17 may 2023 -- 11:30
Aula Tricerri, DIMaI, Firenze
Seminari di Geometria del Dini
Abstract.
One of the open problems in the theory of algebraic cycles is that of the existence of Künneth projectors for any smooth, projective algebraic variety X: in other words, of algebraic cycles on X \times X, acting on (singular, étale...) cohomology H(X) as the projections on each Hi(X). This problem admits a version in families: given a projective, non necessarily smooth morphism f:X -> S between varieties, one asks whether there exist algebraic cycles decomposing the complex Rf 1X into simple factors, called intersection complexes. The modern theory of motives allows one to study these problems by constructing, under suitable hypotheses, certain intersection motives, analogues of their sheaf-theoretic homonyms. The aim of the talk is to present an overview of the applications of these objects in several geometric contexts: quadric bundles, hyper-Kähler varieties, Shimura varieties.