Nonarchimedean and tropical geometry

Nonarchimedean and tropical geometry

by Baker, Matthew Payne, Sam

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This volume grew out of two Simons Symposia on "Nonarchimedean and tropical geometry" which took place on the island of St. John in April 2013 and in Puerto Rico in February 2015. Each meeting gathered a small group of experts working near the interface between tropical geometry and nonarchimedean analytic spaces for a series of inspiring and provocative lectures on cutting edge research, interspersed with lively discussions and collaborative work in small groups. The articles collected here, w…

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Title Statements

  • Full title :Nonarchimedean and tropical geometry / Matthew Baker, Sam Payne, editors.

Summary

This volume grew out of two Simons Symposia on "Nonarchimedean and tropical geometry" which took place on the island of St. John in April 2013 and in Puerto Rico in February 2015. Each meeting gathered a small group of experts working near the interface between tropical geometry and nonarchimedean analytic spaces for a series of inspiring and provocative lectures on cutting edge research, interspersed with lively discussions and collaborative work in small groups. The articles collected here, which include high-level surveys as well as original research, mirror the main themes of the two Symposia. Topics covered in this volume include: Differential forms and currents, and solutions of Monge?Ampère type differential equations on Berkovich spaces and their skeletons; The homotopy types of nonarchimedean analytifications; The existence of "faithful tropicalizations" which encode the topology and geometry of analytifications; Relations between nonarchimedean analytic spaces and algebraic geometry, including logarithmic schemes, birational geometry, and the geometry of algebraic curves; Extended notions of tropical varieties which relate to Huber's theory of adic spaces analogously to the way that usual tropical varieties relate to Berkovich spaces; and Relations between nonarchimedean geometry and combinatorics, including deep and fascinating connections between matroid theory, tropical geometry, and Hodge theory.

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Edition Statements

  • Publication Date:2016.

Publication Statements

  • Publisher(s):Springer
  • Place of Publication:Switzerland
  • Language:eng
  • Publication Date:2016.

Notes

Includes index.

The Physical Object

  • Description:1 online resource (xiv, 526 pages) : : illustrations (some color)

Edition Identifiers

Table of Contents

Preface; 1 Introduction; 2 Contents; Contents; List of Contributors; Forms and Currents on the Analytification of an Algebraic Variety (After Chambert-Loir and Ducros); 1 Introduction; 2 Superforms and Supercurrents on Rr; 3 Superforms on Polyhedral Complexes; 4 Moment Maps and Tropical Charts; 5 Differential Forms on Algebraic Varieties; 6 Currents on Algebraic Varieties; 7 Generalizations to Analytic Spaces; References; The Non-Archimedean Monge-Ampère Equation; 1 Introduction; 2 Metrics on Lines Bundles; 3 The Monge-Ampère Operator; 4 The Complex Monge-Ampère Equation

5 The Non-Archimedean Monge-Ampère Equation6 A Variational Approach; 7 Singular Semipositive Metrics; 8 Energy; 9 Envelopes, Differentiability, and Orthogonality; 10 Curves; 11 Toric Varieties; 12 Outlook; References; Convergence Polygons for Connections on Nonarchimedean Curves; 1 Newton Polygons; 2 PL Structures on Berkovich Curves; 3 Convergence Polygons: Projective Line; 4 A Gallery of Examples; 5 Convergence Polygons: General Curves; 6 Derivatives of Convergence Polygons; 7 Subharmonicity and Index; 8 Ramification of Finite Morphisms; 9 Artin-Hasse Exponentials and Witt Vectors

10 Kummer-Artin-Schreier-Witt…

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Subjects

Conference papers and proceedings Géométrie -- Congrès geometry Geometry -- Congresses MATHEMATICS -- Geometry

Writers

Baker, Matthew Payne, Sam

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