - Dynamics of Institutions and Markets in
Europe” and the Specific Targeted Research Project “CO3 - Common Com-
plex Collective Phenomena in Statistical Mechanics, Society, Economics and
Biology”
Bibliography
[1] Berceanu (B.), Parveen (S.). — Braid groups in complex projective spaces, Adv.
Geom. 12, p. 269-286 (2012).
[2] Birman (J. S.). — Braids, Links, and Mapping Class Groups, Annals of Mathe-
matics 82, Princeton University Press (1974).
[3] Fadell (E.R.). — Homotopy groups of configuration spaces and the string problem
of Dirac, Duke Math. J. 29, p. 231-242 (1962).
[4] Fadell (E.R), Husseini (S.Y.). — Geometry and Topology of Configuration
Spaces, Springer Monographs in Mathematics (2001), Springer-Verlarg Berlin.
[5] Fadell (E.R.), Neuwirth (L.). — Configuration spaces, Math. Scand. 10, p. 111-
118 (1962).
[6] Manfredini (S.), Parveen (S.), Settepanella (S.). — Braid groups in complex
spaces, arXiv: 1209.2839 (2012).
– 359 –


On the Configuration Spaces of Grassmannian Manifolds

Damien Rössler

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
DAMIEN RÖSSLER
Préface
Tome XXIII, no 3 (2014), p. i-vi.
<http://afst.cedram.org/item?id=AFST_2014_6_23_3_r1_0>
© Université Paul Sabatier, Toulouse, 2014, tous droits réservés.
L’accès aux articles de la revue « Annales de la faculté des sci-
ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique
l’accord avec les conditions générales d’utilisation (http://afst.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous quelque
forme que ce soit pour tout usage autre que l’utilisation à fin strictement
personnelle du copiste est constitutive d’une infraction pénale. Toute copie
ou impression de ce fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Préface
Le présent volume a vu le jour à la suite d’une conférence en l’honneur
de Christophe Soulé intitulée ‘Arakelov Geometry and K-theory’ qui s’est
déroulée à Toulouse du 21 au 23 mai 2012. La plupart des articles qui
suivent (mais pas tous) ont été proposés par des intervenants de ce congrès
et il reflète donc dans une certaine mesure le contenu scientifique de la
rencontre. Nous avons reproduit en annexe le programme de la conférence
et la liste des participants. On trouvera aussi à la suite de cette préface
l’affiche de la conférence.
Pendant sa remarquable carrière, C. Soulé s’est intéressé à un spectre
très large de mathématiques. C’est surtout la K-théorie et la cohomologie
des groupes arithmétiques qui l’occupent pendant ses jeunes années, puis à
la fin des années 1980, il se tourne (en collaboration avec H. Gillet) vers la
géométrie d’Arakelov et l’analyse spectrale sur les variétés. Depuis la fin des
années 1990 il s’intéresse aux problèmes mathématiques issus de la biologie
et aux spéculations autour du corps à un élément.
Parmi les nombreuses contributions de C. Soulé aux mathématiques, en
voici quelques-unes qui nous semblent particulièrement intéressantes:
– ses travaux sur la K-théorie des nombres entiers (voir par ex. [4]) ;
– la démonstration (avec H. Gillet) d’une conjecture de J.-P. Serre sur
la positivité des multiplicités d’intersections (voir [3]) ;
– le théorème de Riemann-Roch arithmétique en toute dimension relative
(avec J.-M. Bismut et H. Gillet ; voir [1]) ; il s’agit là d’un résultat spectacu-
laire, où l’analyse spectrale rencontre la géométrie algébrique arithmétique
dans un théorème dont la véritable portée reste encore largement incom-
prise ;
– la démonstration (avec. H. Gillet) d’une conjecture de Serre sur les
invariants universels du groupe de Grothendieck des variétés (voir [2]).
Voici enfin quelques éléments biographiques sur C. Soulé.
C. Soulé est né en 1951 et a été élève de l’école normale supérieure au
début des années 1970. Il est entré au CNRS comme attaché de recherches
en 1974. Il fera toute sa carrière dans cette institution. Il devient directeur
de recherches de deuxième classe en 1986, avant d’être promu à la première
classe en 1990. Il arrive à l’Institut des hautes études scientifiques (IHÉS) en
– i –
Préface
1984, un centre de recherches qu’il n’a jamais quitté depuis, à part pour un
séjour scientifique d’un an à Harvard en 1990. Pendant ce séjour, il donne un
cours sur ses travaux en théorie d’Arakelov. Les notes de ce cours donneront
lieu à un livre qui est encore une référence incontournable.
C. Soulé a été conférencier au congrès international des mathématiciens
à deux reprises. La première fois à Varsovie en 1983 et la deuxième fois
(pour exposer les travaux de V. Voevodsky) à Péking en 2002. Il a par
ailleurs obtenu la médaille de bronze du CNRS en 1979 ainsi que deux prix
de l’Académie des science – le prix J. Ponti en 1985 et le prix Ampère en
1993.
Il a été élu membre correspondant de l’Académie des sciences le 3 mars
1997, puis membre plein le 12 novembre 2001.
C. Soulé n’a eu qu’un petit nombre d’élèves. Il s’agit de Florence Lecomte
et Bruno Kahn (en K-théorie) ainsi que de Laurent Lafforgue, Damian
Rössler, Nicusor Dan et Javier Frésan (en théorie d’Arakelov).
Annexe. Programme de la conférence
Schedule:
Monday
8.45 registration, coffee
9.30 Gillet
10.45 Kahn
14.00 Bost
15.00 registration, coffee
15.45 Kramer
17.00 Moriwaki
Tuesday
9.15 Chen
10.15 coffee
10.45 Scholl
14.00 Ton
15.15 Gangl
17.30 reception in the townhall of Toulouse
– ii –
Préface
Wednesday
9.15 Bismut
10.15 coffee
10.45 Déglise
14.00 Voisin
15.00 coffee
15.30 Burgos
Program:
J.-M. Bismut. — Laplacien hypoelliptique et torsion analytique
Pour une projection propre projective, le théorème de Riemann-Roch-
Grothendieck exprime le caractère de Chern de l’image directe d’un fibré
holomorphe comme intégrale dans la fibre de classes caractéristiques. Les
travaux de Gillet, Lebeau, Soulé, et moi-même ont raffiné ce résultat au
niveau des formes différentielles. Les formes de torsion analytique holomor-
phes explicitent ce raffinement. Dans l’exposé, je montrerai que sans aucune
hypothèse sur l’espace total de la projection, quand l’image directe est lo-
calement libre, la théorie du laplacien hypoelliptique permet d’obtenir une
version du théorème de Riemann-Roch-Grothendieck dans la cohomologie
de Bott-Chern. La torsion analytique associée au laplacien hypoelliptique
possède des propriétés proches de la torsion analytique elliptique. On ob-
tient une formule de comparaison entre métriques de Quillen elliptiques et
hypoelliptiques, qui fait intervenir le genre R de Gillet et Soulé.
J.-B. Bost. — Inégalités en géométrie d’Arakelov et constantes
optimales
Divers résultats de géométrie d’Arakelov prennent la forme d’inégalités re-
liant des degrés d’Arakelov ou des nombres d’intersection arithmétique as-
sociés à des objets tels que des variétés projectives sur des corps de nom-
bres, ou des cycles algébriques ou des fibrés vectoriels sur de telles variétés.
Comparées à leurs avatars concernant des objets sur des corps de fonc-
tions, ces inégalités font intervenir des constantes dépendant des invariants
géométriques des objets concernés. Cet exposé discutera le rôle de ces con-
stantes dans les applications diophantiennes, ainsi que diverses questions
posées par la détermination de leur valeur optimale.
J.-I. Burgos. — On the arithmetic Riemann-Roch theorem
The arithmetic Riemann Roch theorem is one of the cornerstones of Arakelov
theory and is due, among other mathematicians to C. Soulé. In this talk we
will explain how to generalize the holomorphic analytic torsin forms of Bis-
– iii –
Préface
mut and Köhler to non smooth projective morphism and use it to derive an
arithmetic Riemann-Roch theorem for morphisms that are not necessarily
smooth at the generic fibre.
H. Chen. — Corps d’Okounkov et positivité locale en géométrie
d’Arakelov
On explique comment associer à un système linéaire gradué convenable
d’un fibré inversible gros un corps convexe qui décrit la positivité locale
du système linéaire gradué. Ensuite, on discute les applications de cette
construction dans la géométrie d’Arakelov.
F. Déglise. — Une formule de Riemann-Roch résiduelle
La topologie algébrique a depuis longtemps interprété le théorème de Riemann-
Roch dans le cadre de la théorie de l’orientation des théories cohomologiques.
J’expliquerai comment utiliser la théorie homotopique stable de Morel et
Voevodsky pour obtenir cette version du théorème de Riemann-Roch dans
un cadre arithmétique. Chemin faisant, on mettra au jour une forme de ce
théorème qui fait intervenir les résidus plutôt que les morphismes de Gysin.
H. Gangl. — Perfect forms, K-theory and the cohomology of mod-
ular groups
The classical Voronoi cell complex attached to a space of rank N quadratic
forms gives detailed information both on homology with Steinberg coeffi-
cients of GLN (R) for a number ring R, whence also on certain Km(R), and
on the cohomology of GLN (R). We recall how those objects are related and
investigate in particular the cases where R denotes the integers, with N  7
(joint with Elbaz-Vincent and Soule), or the Gaussian integers, with N  4
(joint with Yasaki, Dutour, Schuermann et al.).
H. Gillet. — Higher Derivations and Descent in Characteristic p
In characteristic 0 the Kodaira-Spencer class is the obstruction to a family of
curves being isotrivial, and more generally one knows that if X a projective
variety over an algebraically closed field K of characteristic zero, then it is
defined over the fixed field of all the derivations of K which lift to X. I shall
describe how one may use higher derivations, in the sense of Hasse-Schmidt,
to obtain a similar result in positive characteristic.
B. Kahn. — Sur les conjectures de Hodge et de Tate généralisées
pour les produits de courbes elliptiques
Il est connu qu’un produit de courbes elliptiques sur vérifie la conjecture
de Hodge, et qu’un produit de courbes elliptiques sur un corps fini vérifie
la conjecture de Tate. Qu’en est-il pour les conjectures généralisées cor-
respondantes? La réponse est oui pour un produit de courbes elliptiques
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Préface
’en position générale’. Le premier cas de position non générale est celui de
quatre courbes à multiplication complexe par des corps quadratiques imag-
inaires non linéairement disjoints. Si B désigne le produit de ces courbes,
il existe une variété abélienne A simple de dimension 4, à multiplication
complexe par le composé des corps quadratiques ci-dessus, telle que la con-
jecture de Hodge (resp. de Tate) généralisée pour N1H4(B) soit ’contrôlée’
par certains cycles de Hodge (resp. de Tate) explicites, de codimension 3
sur A2 × B. L’algébricité de ces cycles est – pour l’orateur – une question
ouverte.
J. Kramer. — A remark on a result of Parshin
In our talk we will discuss effective bounds for Faltings’s delta function for
compact Riemann surfaces of genus g > 1 arising in families parametrized
by a compact Riemann surface. As an example our results can be applied
to a construction of Parshin in which he associates to a smooth projective
curve X of genus g > 1, a family of smooth projective curves covering the
original curve with prescribed ramification.
A. Moriwaki. — Generalized Hodge index theorem on arithmetic
varieties
I would like to talk on the generalized Hodge index theorem on arithmetic
varieties, which is closely related to the works of Prof. Soulé. Moreover, I
will give a recent development of the equality condition of the generalized
Hodge index theorem.
A. Scholl. — NA
B. Toën. — Cyclic homology and symplectic structures in derived
algebraic geometry
We use negative cyclic homology to define the notion of shifted symplectic
structure on a commutative dg-algebra, and by descent techniques on any
derived schemes or stacks. The main theorem states that the derived map-
ping stack from an oriented object to a symplectic target carries a canonical
symplectic structures. This result is used to construct many examples: mod-
uli of sheaves on CY varieties, local systems on compact oriented topological
manifolds. . . If times permits I will explain what quantization means in this
setting.
C. Voisin. — Coniveau de la cohomologie et cycles algébriques
Le fait que la taille des groupes de Chow contrôle le coniveau géométrique de
la cohomologie est bien compris grâce aux travaux de Mumford sur les zéros-
cycles et leurs développements ultérieurs. La conjecture de Bloch prédit in-
versement que le coniveau de la cohomologie contrôle la taille des groupes
de Chow. Nous montrons qu’un tel résultat résulte de la conjecture de Lef-
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Préface
schetz standard, pour les variétés obtenues comme intersections complètes
d’hypersurfaces très amples dans une variété à groupes de Chow triviaux.
Participants:
Anthony Blanc (Montpellier)
Jean-Pierre Bourguignon (Paris)
François Chargois (Nancy)
Robin de Jong (Leiden)
Javier Fresan (Paris)
Michael Harris (Paris)
Benjamin Göbel (Hamburg)
Ariyan Javanpeykar (Paris)
Shu Kawaguchi (Kyoto)
Johannes Kolb (Regensburg)
Ulf Khn (Hamburg)
Klaus Knnemann (Regensburg)
Michael Lapidus (Riverside, California)
Florence Lecomte (Strasbourg)
Steffen Müller (Hamburg)
Alberto Navarro (Madrid)
Amilcar Pacheco (Rio de Janeiro)
Jakob Scholbach (Münster)
Christophe Soulé (Paris)
Shun Tang (Bonn)
Kazuhiko Yamaki (Kyoto)
Bibliographie
[1] Gillet (H.) and Soulé (C.). — An arithmetic Riemann-Roch theorem, Invent.
Math. 110, no. 3, p. 473-543, DOI 10.1007/BF01231343 (1992).
[2] Gillet (H.) and Soulé (C.). — Descent, motives and K-theory, J. Reine Angew.
Math. 478, p. 127-176, DOI 10.1515/crll.1996.478.127 (1996).
[3] Gillet (H.) and Soulé (C.). — Intersection theory using Adams operations, In-
vent. Math. 90, no. 2, p. 243-277, DOI 10.1007/BF01388705 (1987).
[4] Soulé (C.). — On the 3-torsion in K4(Z), Topology 39, no. 2, p. 259-265, DOI
10.1016/S0040- 9383(99)00006-3 (2000).
Damian Rössler
– vi –


Préface

Frédéric Déglise

Suite spectrale du coniveau et $t$-structure homotopique

Mark D. Baker

Alan W. Reid

Principal congruence link complements

Slavyana Geninska

On arithmetic Fuchsian groups and their characterizations

Thibaut Delcroix

Les groupes de Burger-Mozes ne sont pas kählériens

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