Problems in the theory of closed Riemann surfaces

1. The Jacobian varieties provide an effective and natural tool for the study of certain problems in the theory of closed Riemann surfaces. Their classical construction is very simple: 1 Let X be a closed Riema.nn surface of genus g ~ i. Let ~ ,...,~g be a basis for the holomorphic differentials, and let yi,72,...,72g be a basis for the first integral homology group, on X. The associated period matrix Z : (lj) is defined by i Ii ~ j w ~j I t s c o l u m n s a r e N l i n e a r l y i n d e p e n d e n t a n d g e n e r a t e a g r o u p o f t r a n s l a t i o n s F o f C g . T h e c o m p l e x t o r u s J ( X ) = c g / F w i l l b e r e f e r r e d t o a s a J a e o b i a n v a r i e t y o f X. Let P be a fixed reference point of X. For an arbitrary point Q { X the point u(Q) 6 C g with coordinates Q i r i u (Q) =J