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H 1 Complex torus, its good compactifications and the ring of conditions

Auteurs : Khovanskii, Askold (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Let $X$ be an algebraic subvariety in $(\mathbb{C}^*)^n$. According to the good compactifification theorem there is a complete toric variety $M \supset (\mathbb{C}^*)^n$ such that the closure of $X$ in $M$ does not intersect orbits in $M$ of codimension bigger than dim$_\mathbb{C} X$. All proofs of this theorem I met in literature are rather involved.
The ring of conditions of $(\mathbb{C}^*)^n$ was introduced by De Concini and Procesi in 1980-th. It is a version of intersection theory for algebraic cycles in $(\mathbb{C}^*)^n$. Its construction is based on the good compactification theorem. Recently two nice geometric descriptions of this ring were found. Tropical geometry provides the first description. The second one can be formulated in terms of volume function on the cone of convex polyhedra with integral vertices in $\mathbb{R}^n$. These descriptions are unified by the theory of toric varieties.
I am going to discuss these descriptions of the ring of conditions and to present a new version of the good compactification theorem. This version is stronger that the usual one and its proof is elementary.

Codes MSC :
14M17 - Homogeneous spaces and generalizations
14M25 - Toric varieties, Newton polyhedra
14T05 - Tropical geometry

 Informations sur la Vidéo Langue : Anglais Date de publication : 22/09/2017 Date de captation : 21/09/2017 Collection : Research talks ; Algebraic and Complex Geometry Format : MP4 Durée : 01:04:57 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2017-09-21_Khovanskii.mp4 Informations sur la rencontre Nom de la rencontre : Perspectives in real geometry / Perspectives en géométrie réelleOrganisateurs de la rencontre : Brugallé, Erwan ; Itenberg, Ilia ; Shustin, EugeniiDates : 18/09/2017 - 22/09/2017 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1782.htmlCitation Data DOI : 10.24350/CIRM.V.19222103 Cite this video as: Khovanskii, Askold (2017). Complex torus, its good compactifications and the ring of conditions. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19222103 URI : http://dx.doi.org/10.24350/CIRM.V.19222103

### Voir aussi

Bibliographie

1. Kazarnovskii, B., & Khovanskii, A. (2017). Newton polyhedra, tropical geometry and the ring of condition for $(\mathbb{C}^*)^n$. - https://arxiv.org/abs/1705.04248

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