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 (9<k<19) or scalar-valued cusp forms (odd k>18) for the modular group. These results are in fact proved in the generality of lattice-polarization.