There are several canonical symplectic geometric constructions which can be performed on smooth manifolds. For instance, the cotangent bundle of a smooth manifold has a canonical symplectic structure, and one can ask whether the symplectomorphism type of the cotangent bundle remembers the smooth topology of the manifold. In the opposite direction any affine 2n-dimensional Weinstein manifold (which is the symplectic counterpart of an affine complex manifold) can be viewed as a cotangent bundle of a possibly singular n-dimensional complex, and one can ask whether symplectic invariants can be described in terms of smooth topology of this complex. I will discuss in the talk the interplay between these two directions.