Products of locally conformal symplectic manifolds

Abstract

Given two locally conformal symplectic (LCS) structures on manifolds $M_1$ and $M_2$, we construct a natural $\R^+$-torsor of locally conformal symplectic structures on a certain covering space $M_1 \boxplus M_2$ of $M_1 \times M_2$. As the smooth construction of $M_1 \boxplus M_2$ is natural from the perspective of flat line bundles, we use this language to phrase the LCS theory. This construction shares many properties with, and in a sense generalizes, the standard symplectic product. Notably, for a Hamiltonian isotopy $ϕ_t$ of an LCS manifold $M$, there is an associated Lagrangian embedding $Γ(ϕ_1) \colon M \hookrightarrow M \boxplus M$, in which certain fixed points of $ϕ_1$ are in bijection with intersection points of $Γ(ϕ_1)$ with the diagonal $Δ= Γ(\mathrm{id})$. Using a Lagrangian intersection of result of the first author and E. Murphy, we may conclude that if $ϕ_t$ is a $C^0$-small Hamiltonian isotopy, then the number of fixed points of $ϕ_1$ is bounded below by the rank of the Novikov theory associated to the Lee class of the LCS structure on $M$. Finally, we end the paper by constructing the suspension of a Lagrangian submanifold along a Hamiltonian isotopy in the LCS theory, again generalizing the symplectic setting.