Morse-Novikov cohomology for blow-ups of complex manifolds

TL;DR

This paper studies Morse-Novikov cohomology on complex manifolds, providing new blow-up formulas, invariance results, and explicit calculations for projective bundles, using sheaf-theoretic methods and classical theorems.

Contribution

It introduces blow-up formulas for Morse-Novikov cohomology on complex manifolds, extending previous results and employing sheaf-theoretic techniques.

Findings

Blow-up formulas for Morse-Novikov cohomology derived

The $ heta$-Lefschetz number is shown to be independent of $ heta$

Explicit calculations of cohomology for projective bundles

Abstract

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.