Theta and Heisenberg Groups

Abstract

For the introductory remarks of this chapter let us assume that L is a very ample line bundle on an abelian variety X = V/Λ and ϕ _L :X ↪ℙ _N the associated embedding. Recall the group K(L) consisting of all x ∈ X with t _x ^* L ≃ L. We will see that the translations of X by elements of K(L) extend to linear automorphisms of ℙ _N . In fact, K(L) is the largest group of translations with this property. This leads to a projective representation ϱ:K(L) → PGL _N (ℂ), with respect to which the embedding ϕ _L is equiv-ariant. It will be an important tool in the investigation of the geometric properties of the embedded abelian variety ϕ _L (X) in ℙ _N .