On the Lefschetz standard conjecture for Lagrangian covered hyper-K\"ahler varieties

We investigate the Lefschetz standard conjecture for degree $2$ cohomology of hyper-K\"ahler 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 fibration. In dimension $4$, we consider the more general case of a Lagrangian covered fourfold $X$, and prove the Lefschetz standard conjecture in degree $2$, assuming $\rho(X)=1$ and $X$ is general in moduli. Finally we discuss various links between Lefschetz cycles and the study of the rational equivalence of points and Bloch-Beilinson type filtrations, giving a general interpretation of a recent intriguing result of Marian and Zhao.