The dual complex of singularities. (English) Zbl 1388.14107

de Fernex, Tommaso; Kollár, János; Xu, Chenyang

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)

Keywords:

dual complex; singularities; minimal model program