Dualité pour les groupes de type multiplicatif sur certains corps de fonctions

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. In this note, we state and sketch the proof of an arithmetic duality theorem for Tate-Shafarevich groups of groups of multiplicative type over K (and more generally of some two-term complexes of tori over K).