Logic Seminar
Zariski-type structures were introduced and studied by Zilber and Hrushovski in their study of strongly minimal sets. A Zariski-type structure is a set X with a collection of compatible Noetherian topologies, one on each Xn, and an assignment of dimension to the closed sets, satisfying certain conditions. For example, taking the complex analytic subvarieties of Mn, for a compact complex analytic manifold M, to be the closed sets and the dimension to be the complex dimension gives a Zariski-type structure. I am showing that a similar statement holds if the manifold is almost complex, i.e., admits an automorphism J of the tangent bundle TM such that J2=−Id, that does not necessarily arise from a holomorphic atlas on the manifold. I will define an almost complex subvariety, and outline the geometric results that are needed to show that the obtained structure is Zariski-type. I will also talk about geometric applications.