  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 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.
