Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

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.