On the Cohomology of a Simple Normal Crossings Divisor

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 ∆(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.