On the topology and analysis of a closed one form. I (Novikov's theory revisited)

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