Spaces with $\mathbb{G}_m$-action, hyperbolic localization and nearby cycles

TL;DR

This paper proves a generalization of hyperbolic localization for algebraic spaces with a $ ext{G}_m$-action over arbitrary bases and shows it commutes with nearby cycles, extending previous results.

Contribution

It extends Braden's hyperbolic localization theorem to families over arbitrary base schemes and demonstrates its compatibility with nearby cycles.

Findings

Hyperbolic localization commutes with nearby cycles.

Braden's theorem holds for algebraic spaces over arbitrary bases.

Generalization of hyperbolic localization to new settings.

Abstract

We study families of algebraic spaces with a fiberwise $\mathbb{G}_m$-action and prove Braden's theorem on hyperbolic localization for arbitrary base schemes. As an application, we obtain that hyperbolic localization commutes with nearby cycles.