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