Torus fibers and the weight filtration

TL;DR

This paper establishes a geometric construction of certain weight filtration pieces in the cohomology of log Calabi--Yau pairs and degenerations, linking torus fibers to mixed Hodge structures.

Contribution

It introduces a novel use of real torus fibers to explicitly construct parts of the weight filtration in the cohomology of log Calabi--Yau pairs and related degenerations.

Findings

Constructs $W_{2k-1}H^k$ using torus fibers for simple normal crossings pairs.

Shows P=W type results for rational surfaces with nodal anticanonical divisors.

Extends results to degenerations of Calabi--Yau varieties and K3 surfaces.

Abstract

We show that if $(X,Y)$ is a simple normal crossings log Calabi--Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of $Y$ which can be used to construct $W_{2k-1}H^k(X \setminus Y;\mathbb{Q})$ for all $k$. We show that an analogous result holds for degenerations of Calabi--Yau varieties. We use this to show that P=W type results hold for pairs $(X,Y)$ consisting of a rational surface $X$ and a nodal anticanonical divisor $Y$, and for K3 surfaces.