Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

TL;DR

This paper constructs explicit indecomposable higher Chow cycles on Fano varieties of lines on cyclic cubic fourfolds, advancing the understanding of algebraic cycles on holomorphic symplectic manifolds.

Contribution

It provides the first explicit examples of indecomposable higher Chow cycles on holomorphic symplectic manifolds, using degeneration techniques and inducing cycles on Hilbert squares of K3 surfaces.

Findings

Constructed explicit indecomposable (2,1)- and (4,1)-cycles on Fano varieties of lines.

Developed a method to induce (p,1)-cycles on Hilbert squares of K3 surfaces.

Observed that restricted cycles to Lagrangian subvarieties are always decomposable in examples.

Abstract

In this paper we initiate the study of higher Chow cycles on holomorphic symplectic manifolds. Our concrete central result is construction of explicit indecomposable (2,1)- and (4,1)-cycles on the Fano varieties of lines on cyclic cubic fourfolds. This is the first explicit example of such cycles on holomorphic symplectic manifolds. The proof of indecomposability is done by degeneration to cuspidal cubic fourfolds. Along the way, we develop a method of inducing (p,1)-cycles on Hilbert squares of K3 surfaces. Finally, we study restriction of (2,1)-cycles to Lagrangian subvarieties, and observe the phenomenon that the restricted cycles are always decomposable in the examples in our hand.