SM Exposés 2017-2018
No. SM-203 07/04/2018 16:30 ~ 18:30 Salle 0A1, IMO(307), Université Paris-Sud
Microlocal sheaf theory and Applications in Symplectic Geometry
Peng ZHOU (周鹏), IHES
Abstract : Let M be a smooth manifold, and S a Whitney stratification of M. A constructible sheaf adapted to S is a sheaf that restricts to a local system on each stratum. This combinatorial nature makes constructible sheaves rigid and easy to handle. Meanwhile, the Nadler-Zaslow theorem says any constructible sheaf on M can be represented by a Lagrangian in T^*M, and vice versa. This brings in flexibility of symplectic geometry into the picture. After giving definition and examples of constructible sheaves, I will present its applications in symplectic geometry and homological mirror symmetry.
No. SM-202 07/04/2018 14:00 ~ 16:00 Salle 0A1, IMO(307), Université Paris-Sud
Théorème de Bertini sur un corps fini
Xiaozong WANG (王晓宗), Université Paris-Sud
Résumé : La version classique du théorème de Bertini prédit l'extence des sous-variétés de codimension 1 ayant les mêmes propriétés géométriques comme par exemple lissité, réducibilité géométrique, irréductibilité géométrique que la variété quasi-projective que l'on considère. Pour trouver des telles sous-variétés, il suffit de plonger cette variété dans un espace projectif et prendre l'intersection de cette variété avec un hyperplan général (un point rationnel dans un ouvert non-vide de l'espace de modules des hyperplans). Sur un corps fini, la finitude du nombre de ces points rationnel pose des problèmes.
En 2004, Bjorn Poonen a proposé une variante dans ce cas en remplaçant les hyperplans par les hypersurfaces de degré suffisamment grand. Dans cet exposé, je vais donner une version généralisée de résultat de Poonen où un plongement dans un espace projectif est remplacé par un plongement dans un schéma projectif muni d'un faisceau ample.
No. SM-201 24/02/2018 16:30 ~ 18:30 Salle 0A4, IMO(307), Université Paris-Sud
La variété de caractère
Cheng SHU (舒成), IMJ-PRG
Résumé : 201_Cheng SHU_Résumé
No. SM-200 24/02/2018 14:00 ~ 16:00 Salle 0A4, IMO(307), Université Paris-Sud
On mod p local-global compatibility for GLn(Qp) in the ordinary case
Zicheng QIAN (钱子诚), Université Paris-Sud
Abstract : Let p be a prime number, n > 2 an integer, and F a CM field in which p splits completely.Assume that a continuous automorphic Galois representation r : Gal(Q/F) → GLn(Fp) is upper- triangular and satisfies certain genericity conditions at a place w above p, and that every subquotient of r|Gal(Qp/Fw) of dimension > 2 is Fontaine–Laffaille generic. In this paper, we show that the isomorphism class of r|Gal(Qp/Fw) is determined by GLn(Fw)-action on a space of mod p algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to r, assuming a weight elimination result which is a theorem of Bao V. Le Hung in his forthcoming paper [LeH].
No. SM-199 27/01/2018 16:30 ~ 18:30 Salle 0A4, IMO(307), Université Paris-Sud
Approximation forte pour certaines variétés abéliennes crevées en points de torsion
Yongqi LIANG (梁永祺), IMJ-PRG
Résumé : 199_Yongqi LIANG_Résumé
No. SM-198 27/01/2018 14:00 ~ 16:00 Salle 0A4, IMO(307), Université Paris-Sud
Représentations cuspidales de GL_N(F), F corps local non archimédien
Romain DESEINE, Université Paris-Sud
Résumé : Depuis les travaux de Howe en 1977, l'étude des représentations irréductibles de GL(N) se fait par leur restriction à certain sous-groupes ouverts compacts, puis suivant cette idée, Bushnell et Kutzko ont pu donner une description complète du dual admissible complexe de GL(N). Un des avantages de cette approche est que tous les arguments sont algébriques et ne font pas intervenir d'arguments complexes, ce qui permet d'avoir pratiquement les mêmes résultats pour le cas modulaire (on n'en parlera pas dans cet exposé).
Dans cet exposé, nous nous limiterons à donner une description des représentations cuspidales de GL(N) et de montrer que ces dernières sont toutes des induites de sous-groupes ouverts compact modulo le centre (nous aurons même un peu plus ...). Je me suis grandement inspiré d'un exposé au séminaire Bourbaki de Guy Henniart, que je ne peux que recommander.
No. SM-197 13/01/2018 16:30 ~ 18:30 Salle 0A7, IMO(307), Université Paris-Sud
Fibration method and rational points
Yisheng TIAN (田乙胜), Université Paris-Sud
Abstract : In this talk, we are interested in the norm forms. If K is a finite extension of a number field k, then we can consider the norm equation N(x)=P(t), where x is an element in K and P(t) is a polynomial over k in one variable. For a smooth projective k-variety X birational to the affine variety defined by the norm equation above, we ask typical questions like whether X satisfies the Hasse principle and weak approximation.
We will see a successful answer due to recent results by Y. Harpaz, A.N. Skorobogatov and O. Wittenberg. More precisely, we will use recent results in additive combinatorics to establish a special case of a historical tool (Schinzel's hypothesis). Then we can use the so-called fibration method to study some typical questions as above. Finally, we will see some explicit examples.
No. SM-196 13/01/2018 14:00 ~ 16:00 Salle 0A7, IMO(307), Université Paris-Sud
On two geometric realizations of the periodic module for the affine Hecke algebra
Changjian SU (苏长剑), IHES
Abstract : It is well known that the affine Hecke algebra can be realized as the bi-Iwahori invariant functions on the p-adic group, or as equivariant K group of the Steinberg variety for the Langlands dual group. In this talk, I will compare the corresponding two geometric realizations of the periodic modules for the affine Hecke algebra. With this, we can have an equivariant K theory proof for the Macdonald formula for the zonal spherical function and Casselman--Shalika formula for the sphecial Whittaker function. Joint work with Changlong Zhong and Gufang Zhao.
No. SM-195 09/12/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Sur la construction du crochet de Deligne (II)
Juanyong WANG (王隽永), Ecole Polytechnique
Résumé : 193_Juanyong WANG_Resumé
No. SM-194 09/12/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
On the stable cohomology of the Satake compactification of A_g and its mixed Hodge structure
Jiaming CHEN (陈家明), Université Paris-Diderot
Abstract : Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification A_g of A_g stabilizes as g tends to infinity and they computed this stable cohomology as a Hopf algebra. In this talk, I will give a relatively simple algebro-geometric proof of their theorem and show that this stable cohomology comes with a mixed Hodge structure of which we determine the Hodge numbers. This is a joint work with Eduard Looijenga.
No. SM-193 25/11/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Sur la construction du crochet de Deligne
Juanyong WANG (王隽永), Ecole Polytechnique
Résumé : 193_Juanyong WANG_Resumé
No. SM-192 25/11/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
On factorization of automorphic periods
Jie LIN (林洁), IHES
Abstract : The question on the factorization of automorphic periods was initiated by Shimura where periods refer to the Petersson inner products of algebraic forms. Essentially, he predicted that periods related to Hilbert modular forms, or more generally to algebraic forms on a division algebra, factorize as products of periods indexed by the split archimedean places of the division algebra. The initial conjecture was first proved by M. Harris and completed by H. Yoshida. However, their methods seem very difficult to generalize to higher ranks. In this talk, we will explain a new and simple proof for general rank. We will also explain how to read this factorization from the point of view of motives, and why it is important in the study of special values of L-functions.
No. SM-191 11/11/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Théorème d’extension de Hartogs :
contre-exemples pour les fibrés holomorphes en droites ; résultats positifs pour les feuilletages holomorphes
Zhangchi CHEN (陈张弛), Université Paris-Sud
Résumé : 191_Zhangchi CHEN_Résumé
No. SM-190 11/11/2017 14:00 ~ 16:。0 Amphithéâtre Léon Motchane, IHES
Deligne pairings and Deligne-Riemann-Roch isomorphism
Mingchen XIA (夏铭辰), ENS
Abstract : Let $X$, $S$ be arithmetic varieties, namely projective flat scheme over $\Spec \mathbb{Z}$, whose fibre at infinity is regular. Let $f:X\rightarrow S$ be a projective smooth morphism between them. Let $\hat{E}$ be a Hermitian vector bundle on $X$. Let $\hat{\lambda}(\hat{E})$ be the determinant line bundle of $\hat{E}$ along $f$ equipped with Quillen metric.
The celebrated arithmetic Riemann-Roch theorem calculates the arithmetic Chern class of $\lambda(E)$ in terms of the arithmetic characteristic classes of $E$ and $T_{X/S}$.
From this piece of information, the isometric class of the line bundle $\hat{\lambda}(\hat{E})$ is determined up to torsion.
We are then interested in reconstructing $\hat{\lambda}(\hat{E})$ directly from the arithmetic characteristic classes.
The main tool is the so-called Deligne pairing proposed by Deligne. Roughly speaking, Deligne pairing is a functorial lift of the usual intersection theory. More precisely, given a good morphism $f:X\rightarrow S$ of pure relative dimension $n$ and a homogeneous Chern polynomial $P$ of degree $n+1$, the Deligne pairing of $P$ is a functoially defined line bundle on $S$, whose $c_1$ represents the fibre integration of $P$.
Equipped with Deligne pairings, we can then prove a functorial version of Riemann-Roch theorem: Let $E$ be a vector bundle on $X$, the Deligne pairing of the homogeneous part of degreee $n+1$ of the Riemann-Roch polynomial of $E$ is exactly the determinant line bundle of $E$ along $f$.
In this talk, I will explain the motivation and construction of Deligne pairings and the proof of Deligne-Riemann-Roch theorem.
If time permitted, I will also explain some unpublished results and conjectures concerning the higher degree generalization of the Deligne-Riemann-Roch isomorphism.
Reference : arXiv:1710.09731
No. SM-189 14/10/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Quiver varieties and constructions of representations of Lie algebras
Wille LIU, Université Paris-Diderot
Abstract : Quiver varieties are moduli spaces of representations of quivers, the most basic examples of which are Hilbert schemes of points on A^2. Cohomological correspondences between Hilbert schemes of points on A^2 can be organized into some representations of the Heisenberg algebra. From another point of view, the cohomology and the cohomological correspondences of the Hilbert schemes allow one to describe geometrically the Heisenberg algebra. This construction can be quantized and be generalized to any quiver varieties, which yields a geometric construction of certain Lie algebras and their quantizations. I will explain this geometric construction, which is due to Maulik and Okounkov.
No. SM-188 14/10/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
O-minimality and unlikely intersections for shimura varieties
Jinbo REN (任金波) , IHES
Abstract : Let S be a connected shimura variety and V\subset S be a subvariety. The André-Oort conjecture asserts that V contains a Zariski dense subset of CM points if and only if V is itself a shimura variety. Roughly speaking, this conjecture gives a description of the distribution of CM points in a shimura variety.
In the first part of my talk, I will outline a proof of the André-Oort conjecture for Siegel modular varieties by using O-mininality in model theory. In the second part, I will explain a generalisation of this idea in describing the distribution of higher dimensional shimura subvarieties.
The second part of my talk is joint work with Christopher Daw.