  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.
