Summary: 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 \(K\)).