Theta Functions and Divisors

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 ϕ_i be a meromorphic function on U_i. If for each pair of indices (i, j) the function ϕi /ϕ_j is holomorphic and invertible on Ui ∩ U_j, then we shall say that the family (U_i, ϕ_i) 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 (U_i, ϕ_i) if ϕi/ϕ is holomorphic invertible on U ∩ U_i. If this is the case, then the pair (U, ϕ) can be adjoined to our family, and again represents a divisor. Two families (U_i, ϕ_i) and (V_k, ψ_k) are said to be equivalent if each pair (V_k, ψ_k) 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.