On the cohomology of a simple normal crossings divisor

Details On the cohomology of a simple normal crossings divisor On the cohomology of a simple normal crossings divisor

2008-11-14

arxiv.org/abs/0811.2246v1

Journal of Algebra, vol. 324, pp. 2676-2691, 2010

Mathematics

Algebraic Geometry

19 pages

Scientific paper

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.

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