Principe local-global pour les corps de fonctions sur des corps locaux supÉrieurs II

— Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. We apply some arithmetic duality theorems to the weak approximation for tori over K and to the study of the obstruction to the local-global principle for K-torsors under a connected linear algebraic group.