The Localization Theorem for the Motivic Homotopy Theory of Complex Analytic Stacks and other Geometric Settings

TL;DR

This paper proves a localization theorem in complex analytic motivic homotopy theory, enabling a six-functor formalism and analytification map, with techniques applicable to various geometric contexts.

Contribution

It establishes a localization theorem for complex analytic stacks and develops general methods applicable to algebraic and differentiable stacks.

Findings

Proved the localization theorem for complex analytic stacks.

Established techniques for other geometric settings.

Enabled a compatible analytification map with six operations.

Abstract

We prove the analog of the Morel-Voevodsky localization theorem over complex analytic stacks, which is used in arXiv:2511.09371 to establish a 6-functor formalism of complex analytic motivic homotopy theory and produce an analytification map that is compatible with the six operations. Along the way, we establish general techniques for proving this theorem over other geometric settings, which also apply, for example, to the settings of algebraic stacks and differentiable stacks.