Semi-prorepresentability of formal moduli problems and equivariant structures

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 semiprorepresentable 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.

Keywords

deformation theory, moduli theory, formal moduli problem, equivariance structure

2010 Mathematics Subject Classification

13D10, 14B10, 14D15, 14F35

Full Text (PDF format)

Received 13 February 2023

Received revised 6 July 2023

Accepted 23 September 2023

Published 18 September 2024