Consider a Kaehler degeneration $\mathcal X\to \Delta$ of smooth manifolds: Here $\Delta$ is the unit disc, $\pi$ a proper flat map, smooth over $\Delta^∗=\Delta−\{0\}$. The general fibres are $X_t=\pi^{−1}(t),\; $ and the central fibre is $ \; X_0=\pi^{−1}(0) $. Assume that all of the fibres admit Kaehler-Einstein metrics: $Ric(\omega_s)=\lambda\omega_s$.
Is there a method to compute $$h^1\left(Π(X_0)\right) $$
where $Π(X_0)$ is the dual graph of the central fibre $X_0$?