# Abstract Cavicchi

**Matthia Cavicchi** will speak on:

**Relative motivic decompositions for families with Tate fibers**

Let f: $X\to S$ be a proper morphism of varieties over a field k. When k is algebraically closed of characteristic zero, or the separable closure of a finite field, the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber says that the total direct image along f of the constant local system on X decomposes, in the derived category, as a direct sum of simple objects over S, called “intersection complexes”. It is expected that such a decomposition arises as the realization of a decomposition of the relative motive of X over S, as a sum of “intersection motives”. The aim of the talk is to introduce this circle of ideas and to explain joint work with F. Déglise and J. Nagel, which establishes the existence of such motivic decompositions for some classes of morphisms $X\to S$, even allowing general regular schemes S as bases.