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. [ABSTRACT FROM AUTHOR]