Differential forms on universal K3 surfaces

TL;DR

This paper investigates the existence and classification of holomorphic differential forms on moduli spaces of pointed K3 surfaces, revealing vanishing results and connections to modular forms.

Contribution

It establishes new vanishing theorems and links between differential forms on K3 moduli spaces and modular forms, extending to lattice-polarized cases.

Findings

No nonzero holomorphic k-forms for 0<k<10 and even k>19

Isomorphism between holomorphic k-forms and modular forms for 9<k<19 or odd k>18

Results apply to lattice-polarized K3 surfaces

Abstract

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.