On the $C^2$-local systolic optimality of Zoll odd-symplectic forms

An odd-symplectic form is a closed and maximally non-degenerate $2$-form on a compact odd-dimensional manifold. It describes the dynamics of an autonomous Hamiltonian system on a regular energy level. It is called Zoll if the induced dynamics is a free circle action, up to a global time reparameterization. This article establishes a normal form theorem for odd-symplectic forms close to a Zoll one and cohomologous to it, which is then used to prove that Zoll odd-symplectic forms are the local maximizers of the associated systolic ratio. This generalizes the known systolic optimality of Zoll contact forms in the $C^3$-topology. As an application, local systolic inequalities are established in symplectic manifolds for hypersurfaces close to Zoll ones. In particular, this applies to certain non-exact twisted cotangent bundles of manifolds of dimension greater than two.