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 III - "Standard conjectures under fibration and an application to hyperkähler varieties"
In the final lecture, we will consider the problem of proving the Lefschetz standard conjecture L(X) for a smooth projective variety X, when there exists a projective morphism f:X->B such that L(B) and L(f) are known. We will discuss the obstacles arising when trying this approach and explain how they can be overcome in the realm of hyperkähler varieties equipped with a Lagrangian fibration, in order to obtain a proof of the standard conjectures for new classes of such varieties.