Constant cycle surfaces on Fano varieties of cubic fourfolds

Huybrechts proved the finiteness of constant cycle curves of fixed order in any linear system $\mathrm{\mid }L\mathrm{\mid }$ on a K3 surface. In this paper, we study constant cycle surfaces on the Fano variety of lines $F(X)$ of a smooth cubic fourfold $X$. Fano surfaces $F(Y)\subset F(X)$ of hyperplane sections $Y\subset X$ are higher-dimensional analogues of curves on K3 surfaces. We prove that there are at most finitely many constant cycle surfaces of the form $F(Y)$ of any fixed order on $F(X)$.