Askold Khovanskii: Complex torus, its good compactifications and the ring of conditions
Abstract: Let X be an algebraic subvariety in (ℂ∗)n. According to the good compactifification theorem there is a complete toric variety M⊃(ℂ∗)n such that the closure of X in M does not intersect orbits in M of codimension bigger than dimℂX. All proofs of this theorem I met in literature are rather involved.
The ring of conditions of (ℂ∗)n was introduced by De Concini and Procesi in 1980-th. It is a version of intersection theory for algebraic cycles in (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 ℝn. 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.
Recording during the thematic meeting "Perspectives in real geometry" the September 21, 2017 at the Centre International de Rencontres Mathématiques (Marseille, France)
Filmmaker: Guillaume Hennenfent
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: And discover all its functionalities:
- Chapter markers and keywords to watch the parts of your choice in the video
- Videos enriched with abstracts, bibliographies, Mathematics Subject Classification
- Multi-criteria search by author, title, tags, mathematical area