Modular Forms

Abstract

We can write a canonical invertible sheaf on space Reas(W) as a fixed real vector space W of dimension 2g. We recall that we have the homomorphic bundle B → Reas(W) on rank g. One simply considers the higher exterior power Λ^gB. The fiber of Λ^g B at J is canonically isomorphic to the g-exterior power of the tangent space of the abelian variety X(J) at the origin. As B is homogeneous with respect to the action of Sym_ℝ(V) so is Λ^gB. Recalling that B is Hermitian we see that Λ^gB is Hermitian in an invariant way but this is not the interesting metric on Λ^g B.