Lagrangian constant cycle subvarieties in Lagrangian fibrations

TL;DR

This paper constructs special subvarieties in hyper-Kähler manifolds with Lagrangian fibrations, showing their independence from certain choices and connecting to rational connectivity of fibers in Calabi-Yau manifolds.

Contribution

It introduces a method to construct Lagrangian constant cycle subvarieties in hyper-Kähler manifolds and proves their independence from specific divisor classes and fibrations.

Findings

Constructed Lagrangian constant cycle subvarieties in hyper-Kähler manifolds.

Proved independence of certain zero-cycle classes from divisor choices.

Established rational connectivity of fibers in dominant meromorphic maps from Calabi-Yau manifolds.

Abstract

We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold $X$ whose general fiber is of dimension strictly between $0$ and $\dim X$ is rationally connected. Using this result, we construct for any hyper-K\"ahler manifold $X$ admitting a Lagrangian fibration a Lagrangian constant cycle subvariety $\Sigma_H$ in $X$ which depends on a divisor class $H$ whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on $\Sigma_H$ in $\mathrm{CH}_0(X)$ depend neither on $H$ nor on the Lagrangian fibration (provided $b_2(X) \ge 8$).