On the topology and analysis of a closed one form. I (Novikov's theory revisited)
We consider systems $(M,ω,g)$ with $M$ a closed smooth manifold, $ω$ a real valued closed one form and $g$ a Riemannian metric, so that $(ω,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical invariant $ρ(ω,g)\in[0,\infty]$ and improve Morse-Novikov theory by showing that the Novikov complex comes from a cochain complex of free modules over a subring $Λ'_{[ω],ρ}$ of the Novikov ring $Λ_{[ω]}$ which admits surjective ring homomorphisms $\ev_s:Λ'_{[ω],ρ}\to\C$ for any complex number $s$ whose real part is larger than $ρ$. We extend Witten-Helffer-Sjöstrand results from a pair $(h,g)$ where $h$ is a Morse function to a pair $(ω,g)$ where $ω$ is a Morse one form. As a consequence we show that if $ρ<\infty$ the Novikov complex can be entirely recovered from the spectral geometry of $(M,ω,g)$.