Corollaries of the identity

Abstract

In this section we will study what happens to Fay’s identities when the 4 points a,b,c,d come together in various stages. The result will be identities involving derivatives of theta functions. First, we need some notation. For the following formulas, let

• a) $\vec z$ ∈ B^g

• b) a,b,c,d ∈ $\tilde X$ with distinct projections to X

• c) ϑ $\left( {\vec z} \right)$ the theta function of X

• d) for every a ∈ X, and local coordinates t on X near a, we expand the differentials of the 1^st kind:

  $$ \omega _i = \left( {\sum\limits_{j = 0}^\infty {v_{ij} \frac{{t^j }} {{j!}}} } \right)dt $$

  and let

  $$ \vec v_j = \left( {v_{lj} , \cdots ,v_{gj} } \right). $$

  (Note that the mapping

  $$ \begin{gathered} \tilde x \to \mathbb{C}^g \hfill \\ x \mapsto \int\limits_a^x {\vec \omega } \hfill \\ \end{gathered} $$

  is given near a by

  $$ t \mapsto \sum\limits_{j = 0}^\infty {\vec v_j \frac{{t^{j + 1} }} {{\left( {j + 1} \right)!}}.)} $$

  We let

  $$ D_a = constant vector field \vec v_0 \cdot \frac{\partial } {{\partial z}}\left( {i.e.,\sum {v_{0i} \frac{\partial } {{\partial z_i }}} } \right) $$
  $$ D'_a = constant vector field \vec v_1 \frac{\partial } {{\partial \vec z}} $$
  $$ D''_a = constant vector field \vec v_2 \frac{\partial } {{\partial \vec z}}. $$

• e) We abbreviate $\int\limits_a^b {\vec \omega }$ to $\int\limits_a^b \cdot$