nXm i and recalling the series expansion for the matrix exponential function, note that exp *(Z1,Z2) = I+ A(Z1,Z2) + (higher order terms) and that exp Z2 exp 4,(z1,Z2) = exp Z1 = I +(Z1 + Z2) + (higher order terms) Writing x(Z1,Z2) = Zl + Z2 .
With this observation, the proof of the theorem is concluded. -25- Local structure of coherent analytic sheaves §2. Over a Riemann surface, any coherent analytic sheaf can be (a) described quite simply in terms of complex vector bundles; the present section will be devoted to a local and semi-local version of this relationship on a general Riemann surface, and the global version over the complex projective line. Theorem 3. On a Riemann surface, every coherent analytic subsheaf of a locally free sheaf is locally free.