Deformations of Pre-symplectic Structures and the Koszul L∞-algebra

We study the deformation theory of pre-symplectic structures, that is, closed 2-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 the 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 that 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.