On the Lefschetz standard conjecture for Lagrangian covered hyper-Kaehler varieties

We will discuss results on the Lefschetz standard conjecture for degree 2 cohomology of hyper-Kaehler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is implied by the SYZ conjecture characterizing classes of divisors associated with Lagrangian fibrations. In the more general case of a Lagrangian covering, we give a numerical criterion implying the Lefschetz standard conjecture in degree 2. We will also discuss potential applications to the study of the Chow ring and the group of 0-cycles.