On exact locally conformally Kähler manifolds

In this article, we explore a distinguished class of complex manifolds known as $\( d_\theta \)$-exact locally conformally Kähler (LCK) manifolds. These manifolds are characterized by the property that their fundamental 2-form $\( \omega \)$ can be expressed as $\( \omega = d_\theta \alpha \)$, where $\( \alpha \)$ is a 1-form on $\( M \)$ and $\( d_\theta = d + \theta \wedge \)$. We establish a key result: if the 1-form $\( \alpha \)$ is holomorphic, then the Morse-Novikov cohomology $\( H_\theta^*(M) \)$ vanishes. Furthermore, we provide sufficient conditions under which a $\( d_\theta \)$-exact LCK manifold admits a Vaisman structure. This work deepens the understanding of the interplay between geometric structures, cohomological properties, and special classes of LCK manifolds.