By R. Narasimhan

Those notes shape the contents of a Nachdiplomvorlesung given on the Forschungs institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. okay. Chandrasekharan and Prof. Jurgen Moser have inspired me to put in writing them up for inclusion within the sequence, released through Birkhiiuser, of notes of those classes on the ETH. Dr. Albert Stadler produced unique notes of the 1st a part of this path, and intensely intelligible class-room notes of the remaining. with no this paintings of Dr. Stadler, those notes should not have been written. whereas i've got replaced a few issues (such because the facts of the Serre duality theorem, the following performed solely within the spirit of Serre's unique paper), the current notes persist with Dr. Stadler's rather heavily. My unique target in giving the direction was once twofold. i needed to offer the fundamental theorems in regards to the Jacobian from Riemann's personal perspective. Given the Riemann-Roch theorem, if Riemann's equipment are expressed in glossy language, they fluctuate little or no (if in any respect) from the paintings of recent authors.

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**Extra resources for Compact Riemann Surfaces**

**Example text**

Then H1(X, ]Eoo) = o. Proof. Let U = {UdiEI be an open covering, and Sij E GE'(Ui n Uj ) be such that {Sij} E Z1(U, ]Eoo). Let {aihEI be a partition of unity relative to U, define Si E GE'(Ui ) by Si = LjEi ajs;j (where ajSij is defined by (ajs;j)(x) = aj(x)s;j(x) if x E U; n Uj' = 0 if x E U; - U; n Uj ). Then, as in the proof of the Mittag-Leffler theorem, j 1 j D 01 - 1 We define the map H (X,E) --- Ai (X)/aGE'(X) as follows: Let {s;j} E Z (U,E) (s;j E HO(Ui n Uj,E)). Let 'Pi E G~(U;) be such that 'Pi - 'Pj = Sij on Ui n Uj .

WEYL'S LEMMA: The regularity theorem for &. Let X be a compact Riemann surface, let IT : E --+ X be a holomorphic vector bundle on X. We equip A~l(X) with the Coo topology (described in §8), viz: the topology of convergence of all derivatives on compact subsets of coordinate neighbourhoods in X. Suppose that F : A~l(X) --+ C is a continuous linear form such that F\&CE(X) = O. Then, there exists a unique 8 E HO(x, E* Q9 Kx) such that F(

Let X be a compact Riemann surface. We introduce a topology of complete metric space (even a Frechet space) on A~l(X) by the following requirement: A sequence cp(v) E A~l (X) converges if and only iffor any U as above, the corresponding sequence ( CPlv) , ... , for any differentiation Dk of order k (Dk = 8x Y;ym , C+ m = k), the sequence {Dk cp;v)} converges uniformly on compact subsets of U]. We introduce, in the same way, a topology on C'E(X). These are called the Coo (or Schwartz) topologies on A~\ C'E.