# On the Functional Independence of Ratios of Theta Functions by Lefschetz S. By Lefschetz S. By Lefschetz S.

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We call the set of all such maps {11 Q .. } with respect to the. } of X according to which 1 both Y and Y' are defined and such that the map U. x CCn -> U. x a::n l l . (v)) 1 2 U. -> i a:n = Matnxn(O::). Note that the maps cp i must satisfy 36 II. 2. 4 (All maps will be required to be differentiable, continuous, holomorphic, or algebraic, according to context. ) Conversely, from a collection of maps {cp i} satisfying II ij

For a proof, see Narasimhan [ 26]. Now we'll give some applications of this theorem. III. B. ::. n(r) -> a: be a holomorphic function. _!!. /f / is a local maximum at 0 then f is constant. Given zECn, define f (u) for small UEC by f (u) z z = f(uz). By the one variable maximum principle, fz is constant. The proposition follows from this. Resolution of singularities easily gives an extension of this to analytic spaces. We can also do this by branched coverings. III. C. 2 maximum at XEX. Then f is constant in a neighborhood of X.

2. 8 This is also a locally free sheaf of rank one, so we have defined a multiplication of line bundles. The inverse of a locally free sheaf of rank one, LI' is Hom JLI'O) because Hom (L1, 0) ® L1 -o 0 -=-> Because of the existence of inverse, locally 0. free sheaves of rank one are called invertible sheaves. } of X and for each i an f. , and so that 1 II .. f. = lJ J 1 f. such that f. , 1 1 J II .. ,O). • g. 1 up Wj, 1 J cp ij a unit in r (UP Wj, 0). J lJ J J = f. 1 on Now from an effective Cartier divisor f.