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 $H^q({\Delta }_X,{\Lambda }^p)\to {\mathrm{gr}}_{2p}^W{H}^{p+q}(X_{\infty },\mathbb{Q})$, 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 $d^{\prime \prime }$-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.

arXiv commentimproved exposition, new material on superforms addded