The geometry of graded cotangent bundles

Given a vector bundle $A\to M$ we study the geometry of graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these objects to classical such as higher Courant algebroids on $A\oplus\bigwedge^{k-1}A^*$ Dirac structures therein, semi-direct products Lie algebroid $A$ with coadjoint representations up homotopy, branes certain AKSZ $\sigma$-models.