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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 )

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    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 rencontre

    Nom du congrès : Perspectives in real geometry / Perspectives en géométrie réelle
    Organisteurs Congrès : Brugallé, Erwan ; Itenberg, Ilia ; Shustin, Eugenii
    Dates : 18/09/2017 - 22/09/2017
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1782.html

    Citation 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


    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