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Iskovskikh Seminar
March 21, 2013 18:00, Moscow, Steklov Mathematical Institute, room 540
 


Dual complex of resolution of singularities

D. A. Stepanov

N. E. Bauman Moscow State Technical University

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Abstract: Dual complex encodes the combinatorics of intersections of the components of exceptional divisor in a resolution of singularities and generalizes to higher dimensions the notion of dual graph of a resolution of a surface singularity. We shall survey some recent results on the dual complex. We shall try to cover the following topics: the proof of homotopy invariance of the dual complex based on the factorization theorem for birational maps and on the theory of Berkovich spaces; calculation of the dual complex of hypersurface singularities with a help of embedded toric resolution; restrictions on the dual complex coming from Hodge theory (dual complex of a complement to an affine set, results of Arapura–Bakhtary–Wlodarzcyk); Payne's example of a rational singularity with the dual complex homeomorphic to $\mathbb RP^2$; Kollar's results on the realizability of any simplicial complex as the dual complex of an isolated singularity.

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