Invariants of analytic \({\mathbb{Z}}_ 2\)-manifolds (Q1063752)

By a (smooth or analytic) \({\mathbb{Z}}_ 2\)-manifold one means a \({\mathbb{Z}}_ 2\)-graded ringed space (X,\({\mathcal O}_ X)\) which is locally isomorphic with a model. In the analytic category a local model is given by an open subset of \({\mathbb{C}}^ p\) provided with a sheaf of holomorphic functions with values in the exterior algebra \(\bigwedge ({\mathbb{C}}^ q)\). A \({\mathbb{Z}}_ 2\)-manifold is called ''simple'' if there exists a vector bundle over the underlying manifold such that the structural sheaf \({\mathcal O}_ X\) is isomorphic to the sheaf of sections of the exterior algebra of this bundle. It is known that smooth \({\mathbb{Z}}_ 2\)-manifolds are simple. This does not hold in the analytic case. In fact, the author constructs a sequence of cohomological invariants allowing one to distinguish non-isomorphic \({\mathbb{Z}}_ 2\)-manifolds defined over the same ordinary manifold. The invariants arise as obstructions to extend isomorphisms of approximations of corresponding structure sheaves to higher order. Finally he uses the invariants to verify that a certain \({\mathbb{Z}}_ 2\)-manifold over a non-singular quadric in \({\mathbb{P}}^ 5\) is not simple.