[Paper Review] A geometric theory of non-Archimedean analytic stacks

Martin Ulirsch|arXiv (Cornell University)|Oct 8, 2014

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper establishes a non-Archimedean analogue of the logarithmic retraction on complex toric varieties by showing that the Kajiwara-Payne tropicalization map arises as a quotient of the analytification of a toric variety by its big affinoid torus. It develops foundational tools for non-Archimedean analytic stacks, particularly analytic groupoids and their quotients, and identifies the tropicalization map as a stack quotient in the analytic category.

ABSTRACT

For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analytification, and the underlying topological spaces of analytic stacks.

Motivation & Objective

• To extend the geometric retraction from complex toric varieties to the non-Archimedean setting.
• To develop foundational tools for non-Archimedean analytic stacks, focusing on analytic groupoids and quotients.
• To establish a precise identification of the Kajiwara-Payne tropicalization map as a stack quotient in the analytic category.
• To clarify the topological and geometric structure of analytic stacks via analytification and quotient constructions.

Proposed method

• Constructing a geometric theory of non-Archimedean analytic stacks, with emphasis on analytic groupoids and their quotients.
• Defining the analytification of algebraic stacks in the non-Archimedean context.
• Using the big affinoid torus action on the analytification of a toric variety to form a quotient stack.
• Identifying the underlying topological space of the quotient stack with the tropicalization via the Kajiwara-Payne map.
• Proving that the tropicalization map is isomorphic to the quotient stack of the analytification by the big affinoid torus.
• Establishing compatibility between the stack-theoretic quotient and the classical tropicalization construction.

Experimental results

Research questions

• RQ1 How can the logarithmic retraction on complex toric varieties be generalized to the non-Archimedean setting?
• RQ2 What is the stack-theoretic interpretation of the Kajiwara-Payne tropicalization map in the analytic category?
• RQ3 How do analytic groupoids and their quotients behave in non-Archimedean analytic geometry?
• RQ4 What is the relationship between the analytification of a toric variety and its tropicalization as a stack quotient?
• RQ5 How can the underlying topological space of a non-Archimedean analytic stack be described in terms of geometric and groupoid structures?

Key findings

• The Kajiwara-Payne tropicalization map is identified as the quotient of the analytification of a toric variety by its big affinoid torus in the category of analytic stacks.
• A geometric theory of non-Archimedean analytic stacks is developed, with a focus on analytic groupoids and their quotients.
• The analytification of a toric variety admits a natural action by its big affinoid torus, leading to a stack quotient structure.
• The underlying topological space of the quotient stack is homeomorphic to the tropicalization of the toric variety.
• The retraction induced by the tropicalization map corresponds to the stack quotient construction in the non-Archimedean setting.
• The construction provides a non-Archimedean analogue of the complex logarithmic retraction, extending classical results to the analytic stack framework.