Conformal symplectic geometry of cotangent bundles

TL;DR

This paper extends the Arnol'd conjecture to conformal symplectic manifolds, establishing lower bounds on intersection points of Lagrangian submanifolds based on Morse-Novikov homology, and provides an accessible overview of conformal symplectic geometry.

Contribution

It proves a version of the Arnol'd conjecture for conformal symplectic manifolds and relates intersection counts to Morse-Novikov homology, introducing new insights into this geometric setting.

Findings

Non-zero Morse-Novikov homology prevents disjoining Lagrangians by small Hamiltonian isotopies.

Number of intersection points is at least the sum of Betti numbers of Morse-Novikov homology.

Provides a concise exposition of conformal symplectic geometry for familiar symplectic or contact geometers.

Abstract

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $\beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $\beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.