Gabriella Clemente (Universite Paris Saclay) : Algebraic aspects of almost-complex manifolds

The purpose of this talk is to discuss algebro-geometrical and topological tools for the study of almost-complex geometry. The plan is to start with a review of some basic examples of almost-complex manifolds: the Cayley hypersurfaces, which carry an almost-complex structure whose integrability is controlled by algebraic features of the octonions; and some 4-dimensional non-complex examples, which are related to the Kodaira classification of compact complex surfaces. The octonionic embedding of the 6-sphere will then be used to present a strategy to tackle the question of non-existence of a complex structure via universal embeddings and bundle constructions, and their homotopy groups. This will lead to a second strategy to tackle the more general problem of non-existence of a complex structure up to deformation. Some candidate complex vector bundle invariants that are needed to advance in that direction will be discussed. The end of the talk will be devoted to potential uses of real algebraic geometry for almost-complex manifolds, and to almost- analogs of familiar classes of complex varieties, such as hyperKaehler manifolds.