Normal forms of $\mathbb Z$-graded $Q$-manifolds

Abstract

Following recent results of A.K. and V.S. on $\mathbb Z$-graded manifolds, we give several local and global normal forms results for $Q$-structures on those, i.e. for differential graded manifolds. In particular, we explain in which sense their relevant structures are concentrated along the zero-locus of their curvatures, especially when the negative part is of Koszul--Tate type. We also give a local splitting theorem.