Derived logarithmic deformation theory and moduli stacks of derived logarithmic structures

TL;DR

This paper develops the deformation theory and moduli stacks for derived and spectral logarithmic structures, extending classical log geometry into derived and spectral contexts with new representability results.

Contribution

It introduces the deformation theory for animated and $ ext{E}_$-log rings and constructs moduli stacks for derived and spectral log structures, establishing their representability.

Findings

Construction of $$-root stacks in derived and spectral settings

Development of deformation theory for animated log rings

Establishment of representability of derived and spectral log stacks

Abstract

This paper investigates the derived and spectral analogs of logarithmic geometry. We develop the deformation theory for animated log rings and $\mathbb{E}_\infty$-log rings and examine the corresponding theories of derived and spectral log stacks. Furthermore, we define moduli stacks for derived and spectral log structures and establish their representability. As an application, we will construct $\infty$-root stacks in the derived and spectral settings and study the associated geometric properties.