A non-desarguesian projective plane

Hrushovski’s construction of “new” strongly minimal structures and more generally “new” stable structures proved very effective in providing a number of examples to classification problems in stability theory. For example, J.Baldwin used this method to construct a non-desarguesian projective plane of Morley rank 2 (see e.g. [3]). But there is still a classification problem of similar type which resists all attempt of solution, the Algebraicity (or CherlinZilber) Conjecture. At present there is a growing belief that there must exists a simple group of finite Morley rank which is not isomorphic to a group of the form G(F) for G an algebraic group and F an algebraically closed field (a bad group). The second author developed an alternative interpretation of the “new” stable structures obtained by Hrushovski’s construction, see e.g. [5]. In this interpretation the universe M of the structure is represented by a complex manifold and relation by some subsets of M explained in terms of the analytic structure on M. In this interpretation Hrushovski’s predimension inequality corresponds to a form of (generalised) Schanuel’s conjecture. We argue that looking for stable structures of analytic origin is potentially a better way of producing new stable structures. Below we briefly explain a construction of a new non-desarguesian projective plane that originates in a complex analytic structure. The new, in comparison with previous examples of e.g. “green fields” (see [6]) is that we have to use a non-trivial collapse procedure.