On a local systolic inequality for odd-symplectic forms

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The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let \Omega be an odd-symplectic form on an oriented closed manifold \Sigma of odd dimension. We say that \Omega is Zoll if the trajectories of the flow given by \Omega are the orbits of a free S^1 -action. After defining the volume of \Omega and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided \Omega 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 S^1 -action yields a flat S^1 -bundle or when \Omega 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 C^1 -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].