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Extended skeletons of poly-stable pairs

Thomas Fenzl

Abstract

We introduce the notion of poly-stable pairs of formal schemes over the valuation ring of a non-archimedean field. For such pairs we define and investigate the dual intersection complex. We proceed to develop the so called extended skeleton of a poly-stable pair via an approximation process using the classical skeletons constructed by Berkovich. This is essentially a generalization of a construction by Gubler, Rabinoff and Werner from the strictly semi-stable case to the arbitrary poly-stable case and we extend their results.

Extended skeletons of poly-stable pairs

Abstract

We introduce the notion of poly-stable pairs of formal schemes over the valuation ring of a non-archimedean field. For such pairs we define and investigate the dual intersection complex. We proceed to develop the so called extended skeleton of a poly-stable pair via an approximation process using the classical skeletons constructed by Berkovich. This is essentially a generalization of a construction by Gubler, Rabinoff and Werner from the strictly semi-stable case to the arbitrary poly-stable case and we extend their results.

Paper Structure

This paper contains 19 sections, 60 theorems, 47 equations.

Table of Contents

  1. Preliminaries
  2. Analytic geometry
  3. Formal geometry
  4. Standard schemes and poly-stability
  5. Stratification
  6. Building blocks
  7. Dual intersection complexes
  8. Poly-simplices
  9. Canonical polyhedra
  10. The strict case
  11. The general case
  12. Extended Skeletons
  13. Classical construction
  14. Standard pairs
  15. Building blocks
  16. ...and 4 more sections

Key Result

Theorem A

Let $(\mathfrak{X}, \mathfrak{H})$ be a poly-stable pair and $Z := \mathfrak{X}_{\eta} \setminus \mathfrak{H}_{\eta}$. The extended skeleton $S(\mathfrak{X}, \mathfrak{H})$ is a subset of $Z$ satisfying the following properties:

Theorems & Definitions (177)

  • Theorem A
  • Theorem B
  • Definition \oldthetheorem
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  • Remark \oldthetheorem
  • Example \oldthetheorem
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  • ...and 167 more