By R. Narasimhan

ISBN-10: 3034886179

ISBN-13: 9783034886178

ISBN-10: 3764327421

ISBN-13: 9783764327422

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. ok. Chandrasekharan and Prof. Jurgen Moser have inspired me to put in writing them up for inclusion within the sequence, released by way of Birkhiiuser, of notes of those classes on the ETH. Dr. Albert Stadler produced exact notes of the 1st a part of this path, and extremely intelligible class-room notes of the remaining. with no this paintings of Dr. Stadler, those notes do not have been written. whereas i've got replaced a few issues (such because the facts of the Serre duality theorem, right here performed totally within the spirit of Serre's unique paper), the current notes keep on with Dr. Stadler's particularly heavily. My unique objective in giving the path used to be twofold. i needed to offer the elemental theorems in regards to the Jacobian from Riemann's personal perspective. Given the Riemann-Roch theorem, if Riemann's equipment are expressed in sleek language, they fluctuate little or no (if in any respect) from the paintings of recent authors.

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**Example text**

2, these sections cannot have common zeros. (s;g} : ... : Note. If L is a holomorphic line bundle on X and HO(X, L) f. pL: X --+ JlDN (N = dimHO(X,L) -1) on all of X. If So, ... ,SN is a basis as above and A = {x E X I S j (x) = Vj }, the map is defined as above on X - A. p L I U - {a} is the point in JlDN with homogeneous coordinates (10 : ... : fN) where fo, .. ·, fN are holomorphic functions on U and are not outside a. We can write fj = Zk gj where k = minj ord a (1j). p L on U is then given by the point in JlDN with homogeneous coordinates (go: ...

Then h 0 sex) = (x,f(x)), x E U, where f is an n-tuple of meromorphic functions on U. There is an integer k such that f = zk g, where g is an n-tuple of functions holomorphic near a, and g(a) "I- o. We set k = orda(s). One checks easily that this is independent of the local coordinate and of the local trivialisation. e. the divisor L f-+ ord a (s), orda(s)a , aEX is called the divisor of s and denoted by (s) or dives). e. vector bundle of rank 1), and we turn now to this construction. Let X be a Riemann surface and D a divisor on X; let D = LPEX npP (where np = D(P) E Z and the sum is locally finite).

If a E X, and i is such that a E Ui , set resa(w) = resa(wi) [this is independent of i because Wi - Wj is holomorphic on Ui n Uj]. Ui = [}Oi = dO i represents the cohomology class D({Wij}) in A~~(X) under the Dolbeault isomorphism. We have Lemma. Jx D ({Wij}) = 27ri LaEX resa(w) . Proof. Let (3 = Wi - 0i on Ui ; (3 is a Coo form on X - {al,"" an}, where S = {al,"" aT} is the set of poles of w (S n Ui = set of poles of Wi). Since dWi = 0 on Ui - S, we have cp = D({Wij}) = -d(3 on X - S. e = 1, ...

### Compact Riemann Surfaces by R. Narasimhan

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