Semi-prorepresentability of formal moduli problems and equivariant structures

TL;DR

This paper extends classical deformation theory to derived settings, providing criteria for semi-prorepresentability and equivariant structures, with applications to algebraic schemes and complex manifolds.

Contribution

It introduces a generalized notion of semi-universality in derived deformation theories and establishes criteria for semi-prorepresentability and equivariant structures.

Findings

Criteria for semi-prorepresentability of formal moduli problems

Conditions for existence of G-equivariant structures

Applications to algebraic schemes and complex manifolds

Abstract

We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semi-prorepresentable is produced. This can be seen as an analogue of Schlessinger's conditions for a functor of Artinian rings to have a semi-universal element. We also give a sufficient condition for a semi-prorepresentable formal moduli problem to admit a $G$-equivariant structure in a sense specified below, where $G$ is a linearly reductive group. Finally, by making use of these criteria, we derive many classical results including the existence of ($G$-equivariant) formal semi-universal deformations of algebraic schemes and that of complex compact manifolds.