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EULER CHARACTERISTIC AND SIGNATURE OF REAL SEMI-STABLE DEGENERATIONS

Published online by Cambridge University Press:  21 February 2022

Erwan Brugallé Open the ORCID record for Erwan Brugallé [Opens in a new window]
Erwan Brugallé*
Affiliation:
Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, F-44322 Nantes, Cedex 3, France
*
erwan.brugalle@math.cnrs.fr
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Abstract

We give a motivic proof of the fact that for nonsingular real tropical complete intersections, the Euler characteristic of the real part is equal to the signature of the complex part. This was originally proved by Itenberg in the case of surfaces in $\mathbb {C}P^{3}$, and has been successively generalized by Bertrand and by Bihan and Bertrand. Our proof, different from previous approaches, is an application of the motivic nearby fiber of semistable degenerations. In particular, it extends the original result by Itenberg, Bertrand, and Bihan to real analytic families admitting a $\mathbb {Q}$-nonsingular tropical limit.

Keywords

topology of real algebraic varietiessemistable degenerationscombinatorial patchworkingtropical geometry

MSC classification

Primary: 14P25: Topology of real algebraic varieties

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 5 , September 2023 , pp. 2447 - 2454
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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