Algebraicity of conformal factors on manifolds with finite-dimensional (co)homology

Determine whether, for manifolds with finite-dimensional (co)homology groups, the set of conformally symplectic factors R = {η > 0 : there exists a diffeomorphism f: M → M with f*ω = η ω for a non-exact symplectic form ω} consists only of algebraic numbers. In particular, ascertain broad conditions on M under which every admissible factor η must be algebraic, generalizing the explicit computations on M = R^d × T^d where f^{#2} acts on H^2(M) and forces η to be an algebraic eigenvalue.

Background

In Section 4.2 the authors paper how the topology of the manifold constrains the set of possible conformal factors for conformally symplectic diffeomorphisms. For a non-exact symplectic form ω, the pullback relation f*ω = η ω implies that η must be an eigenvalue of the induced map f^{#2} on H^2(M), providing topological obstructions. In explicit computations on M = R^d × T^d, they show that η must be an algebraic number (indeed a product of eigenvalues of the induced action on H^1), answering a question posed in prior work.

Motivated by these examples, the authors raise a broader conjecture about the algebraicity of the set R of possible conformal factors on "many" manifolds with finite-dimensional (co)homology theories. Establishing this would link the existence of conformally symplectic dynamics with global algebraic-topological properties of M.