Fano schemes and Moishezon manifolds

In this Note, we give a new family of Moishezon manifolds with Picard group of rank one. These manifolds are bimeromorphically equivalent to some Fano schemes of k-planes contained in a hypersurface in projective space. This construction is used in particular to test and show in dimension 7 and 8 the optimality of a previous result due to the first author We also give a simple method to compute some of the rational cohomology groups of these Fano schemes.