Algebraic representatives and intermediate Jacobians over perfect fields
-- Sebastian Casalaina-Martin, September 15, 2017

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Intermediate Jacobians and Abel--Jacobi maps provide a powerful tool for the study of complex projective manifolds.  In this talk I will discuss recent work with J. Achter and C. Vial showing that the image of the Abel--Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers.  As a consequence we obtain a new proof of a result of Deligne on intermediate Jacobians of complete intersections.  In positive characteristic, over algebraically closed fields, algebraic representatives and regular homomorphisms provide a replacement for the intermediate Jacobian and Abel--Jacobi map. I will discuss recent progress, again with Achter and Vial, extending this theory to the case of perfect fields.  Finally, time permitting I will discuss some applications.