Morse-Novikov cohomology for blow-ups of complex manifolds

The weight $\theta$-sheaf ${\underset{‾}{\mathbb{R}}}_{X,\theta }$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of K"{u}nneth and Leray-Hirsch types. As applications, we prove that the $\theta$-Lefschetz number is independent of $\theta$ and calculate the Morse-Novikov cohomologies of projective bundles. Based on these results, we give two blow-up formulae on (\emph{not necessarily compact}) complex manifolds, where the self-intersection formulae play a key role in establishing the explicit expressions for them.