Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Numerical semistability of projective toric varieties

Numerical semistability is a notion of Geometric Invariant Theory (GIT) stability defined in terms of the inclusion of weight polytopes, such as the Chow and Hurwitz polytopes. A theorem of Paul states that the K-energy of a smooth linearly normal projective variety X ⊂ P^N, restricted to the space of Bergman metrics, is bounded below if and only if X is “semistable”. In this talk, we present a necessary and sufficient criterion for a smooth toric variety X_P to be numerically semistable. Our approach builds on the theory of A-resultants, A-discriminants and Hurwitz forms developed by Gelfand-Kapranov-Zelevinsky, and Sturmfels. As an application, we study smooth polarized toric varieties (X_P,L_P) and clarify the relationship between asymptotic numerical semistability and K-semistability.