Morse–Novikov cohomology for blow-ups of complex manifolds

The weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ 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 $\theta$ calculate projective bundles. Based on these results, two blow-up formulae (\emph{not necessarily compact}) complex manifolds, where self-intersection play a key role in establishing explicit expressions for them.