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 K 3 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≤ k ≤ 18 ) or scalar-valued cusp forms (odd k≥ 19 ) for the modular group. These results are in fact proved in the generality of lattice-polarisation.