Symplectic $ {\mathbb Z}_2^n $-manifolds

Roughly speaking, $\mathbb{Z}_2^n$-manifolds are `manifolds' equipped with $\mathbb{Z}_2^n$-graded commutative coordinates the sign rule being determined by scalar product of their $\mathbb{Z}_2^n$-degrees. We examine notion a symplectic $\mathbb{Z}_2^n$-manifold, i.e., $\mathbb{Z}_2^n$-manifold two-form that may carry non-zero $\mathbb{Z}_2^n$-degree. show basic notions and results geometry generalise to `higher graded' setting, including generalisation Darboux's theorem.