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# Documents  Benoist, Olivier | enregistrements trouvés : 12

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## ​On the Lüroth problem for real varieties Benoist, Olivier | CIRM H

Post-edited

Research talks

The Lüroth problem asks whether every unirational variety is rational. Over the complex numbers, it has a positive answer for curves and surfaces, but fails in higher dimensions. In this talk, I will consider the Lüroth problem for real algebraic varieties that are geometrically rational, and explain a counterexample not accounted for by the topology of the real locus or by unramified cohomology. This is joint work with Olivier Wittenberg.

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Post-edited

Research talks

Let $X$ be a compact Kähler manifold. The so-called Kodaira problem asks whether $X$ has arbitrarily small deformations to some projective varieties. While Kodaira proved that such deformations always exist for surfaces. Starting from dimension 4, there are examples constructed by Voisin which answer the Kodaira problem in the negative. In this talk, we will focus on threefolds, as well as compact Kähler manifolds of algebraic dimension $a(X) = dim(X) -1$. We will explain our positive solution to the Kodaira problem for these manifolds. Let $X$ be a compact Kähler manifold. The so-called Kodaira problem asks whether $X$ has arbitrarily small deformations to some projective varieties. While Kodaira proved that such deformations always exist for surfaces. Starting from dimension 4, there are examples constructed by Voisin which answer the Kodaira problem in the negative. In this talk, we will focus on threefolds, as well as compact Kähler manifolds of algebraic dimension $a(X) = ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Rationality of Fano 3-folds over non-closed fields Kuznetsov, Alexander | CIRM H Post-edited Research talks In the talk I will discuss rationality criteria for Fano 3-folds of geometric Picard number 1 over a non-closed field$k$of characteristic 0. Among these there are 8 types of geometrically rational varieties. We prove that in one of these cases any variety of this type is k-rational, in four cases the criterion of rationality is the existence of a$k$-rational point, and in the last three cases the criterion is the existence of a$k$-rational point and a k rational curve of genus 0 and degree 1, 2, and 3 respectively. The last result is based on recent results of Benoist-Wittenberg. This is a joint work with Yuri Prokhorov. In the talk I will discuss rationality criteria for Fano 3-folds of geometric Picard number 1 over a non-closed field$k$of characteristic 0. Among these there are 8 types of geometrically rational varieties. We prove that in one of these cases any variety of this type is k-rational, in four cases the criterion of rationality is the existence of a$k$-rational point, and in the last three cases the criterion is the existence of a$k$-rational ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Random section of line bundles over real Riemann surfaces Ancona, Michele | CIRM H Multi angle Research talks Given a line bundle$L$over a real Riemann surface, we study the number of real zeros of a random section of$L$. We prove a rarefaction result for sections whose number of real zeros deviates from the expected one. Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Algebraic models of the line in the real affine plane Mangolte, Frédéric | CIRM H Multi angle Research talks We study the following real version of the famous Abhyankar-Moh Theorem: Which real rational map from the affine line to the affine plane, whose real part is a non-singular real closed embedding of$\mathbb{R}$into$\mathbb{R}^2$, is equivalent, up to a birational diffeomorphism of the plane, to the linear one? We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are plenty of non-equivalent smooth rational closed embeddings up to birational diffeomorphisms. Some of these are simply detected by the non-negativity of the real Kodaira dimension of the complement of their images. But we also introduce finer invariants derived from topological properties of suitable fake real planes associated to certain classes of such embeddings. (Joint Work with Adrien Dubouloz). We study the following real version of the famous Abhyankar-Moh Theorem: Which real rational map from the affine line to the affine plane, whose real part is a non-singular real closed embedding of$\mathbb{R}$into$\mathbb{R}^2$, is equivalent, up to a birational diffeomorphism of the plane, to the linear one? We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Sums of three squares and Noether-Lefschetz loci Benoist, Olivier | CIRM H Multi angle Research talks;Algebraic and Complex Geometry;Number Theory It is a theorem of Hilbert that a real polynomial in two variables that is nonnegative is a sum of 4 squares of rational functions. Cassels, Ellison and Pfister have shown the existence of such polynomials that are not sums of 3 squares of rational functions. In this talk, we will prove that those polynomials that may be written as sums of 3 squares are dense in the set of nonnegative polynomials. The proof is Hodge-theoretic. Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Special rational fibrations in Fano 4-folds Casagrande, Cinzia | CIRM H Multi angle Research talks Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number$b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a contraction). The main result that we want to present is the following: let$X$be a Fano 4-fold having a nonconstant rational contraction$X --> Y$of fiber type. Then either$b_{2}(X)$is at most 18, with equality only for a product of surfaces, or$Y$is$\mathbb{P}^{1}$or$\mathbb{P}^{2}$. The proof is achieved by reducing to the case of "special" rational contractions of fiber type. We will explain this notion and give an idea of the techniques that are used. Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number$b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Gushel-Mukai varieties and their periods Debarre, Olivier | CIRM H Multi angle Research talks Gushel-Mukai varieties are defined as the intersection of the Grassmannian Gr(2, 5) in its Plücker embedding, with a quadric and a linear space. They occur in dimension 6 (with a slighty modified construction), 5, 4, 3, 2 (where they are just K3 surfaces of degree 10), and 1 (where they are just genus 6 curves). Their theory parallels that of another important class of Fano varieties, cubic fourfolds, with many common features such as the presence of a canonically attached hyperkähler fourfold: the variety of lines for a cubic is replaced here with a double EPW sextic. There is a big difference though: in dimension at least 3, GM varieties attached to a given EPW sextic form a family of positive dimension. However, we prove that the Hodge structure of any of these GM varieties can be reconstructed from that of the EPW sextic or of an associated surface of general type, depending on the parity of the dimension (for cubic fourfolds, the corresponding statement was proved in 1985 by Beauville and Donagi). This is joint work with Alexander Kuznetsov. Gushel-Mukai varieties are defined as the intersection of the Grassmannian Gr(2, 5) in its Plücker embedding, with a quadric and a linear space. They occur in dimension 6 (with a slighty modified construction), 5, 4, 3, 2 (where they are just K3 surfaces of degree 10), and 1 (where they are just genus 6 curves). Their theory parallels that of another important class of Fano varieties, cubic fourfolds, with many common features such as the ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Algebraic and arithmetic aspects of twistor spaces Huybrechts, Daniel | CIRM H Multi angle Research talks I will recall the well-known notion of twistor spaces for K3 surfaces (and Hyperkähler manifolds) and discuss some natural questions relating to the algebraic and arithmetic geometry of their fibres. Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Cayley-Bacharach theorems with excess vanishing Lazarsfeld, Robert | CIRM H Multi angle Research talks A classical result usually attributed to Cayley and Bacharach asserts that if two plane curves of degrees c and d meet in cd points, then any curve of degree (c + d - 3) passing through all but one of these points must also pass through the remaining one. In the late 1970s, Griffiths and Harris showed that this is a special case of a general result about zero-loci of sections of a vector bundle. Inspired by a recent paper of Mu-Lin Li, I will describe a generalization allowing for excess vanishing. Multiplier ideals enter the picture in a natural way. Time permitting, I will also explain how a result due to Tan and Viehweg leads to statements of Cayley-Bacharach type for determinantal loci. This is joint work with Lawrence Ein. A classical result usually attributed to Cayley and Bacharach asserts that if two plane curves of degrees c and d meet in cd points, then any curve of degree (c + d - 3) passing through all but one of these points must also pass through the remaining one. In the late 1970s, Griffiths and Harris showed that this is a special case of a general result about zero-loci of sections of a vector bundle. Inspired by a recent paper of Mu-Lin Li, I will ... 14F05 Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Examples of Kähler groups Eyssidieux, Philippe | CIRM H Multi angle Research talks Malgré les succès de la théorie de Hodge non abélienne de Corlette-Simpson pour exclure que de nombreux groupes de présentation finie soient groupes fondamentaux de variétés projectives lisses (ou des groupes Kähleriens), les techniques de construction manquent. La construction de Campana du groupe fondamental orbifold d'une paire orbifolde permet de considérer le groupe fondamental des compactifications orbifolds d'une variété (ou champ) quasiprojective lisse donnée$U$qui, si quelques précautions sont prises et sous des hypothèses raisonnables - mais pas toujours faciles a vérifier, est un groupe Kählerien. En choisissant bien la variété$U$, les groupes obtenus sont potentiellement intéressants et on utilise souvent des techniques inattendues pour établir les propriétés de leurs représentations linéaires. L'exposé fera un survey de cas particulièrement intrigants ou, par exemple,$U\$ est un complément d'arrangement de droites, une variété localement complexe hyperbolique non compacte ou un espace de modules de courbes pointées. Malgré les succès de la théorie de Hodge non abélienne de Corlette-Simpson pour exclure que de nombreux groupes de présentation finie soient groupes fondamentaux de variétés projectives lisses (ou des groupes Kähleriens), les techniques de construction manquent. La construction de Campana du groupe fondamental orbifold d'une paire orbifolde permet de considérer le groupe fondamental des compactifications orbifolds d'une variété (ou champ) ...

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## Fano fibrations in positive characteristic Fanelli, Andrea | CIRM H

Multi angle

Research talks

In this talk, starting from the perspective of characteristic zero, I will discuss pathologies for the generic fibre of Fano fibrations in characteristic p.
The new approach of the joint project with Stefan Schröer has two goals:
- controlling these pathological phenomena; and
- describing new examples.
I'm going to focus on dimension 3, motivated by the recent progress in Mori theory in positive characteristic.

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