Morse-Novikov cohomology of locally conformally K\"ahler surfaces

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally K\"ahler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally K\"ahler (and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We prove the nonexistence of LCK metrics with potential and more generally, of $d_\theta$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.