  We study the deformation theory of pre-symplectic structures, i.e. closed
two-forms of fixed rank. The main result is a parametrization of nearby
deformations of a given pre-symplectic structure in terms of an
$L_\infty$-algebra, which we call Koszul $L_\infty$-algebra. This
$L_\infty$-algebra is a cousin of the Koszul dg Lie algebra associated to a
Poisson manifold. In addition, we show that a quotient of the Koszul
$L_{\infty}$-algebra is isomorphic to the $L_\infty$-algebra which controls the
deformations of the underlying characteristic foliation. Finally, we show that
the infinitesimal deformations of pre-symplectic structures and of foliations
are both obstructed.
