On the cohomology of a simple normal crossings divisor and its dual complex, with applications to isolated singularities

Parsa Bakhtary, Purdue University

Abstract

We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) in terms of the simplicial cohomologies of the dual complex with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of the SNC divisor and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also discuss relevant joint work with D. Arapura and J. W lodarczyk which gives bounds on the Betti numbers of the dual complex associated to a resolution of an isolated singularity. A consequence of this result is the vanishing of higher cohomologies of the dual complex of an isolated rational singularity, which in turn implies that the rationalization of the dual complex associated to a resolution of an isolated rational hypersurface singularity of dimension at least three is contractible.

Degree

Ph.D.

Advisors

Wlodarczyk, Purdue University.

Subject Area

Mathematics

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