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