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The dual complex of singularities. (English) Zbl 1388.14107

Oguiso, Keiji (ed.) et al., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-046-4/hbk). Advanced Studies in Pure Mathematics 74, 103-129 (2017).
Summary: The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a “minimal” representative of the homotopy class that is well defined up-to piecewise linear homeomorphism. This is derived from a more global result concerning dual complexes of dlt pairs.
As an application, we also show that the dual complex of a log terminal singularity as well as the one of a simple normal crossing degeneration of a family of rationally connected manifolds are contractible.
For the entire collection see [Zbl 1388.14012].

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14D06 Fibrations, degenerations in algebraic geometry
14E30 Minimal model program (Mori theory, extremal rays)
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