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

TL;DR

This paper explores the Lefschetz standard conjecture for hyper-K"ahler varieties with Lagrangian structures, linking it to the SYZ conjecture and proving it in specific cases, with implications for algebraic cycles.

Contribution

It establishes the Lefschetz standard conjecture for certain hyper-K"ahler varieties with Lagrangian fibrations and covers, connecting it to the SYZ conjecture and moduli conditions.

Findings

Lefschetz conjecture follows from SYZ conjecture in Lagrangian fibrations.

Proved the conjecture for fourfolds with  ho(X)=1 in general moduli.

Discussed links to rational equivalence and Bloch-Beilinson filtrations.

Abstract

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.