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Berkovich Skeleta and Birational Geometry

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Book cover Nonarchimedean and Tropical Geometry

Part of the book series: Simons Symposia ((SISY))

Abstract

We give a survey of joint work with Mircea Mustaţă and Chenyang Xu on the connections between the geometry of Berkovich spaces over the field of Laurent series and the birational geometry of one-parameter degenerations of smooth projective varieties. The central objects in our theory are the weight function and the essential skeleton of the degeneration. We tried to keep the text self-contained, so that it can serve as an introduction to Berkovich geometry for birational geometers.

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Notes

  1. 1.

    To be precise, \(\vert \Delta (\mathcal{X}_{k})\vert \) is not a simplicial complex in the strict sense, because we allow, for instance, multiple edges between two vertices. This has no importance for the present exposition (and can always be remediated by blowing up \(\mathcal{X}\) at connected components of the subvarieties E J , which gives rise to a stellar subdivision of the corresponding face of \(\vert \Delta (\mathcal{X}_{k})\vert \)). In any case, \(\vert \Delta (\mathcal{X}_{k})\vert \) is the topological realization of a finite simplicial set.

  2. 2.

    However, the class of poly-stable formal schemes is much larger than the class of sncd-models with reduced special fiber.

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Acknowledgements

I am grateful to Sam Payne for helpful comments on an earlier version of this text.

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Authors and Affiliations

  1. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Heverlee, Belgium

    Johannes Nicaise

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  1. Johannes Nicaise
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Correspondence to Johannes Nicaise .

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Editors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Matthew Baker

  2. Department of Mathematics, Yale University, New Haven, Connecticut, USA

    Sam Payne

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Nicaise, J. (2016). Berkovich Skeleta and Birational Geometry. In: Baker, M., Payne, S. (eds) Nonarchimedean and Tropical Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-30945-3_7

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