Morse-Novikov cohomology on complex manifolds

TL;DR

This paper studies Dolbeault-Morse-Novikov cohomology on complex manifolds, establishing invariants, formulas, and relations to other cohomologies, and examines their stability under complex structure deformations.

Contribution

It introduces sheaf-theoretic interpretations, proves bimeromorphic invariants, and derives formulas like Leray-Hirsch and blow-up for Dolbeault-Morse-Novikov cohomology.

Findings

Established bimeromorphic invariants for Dolbeault-Morse-Novikov cohomology.

Proved Leray-Hirsch theorem and blow-up formula in this context.

Analyzed stability of cohomology dimensions under complex structure deformations.

Abstract

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_\eta(X) as the cohomology of the sheaf \Omega_{X,\eta}^p of \eta-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we consider the relations between Morse-Novikov cohomology and Dolbeault-Morse-Novikov cohomology, moreover, investigate stabilities of their dimensions under the deformations of complex structures. In some aspects, Morse-Novikov and Dolbeault-Morse-Novikov cohomology behave similarly with de Rham and Dolbeault cohomology.