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.