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).

In the first lecture, we will give a quick review of the relation between algebraic cycles and singular cohomology on a complex, smooth projective variety, and explain the statements and the relevance of Grothendieck's standard conjectures on algebraic cycles. We will focus in particular on the standard conjecture of Lefschetz type L(X) for a smooth projective X, which implies all the other standard conjectures on X when one works over the complex numbers.