Serre duality on complex supermanifolds (Q1093050)

Let \({\mathcal F}\) be a locally free \({\mathcal O}\)-module on a compact complex supermanifold M of dimension (m\(| n)\), and let \({\mathcal B}er_{{\mathcal O}}\) be the Berezinian sheaf on M. It is proved that the duality exists between \(H^ q(M,{\mathcal F})\) and \(H^{m-q}(M,{\mathcal B}er_{{\mathcal O}}\otimes {\mathcal F}^*)\) (this is the precise analogue of Serre's original duality theorem). The proof is in the spirit of Serre's original proof, and it uses the natural duality between smooth functions and distributions.