Automorphic forms in the geometry of orthogonal Shimura varieties

Orthogonal Shimura varieties arise naturally as moduli spaces in algebraic geometry, one example being the moduli space of quasi-polarized K3 surfaces. Their geometry is closely related to the theory of automorphic forms. I will discuss two instances of this relationship. First, the obstruction space for extending a line bundle on an orthogonal Shimura variety to its Baily-Borel compactification can be identified with a certain space of theta series for a symplectic group. By showing that these theta series span the space of cusp forms we conclude that the Picard group of this Baily-Borel compactification has rank one and is spanned by a multiple of the hodge line bundle if the underlying lattice splits two hyperbolic planes globally and three locally. In the second part, I will discuss a result in which Eisenstein series for orthogonal groups are used to study the birational geometry of orthogonal Shimura varieties. We construct pluricanonical forms from Eisenstein series, yielding new results on when such varieties are of general type.