Lifting of Holomorphic Actions on Complex Supermanifolds

Abstract

We study the problem of lifting analytic actions of a Lie group $G$ to a non-split complex analytic supermanifold $(M, \mathcal{O})$ from its retract $(M, \mathcal{O}_{\mathrm{gr}})$. In the case when $G$ is compact (or complex reductive), two criteria for lifting a Lie group action are found. The first one is invariance of the Čech 1-cocycle with values in a special automorphism sheaf of $(M, \mathcal{O}_{\mathrm{gr}})$ determining the non-split supermanifold $(M, \mathcal{O})$, while the second one is invariance of a certain differential form of a special kind which can also be viewed as a global derivation of a sheaf of differential forms on $M$. If the action is transitive on $M$, then the second criterion allows to reduce the lifting problem to the study of invariants of a finite dimensional linear representation.