gross showed that to every hermitian symmetric tube domain we may associate a canonical variation of hodge structure (vhs) of calabi-yau type. the construction is representation theoretic, not geometric, in nature, and it is an open question to realize this abstract vhs as the variation induced by a family of polarized, algebraic calabi-yau manifolds. in order for a geometric vhs to realize gross's vhs it is necessary that the invariants associated to the two vhs coincide. for example, the hodge numbers must agree. the later are discrete/integer invariants. characteristic forms are differential-geometric invariants associated to vhs (introduced by sheng and zuo). remarkably, agreement of the characteristic forms is both necessary and sufficient for a geometric vhs to realize one of gross's vhs. that is, the characteristic forms characterize gross's calabi-yau vhs. i will explain this result, and discuss how characteristic forms have been used to study candidate geometric realizations of gross's vhs.