Combinatorial part of the cohomology of the nearby fibre

Abstract

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.