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|a Nonarchimedean and tropical geometry /
|c Matthew Baker, Sam Payne, editors.
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| 264 |
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|a Switzerland :
|b Springer,
|c 2016.
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| 300 |
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|a 1 online resource (xiv, 526 pages) :
|b illustrations (some color)
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|a Simons symposia,
|x 2365-9564
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|a 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
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| 505 |
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|a 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
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|a 10 Kummer-Artin-Schreier-Witt Theory11 Automorphisms of a Formal Disc; Appendix 1: Convexity; Appendix 2: Thematic Bibliography; References; About Hrushovski and Loeser's Work on the Homotopy Type of Berkovich Spaces; 1 Introduction; 2 Model Theory of Valued Fields: Basic Definitions; 3 Hrushovski and Loeser's Fundamental Construction; 4 Homotopy Type of and Links with Berkovich Spaces; 5 An Application of the Definability of for C a Curve; References; Excluded Homeomorphism Types for Dual Complexes of Surfaces; 1 Introduction; 2 Tropical Complexes and Tropical Surfaces; 3 Degenerations
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|a 4 Proof of the Main TheoremsReferences; Analytification and Tropicalization Over Non-archimedean Fields; 1 Introduction; 2 Berkovich Spaces and Tropicalizations; 2.1 Notation and Conventions; 2.2 Berkovich Spaces; 2.3 Tropicalization; 3 The Case of Curves; 4 Tropical Grassmannians; 4.1 The Setting; 4.2 A Section of the Tropicalization Map; 4.3 Sketch of Proof in the Dense Torus Orbit; 5 Skeleta of Semistable Pairs; 5.1 Integral Affine Structures; 5.2 Semistable Pairs; 5.3 Skeleta; 6 Functions on the Skeleton; 7 Faithful Tropicalizations; 7.1 Finding a Faithful Tropicalization for a Skeleton
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| 505 |
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|a 7.2 Finding a Copy of the Tropicalization Inside the Analytic SpaceReferences; Berkovich Skeleta and Birational Geometry; 1 Introduction; 2 The Berkovich Skeleton of an sncd-Model; 2.1 Birational Points; 2.2 Models; 2.3 Divisorial and Monomial Points; 2.4 The Berkovich Skeleton; 2.5 The Deformation Retraction in a Basic Example; 3 Weight Functions and the Kontsevich-Soibelman Skeleton; 3.1 The Work of Kontsevich and Soibelman; 3.2 Log Discrepancies in Birational Geometry; 3.3 Definition of the Kontsevich-Soibelman Skeleton; 3.4 Definition and Properties of the Weight Function
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| 500 |
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|a Includes index.
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|a Online resource; title from PDF title page (SpringerLink, viewed August 30, 2016).
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|a 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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|a Geometry
|v Congresses.
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|a Electronic books.
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| 655 |
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|a Conference papers and proceedings.
|2 fast
|0 (OCoLC)fst01423772
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| 700 |
1 |
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|a Baker, Matthew,
|e editor.
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| 700 |
1 |
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|a Payne, Sam,
|e editor.
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| 711 |
2 |
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|a Simons Symposia on "Nonarchimedean and Tropical Geometry"
|d (2013 :
|c Saint John, Antigua and Barbuda)
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| 711 |
2 |
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|a Simons Symposia on "Nonarchimedean and Tropical Geometry"
|d (2015 :
|c Puerto Rico)
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|t Nonarchimedean and tropical geometry.
|d Switzerland : Springer, 2016
|z 3319309447
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