Apparaît dans la collection : 2018 - T1 - WS3 - Model theory and applications

Let G be a definable two-dimensional group in an o-minimal structure M and let D be a strongly minimal expansion of G, whose atomic relations are definable in M. We prove that if D is not locally modular then G is definably isomorphic to a one dimensional algebraic group A over a D-definable algebraically closed field K. Moreover, D is precisely the structure which K induces on A. The result generalizes a theorem of Hasson and Kowalski on expansions of (C, +) and gives another indication to the so-called Zilber’s Trichotomy conjecture, for strongly minimal structures that are definable in o-minimal ones. The proof of the result combines the geometric constraints of o minimality, together with the combinatorial restrictions of strong minimality, in order to recover “complex intersection theory” which allows us to define a field. (joint work with Pantelis Eleftheriou and Assaf Hasson)