Theta Functions and Divisors

Abstract

Let M be a complex manifold. In the sequel, M will either be Cn or Cn/D where D is a lattice (discrete subgroup of real dimension 2n). Let U i be an open covering of M, and let ϕi be a meromorphic function on Ui. If for each pair of indices (i, j) the function ϕi /ϕj is holomorphic and invertible on Ui ∩ Uj, then we shall say that the family (Ui, ϕi) represents a divisor on M. If this is the case, and (U, ϕ) is a pair consisting of an open set U and a meromorphic function ϕ on U, then we say that (U, ϕ) is compatible with the family (Ui, ϕi) if ϕi/ϕ is holomorphic invertible on U ∩ Ui. If this is the case, then the pair (U, ϕ) can be adjoined to our family, and again represents a divisor. Two families (Ui, ϕi) and (Vk, ψk) are said to be equivalent if each pair (Vk, ψk) is compatible with the first family. An equivalence class of families as above is called a divisor on M. Each pair (U, ϕ) compatible with the families representing the divisor is also said to represent the divisor on the open set U.