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Documents  14J45 | enregistrements trouvés : 8

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Post-edited  Algebraicity of the metric tangent cones
Wang, Xiaowei (Auteur de la Conférence) | CIRM (Editeur )

We proved that any K-semistable log Fano cone admits a special degeneration to a uniquely determined K-polystable log Fano cone. This confirms a conjecture of Donaldson-Sun stating that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. This is a joint work with Chi Li and Chenyang Xu.

14J45 ; 32Q15 ; 32Q20 ; 53C55

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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 ...

05-XX ; 41-XX ; 62-XX ; 14J45

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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.

14J45 ; 14E30 ; 14G17

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A Gushel-Mukai variety is a Fano variety of coindex 3, Picard number 1, and degree 10. I will discuss classification of these Fano varieties, their moduli spaces, and their relation to EPW sextics. This is a joint work with Olivier Debarre.

14H10 ; 14J45 ; 14E08

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Wang et Zhu ont caractérisé l'existence de métriques de Kähler-Einstein sur les variétés toriques Fano en termes du barycentre du polytope associé. L'objectif de cet exposé est de présenter un résultat similaire pour les compactifications $G \times G$-équivariantes Fano d'un groupe réductif $G$. Je présenterai le polytope moment associé à une telle variété et comment le barycentre de ce polytope par rapport à la mesure de Duistermaat-Heckman est lié à l'existence de métriques de Kähler-Einstein. La condition nécessaire et suffisante d'existence de métriques de Kähler-Einstein ainsi obtenue est vérifiable en pratique et donne de nouveaux exemples de variétés de Kähler-Einstein Fano (par exemple la compactification magnifique du groupe semisimple adjoint PSL$(3, \mathbb{C})$).
Wang et Zhu ont caractérisé l'existence de métriques de Kähler-Einstein sur les variétés toriques Fano en termes du barycentre du polytope associé. L'objectif de cet exposé est de présenter un résultat similaire pour les compactifications $G \times G$-équivariantes Fano d'un groupe réductif $G$. Je présenterai le polytope moment associé à une telle variété et comment le barycentre de ce polytope par rapport à la mesure de Duistermaat-Heckman est ...

32Q20 ; 14J45 ; 53C55 ; 32Q10 ; 14M27

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Multi angle  Orbital degeneracy loci
Benedetti, Vladimiro (Auteur de la Conférence) | CIRM (Editeur )

I will present a joint work with Sara Angela Filippini, Laurent Manivel and Fabio Tanturri (arXiv: 1704.01436). We introduce a new class of varieties, called orbital degeneracy loci. The idea is to use any orbit closure in a representation of an algebraic group to generalise the classical construction of degeneracy loci of morphisms between vector bundles, and of zero loci as well. After giving the definition of an orbital degeneracy locus, I will explain how to control the canonical bundle of these varieties: under some Gorenstein condition on the orbit closure, it is possible to construct examples of varieties with trivial canonical bundle or of Fano type. Finally, if time will permit, I will give some explicit examples of such degeneracy loci, which allow to construct many Calabi-Yau varieties of dimension three and four, and some new Fano fourfolds.
I will present a joint work with Sara Angela Filippini, Laurent Manivel and Fabio Tanturri (arXiv: 1704.01436). We introduce a new class of varieties, called orbital degeneracy loci. The idea is to use any orbit closure in a representation of an algebraic group to generalise the classical construction of degeneracy loci of morphisms between vector bundles, and of zero loci as well. After giving the definition of an orbital degeneracy locus, I ...

14M12 ; 14C05 ; 14M15 ; 14J60 ; 14J32 ; 14J45

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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 ...

14J45 ; 14J35 ; 14E30

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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 ...

14J35 ; 14J40 ; 14J45 ; 14M15 ; 14D07 ; 32G20

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