Hodge cycles for cubic hypersurfaces

Abstract

We study an algebraic cycle of the form $Z_0= r {\mathbb P}^{\frac{n}{2}}+\check r \check{\mathbb P}^{\frac{n}{2}}$, $r \in{\mathbb N},\check r \in{\mathbb Z},\ \ 1\leq r , |\check r |\leq 10,\ \ \gcd ( r ,\check r )=1$, inside the cubic Fermat variety $X_0$ of even dimension $n\geq 4$ and with $\dim\left ({\mathbb P}^{\frac{n}{2}}\cap \check{\mathbb P}^{\frac{n}{2}}\right)=m$. We take a smooth deformation space $\sf S$ of $X_0$ such that the triple $(X_0, {\mathbb P}^\frac{n}{2}, \check{\mathbb P}^\frac{n}{2})$ becomes rigid. For $m=\frac{n}{2}-2$ and for many examples of $N\in{\mathbb N}$ and $n$ we show that the $N$-th order Hodge locus attached to $Z_0$ is smooth and reduced of positive dimension if and only if $( r ,\check r )=(1,-1)$. In this case, the underlying algebraic cycles are conjectured to be cubic ruled cycles. For $m=\frac{n}{2}-3$ the same happens for all choices of coefficients $ r $ and $\check r $ and we do not know what kind of algebraic cycles might produce such Hodge cycles. The first case gives us a conjectural description of a component of the Hodge locus, and the second case gives us strong computer assisted evidences for the existence of new Hodge cycles for cubic hypersurfaces. Whereas the well-known construction of Hodge cycles due to D. Mumford and A. Weil for CM abelian varieties, and Y. Andr\'e's motivated cycles can be described in theoretical terms, the full proof of the existence of our Hodge cycle seems to be only possible with more powerful computing machines.