I am considering several examples of compact complex threefolds $X$ such that $rk H^2(X)=3$. Note that we have a cubic form on $H^2(X, \mathbb Q)$ which comes from the cup product. I calculated the Aronhold S-invariants for those cubic forms and found that they are all zero, which is very curious to me.
So let me ask some questions:
If S-invariants for a ternary cubic form is zero, what does it mean to the cubic form?
Is it a general phenomenon that S-invariants for cubic forms on such $X$'s are zero? If it not, what does zero S-invariant possibly mean to $X$?