Combinatorial part of the cohomology of the nearby fibre

TL;DR

This paper introduces a sheaf of graded algebras on the dual intersection complex of a degeneration's central fibre, establishing a link between combinatorial data and the cohomology of the nearby fibre, especially for K3 surface degenerations.

Contribution

It constructs a sheaf of graded algebras on the dual intersection complex and proves an injectivity result relating combinatorial cohomology to the nearby fibre cohomology, extending previous work.

Findings

The sheaf $ ext{Lambda}^ullet$ encodes combinatorial and geometric information.

Injective map from combinatorial cohomology to graded pieces of the nearby fibre cohomology.

For Type III Kulikov degenerations of K3 surfaces, the sheaf recovers the affine structure with singularities.

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.