Differential forms on universal K3 surfaces

We give a vanishing and classification result for holomorphic differential forms on smooth projective models of the moduli spaces of pointed K3 surfaces. We prove that there is no nonzero holomorphic k-form for $0<k<10$ and for even $k>19$ . In the remaining cases, we give an isomorphism between the space of holomorphic k-forms with that of vector-valued modular forms ( $10\leq k \leq 18$ ) or scalar-valued cusp forms (odd $k\geq 19$ ) for the modular group. These results are in fact proved in the generality of lattice-polarisation.