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

Abstract

We investigate the Lefschetz standard conjecture for degree 2 cohomology of hyper-Kähler 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 ρ(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.