Morse–Novikov cohomology for blow-ups of complex manifolds

Lingxu Meng

Vol. 320 (2022), No. 2, 365–390

DOI: 10.2140/pjm.2022.320.365

Abstract

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

Keywords

Morse–Novikov cohomology, blow-up, weight $\theta$-sheaf, self-intersection formula, Leray–Hirsch theorem

Mathematical Subject Classification

Primary: 53C56

Secondary: 55N35, 32Q55

Milestones

Received: 22 September 2021

Revised: 14 June 2022

Accepted: 13 September 2022

Published: 15 February 2023