Zoom link: https://huji.zoom.us/j/82898386372?pwd=moVqtfYoiSkyAgUZcD9KQbMo5QDgYG.1
Meeting ID: 828 9838 6372
Passcode: 320017
Title: Reconstructing semi-abelian varieties from subvarieties
Abstract: Let C be a smooth projective curve of genus at least 2 over an algebraically closed field K, and let J be its Jacobian variety. In 2012 Zilber proved, roughly, that the full algebro-geometric structure of J can be recovered from the (abstract) group of K-rational points of J augmented by a subset for the image of C(K) under the Abel-Jacobi map.Zilber's "secret sauce" for the proof was a trichotomy theorem due to Rabinovich, The recent solution of the full Restricted Trichotomy for ACF allows a generalisation of Zilber's results to arbitrary (semi) abelian varieties, In the talk I will give a characterisation of when a (semi) abelian variety can be reconstructed from its pure abelian group structure, augmented by a predicate for a closed subvariety. More generally, I will describe a unique decomposition theorem for pairs (A,X) of an abelian variety A and a closed subvariety X into simple pairs (Ai, Xi) each of which consisting of an abelian subvariety Ai and a subvariety Xi allowing to reconstruct Ai.
No knowledge of algebraic geometry is needed, No model theory beyond that of the classical theory of groups of finite Morley Rank is needed and I will try to review those parts of it needed for the talk.
Joint work with Ben Castle.