Given a complex, smooth projective variety, one can study it through invariants of different nature. A natural bridge from invariants of algebraic-geometric nature (groups of algebraic cycles) to invariants of topological nature (singular cohomology) is provided by the cycle class map. One of the longstanding open problems of algebraic geometry consists precisely in understanding the image of the cycle class map, and Grothendieck's standard conjectures point out some natural classes that should lie in it. The goal of this three-lecture series is to give an introduction to this circle of ideas and to explain the recent proof of new cases of the standard conjectures, through an approach based on studying how the desired classes behave when algebraic varieties move in families, with possibly singular fibers (joint work of the lecturer with G. Ancona, R. Laterveer and G. Saccà). 

Lecture II - "Projective families of algebraic varieties and the decomposition theorem"

In the second lecture, we will move to the relative setting and explain the additional structures existing on the cohomology of a smooth projective variety, when it admits a fibration in projective varieties over a base. We will discuss relative cycles and the BBDG decomposition theorem for a projective morphism f, and use these to formulate a relative version, denoted L(f), of the Lefschetz standard conjecture.