An-Khuong Doan: (Formal) moduli problems and equivariant structures

Abstract

It is well-known that any projective variety \(X_0\) admits a formal semi-universal deformation containing all information about its small deformations. In 1980, D. S. Rim proved that if further \(G\) is a linearly reductive group acting algebraically on \(X_0\) then this semi-universal deformation can be equipped with a \(G\)-equivariant structure extending the given \(G\)-action on \(X_0\). In this talk, we first give an example showing that the reductiveness assumption in Rim’s result is really optimal. Next, we prove an analytic version of Rim’s result when \(X_0\) is a complex compact manifold. Finally, we generalize Rim’s result in the framework of derived algebraic geometry.