On the cohomology of a simple normal crossings divisor

TL;DR

This paper provides a formula linking the cohomology of sheaves on simple normal crossings divisors to the simplicial cohomology of their dual complexes, enabling new insights into Hodge theory and singularity resolutions.

Contribution

It introduces a novel cohomology decomposition formula for SNC divisors and explores its implications in Hodge theory and singularity resolution.

Findings

Established a cohomology decomposition formula for SNC divisors.

Derived a Hodge decomposition for SNC divisors.

Connected the formula to vanishing results in singularity theory.

Abstract

We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $\Delta(D)$ with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of $D$ and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity.