Cohomological obstruction to a Lagrangian section ($[\sigma ]\ne 0⟹\text{no global Lagrangian section}$)

If $[\sigma ]\ne 0$ in de Rham cohomology, a twisted cotangent symplectic form has no Lagrangian submanifold whose projection is a diffeomorphism onto the entire base. Such a submanifold would be a global one-form graph, and the twisted Lagrangian graph criterion would make $\sigma$ exact.