Deformation of singularities and additive invariants

Journal of Singularities
volume 13 (2015), 11-41

Received 28 October 2013. Received in revised form 4 September 2014.

DOI: 10.5427/jsing.2015.13b

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Abstract:

In this survey on local additive invariants of real and complex definable singular germs we systematically present classical or more recent invariants of different nature as emerging from a tame degeneracy principle. For this goal, we associate to a given singular germ a specific deformation family whose geometry degenerates in such a way that it eventually gives rise to a list of invariants attached to this germ. Complex analytic invariants, real curvature invariants and motivic type invariants are encompassed under this point of view. We then explain how all these invariants are related to each other as well as we propose a general conjectural principle explaining why such invariants have to be related. This last principle may appear as the incarnation in definable geometry of deep finiteness results of convex geometry, according to which additive invariants in convex geometry are very few.

Author(s) information:

Georges Comte
Laboratoire de Mathématiques
de l'Université de Savoie
UMR CNRS 5127
Bâtiment Chablais, Campus scientifique
73376 Le Bourget-du-Lac cedex, France
email: georges.comte@univ-savoie.fr