On a local systolic inequality for odd-symplectic forms

The aim of this paper is to formulate a local systolic inequality for oddsymplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let Ω be an odd-symplectic form on an oriented closed manifold ∑ of odd dimension. We say that Ω is Zoll if the trajectories of the flow given by Ω are the orbits of a free S1-action. After defining the volume of Ω and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided Ω is Zoll. This builds the equality case of a conjectural systolic inequality for odd-symplectic forms close to a Zoll one. We prove the conjecture when the S1-action yields a flat S1-bundle or when Ω is quasi-autonomous. Together with previous work [BK19a], this establishes the conjecture in dimension three. This new inequality recovers the local contact systolic inequality (recently proved in [AB19]) as well as the inequality between the minimal action and the Calabi invariant for Hamiltonian isotopies C1-close to the identity on a closed symplectic manifold. Applications to the study of periodic magnetic geodesics on closed orientable surfaces is given in the companion paper [BK19b].