Complex geometry seminar

A fundamental result in Complex Geometry is the Kobayashi-Hitchin correspondence, stating that a holomorphic vector bundle on a Kähler manifold is poly-stable (as defined by Mumford, Takemoto) if and only if it admits a Hermitian metric solving the Hermite-Einstein equation. It has now become clear that there exist many possible different stability notions for vector bundles, that are of great interest in Algebraic Geometry and String Theory. It is natural to wonder if these stabilities are also tied to the existence of Hermitian metrics with special curvature properties. In this talk, based on joint work with Julien Keller (arXiv:2405.03312[math.DG]), we will consider a class of “polynomial” equations for the curvature of rank 2 holomorphic vector bundles on compact projective surfaces, and a corresponding class of polynomial stability conditions for these bundles. We will then explain how each of these stability conditions is related to the existence of a Hermitian metric satisfying the corresponding equation. This refines and partially confirms a conjectural correspondence between Bridgeland stability conditions and PDEs on holomorphic vector bundles, formulated by Dervan, McCarthy, and Sektnan.

Variétés de Fano avec un lieu de base anticanonique

We discuss some recent results concerning complex manifolds whose
universal coverings admit many bounded holomorphic functions.

Let $X$ be a quasi-projective manifold whose universal covering $M$ is
a strongly Carathéodory hyperbolic manifold. We will see that any
(quasi-)projective subvariety of $X$ is of (log-)general type. The
result is consistent with the prediction of a conjecture of Lang. We
will also see that $M$ has many interesting geometric and analytic
properties. Examples of $X$ include finite volume quotients of bounded
symmetric domains, moduli space of hyperbolic Riemann surfaces, etc.

Next we consider holomorphic maps $f:S=\Omega/\Gamma \rightarrow N$
from a finite volume quotient of bounded symmetric domain $Omega$ of
rank $\geq 2$ to a complex manifold $N$, where the universal covering
$\widetilde{N}$ of $N$ has sufficiently many bounded holomorphic
functions. We will see that the inverse $F^{-1}$ of the lifting
$F:\Omega\rightarrow \widetilde{N}$ of $f$ extends to a bounded
holomorphic map $R:\widetilde{N}\rightarrow \mathbb{C}^n$. This gives
another proof that $F$ must be a holomorphic embedding and lead to
certain rigidity result when $N$ satisfies some natural additional
geometric properties.

Voisin’s work, which constructs a series of K-trivial varieties from cubic hypersurfaces, and self-rational maps on them, called the Voisin maps, will be the focus here. Notable among these is the Fano variety of lines of a cubic fourfold, a dimension 4 hyper-Kähler manifold. The Voisin map in this case has been extensively studied. We’ll examine higher-dimensional examples, which are all strict Calabi-Yau manifolds. This session aims to study the geometry of these manifolds and apply their structural insights to the conjectures on algebraic cycles such as the generalised Bloch conjecture. The results presented here are written in a recent preprint paper: arxiv 2404.10138.

Title: Random walks on affine buildings of type \tilde{A}_2.

Summary:

Let $G$ be a group acting on a building $X$ of type $\tilde{A}_2$ and let ${Z_n}$ be a random walk on the group G, generated by an admissible measure $\mu$. The purpose of the talk is to investigate some properties of the measured dynamical system ${Z_n o}$, for $o$ a point of the building $X$. Using tools from boundary theory and the geometry of such buildings, we can prove that there exists a unique $\mu$-stationary measure supported on the chambers of the spherical building at infinity. If time allows it, we will discuss some applications about the asymptotic properties of the random walk ${Z_n o}$. I will try to introduce most notions: (affine) buildings and their boundaries, random walks and stationary measures, the Poisson-Furstenberg boundary and some of its ergodic properties.

Support cohomologique des modules basculants pour les groupes algébriques réductifs

Il est connu depuis les années 1970 que de nombreuses informations concernant la théorie des représentations des groupes algébriques réductifs sur des corps de caractéristique positive peuvent s’exprimer en terme de la combinatoire du groupe de Weyl affine associé. Une forme subtile de cette relation a été conjecturée par Humphreys dans les années 1990, qui exprime le support cohomologique des représentations basculantes indécomposables en termes d’orbites nilpotentes associées aux cellules de Kazhdan-Lusztig bilatères (via une bijection de Lusztig). Dans cet exposé je présenterai des résultats obtenus en direction de cette conjecture, en collaboration avec Pramod Achar et William Hardesty.

Title: The E3-structure on the spherical Hecke category
of a reductive group

Abstract: let G be a reductive group over the complex numbers, e.g. GLn,C . The
notion of affine Grassmannian associated to G leads to the introduction of
a monoidal dg/∞-category Sph(G), called the spherical category of G, which
plays an important role in the Geometric Langlands program. For example,
its behaviour provides important constraints in the formulation of the Geometric Langlands Conjecture.

This monoidal ∞-category is not symmetric
monoidal (although its homotopy category is), but it admits a t-structure whose
heart is symmetric monoidal: more precisely, by the Geometric Satake Theorem
(Ginzburg, Mirkovic–Vilonen) the heart is monoidal-equivalent to the category
of representations of the Langlands dual of G with its (symmetric monoidal)
tensor product.
In this talk, I will present how to upgrade the existing E1 -monoidal struc-
ture on Sph(G) to an E3 -monoidal one, which formally recovers the symmetric
monoidal structure of the heart. The construction implements ideas of Jacob
Lurie and uses a topologically-flavoured presentation of Sph(G), namely as an
∞-category of equivariant constructible sheaves over a stratified space.

Titre : Autour d'un théorème de Deligne-Lusztig

Résumé : Soit $G$ un groupe réductif sur un corps fini $\mathbb{F}_q$.
 La théorie des représentations du groupe $G(\mathbb{F}_q)$ est souvent
 comprise en étudiant la cohomologie des variétés de Deligne-Lusztig. 
Le théorème fondamental qui motive la construction initiale de Deligne et 
Lusztig est que toutes les représentations 
irréductibles de $G(\mathbb{F}_q)$ apparaissent dans la cohomologie
 de ces variétés. 
Dans cet exposé, je presenterai une simplification de la
 preuve de ce théorème et je placerai cette construction 
au sein d'un programme dont le but est de reconstruire la 
théorie par l'introduction systématiques de méthodes 
géométriques et catégoriques. 

Titre et résumés à venir.