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Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 31–44 | Cite as

Igusa Zeta Functions and the Non-archimedean SYZ Fibration

  • Johannes NicaiseEmail author
Article
  • 91 Downloads

Abstract

We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.

Keywords

Igusa zeta functions Non-archimedean geometry Mirror symmetry 

Mathematics Subject Classification (2010)

14E30 14J33 11S40 

Notes

Acknowledgements

I would like to thank the organizers of the VIASM Annual Meeting 2017 for the invitation to deliver a lecture at the meeting and to write this survey article for the proceedings. The results presented here are a result of joint work with Mircea Mustaţă and Chenyang Xu, and it is a pleasure to thank them both for the pleasant and interesting collaboration. I am also indebted to Wim Veys for sharing his ideas on the conjecture that constitutes the main subject of this text.

Funding Information

The author is supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.South Kensington CampusImperial College LondonLondonUK
  2. 2.Department of MathematicsKU LeuvenHeverleeBelgium

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