Abstract
The theory of real smooth supermanifolds in the Berezin approach is not well adapted to the requirements of quantization since hermitian conjugation, which is the classical limit of taking operator adjoint in quantized theories, is not compatible with the sign rule of Z -graded algebra. In this paper, we give the modifications of the theory necessary to reconcile it with hermitian conjugation. In particular, we introduce a second sign rule, and we show how to modify the usual definition of supermanifolds, getting hermitian supermanifolds. Almost all theorems of usual super differential geometry together with their proofs carry over to the new, hermitian setting. We also consider complex supermanifolds and hermitian metrics on complex vector bundles.
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