  Let $f: X \to S$ be a unipotent degeneration of projective complex manifolds
over a disc such that the reduction of the central fibre $Y=f^{-1}(0)$ is
simple normal crossings, and let $X_\infty$ be the canonical nearby fibre.
Building on the work of Kontsevich, Tschinkel, Mikhalkin and Zharkov, I
introduce a sheaf of graded algebras $\Lambda^\bullet$ on the dual intersection
complex of $Y$, denoted $\Delta_X$. I show that there exists a map
$H^q(\Delta_X, \Lambda^p) \to \mathrm{gr}^W_{2p} H^{p+q}(X_\infty,
\mathbb{Q})$, where $W$ is the monodromy weight filtration, which is injective
whenever there exists a class $\omega \in H^2(Y)$ which is combinatorial and
Lefschetz, a certain technical condition. When $f$ is a Type III Kulikov
degeneration of $K3$ surfaces, the sheaf $\Lambda^1$ recovers the affine
structure with singularities of Engel and Friedman on $\Delta_X$. In this case,
I show that existence of such class follows from the existence of a positive
$d''$-closed $(1,1)$-superform or supercurrent in the sense of Lagerberg on
$\Delta_X$. The latter is established in the case of simple affine structure
singularities in \cite{hessian}, in fact, the cohomology of sheaves $\Lambda^p$
coincides with the full nearby fibre cohomolgy then.
