Séminaire de Géométrie Tropicale
Mercredi 27 mai 16h15 salle 1525-502
Mounir Nisse (KIAS)
On higher convexity for complement of tropical varieties and higher solidness of amoebas
Résumé :
Abstract: We consider Henriques' homological higher convexity for complements of tropical varieties, establishing it for complements of abstract tropical hypersurfaces and abstract tropical curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseux series.
We conjecture that the complement of a tropical variety has this higher convexity, and prove a weak form of this conjecture for the nonarchimedean amoeba of a variety over the complex Puiseux field. One of our main tools is Jonsson's limit theorem for tropical varieties. Moreover, we show that if a variety of codimension k+1 is a complete intersection or if it has dimension 1, then the natural map on reduced homology of dimension k from the complement of its logarithmic limit set to the complement of its amoeba is injective. When this map is an isomorphism, we say that the amoeba is k-solid, and we prove that amoebas of lines are solid in this sense. The first result is an analog of the order map for unbounded components of the complement of a hypersurface amoeba, and the second is analog of the solidity of hyperplane amoebas.
This is joint work with Frank Sottile.