## Compact Riemann Surfaces (Lectures in Mathematics. ETH - download pdf or read online

By R. Narasimhan

ISBN-10: 0817627421

ISBN-13: 9780817627423

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 through 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, the following performed solely within the spirit of Serre's unique paper), the current notes stick with Dr. Stadler's quite heavily. My unique goal in giving the direction used to be twofold. i needed to provide the fundamental theorems concerning the Jacobian from Riemann's personal perspective. Given the Riemann-Roch theorem, if Riemann's equipment are expressed in sleek language, they vary little or no (if in any respect) from the paintings of recent authors.

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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 by way of Birkhiiuser, of notes of those classes on the ETH.

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**Example text**

Springer-Verlag). We mention only one consequence, a famous theorem of Max Noether. curve in jp'n d -1 = N(n -1) + E. 1 1) + NE. g::; 2N(N -1)(71 - Proof. Let T be a large positive integer. Then h Roch theorem NOETHER'S THEOREM. Let X be a compact Riemann surface of genus g ~ 3. Suppose that X is not hyperelliptic. Then, if J{x is the canonical line bundle of X and m ~ 2, the natural map 1 (( T + lV)D) = 0, and by the Riemann- Proof. vVe consider X C jp'g-l as the canonical curve. Then the hyperplane section D is a canonical divisor, hence deg D = deg J{ = 2g - 2.

U Ifl 2 t W /\ lx UjWk = t(Av(Wj)BvCWk) 8t; + iy, dz /\ dz = -2i 1 w. we have setting J Ifl 2 d:r /\ dy . W = f = Bj(Wk) dz on U, Thus, B is symmetric. Prop. 1, - Bv(Wj)Av(Wk)) v=1 Now, let - Bk(wj) CI, ... EckWk) <0 //=1 1m L CvCkBvk < 0 , CvCkBvk > 0 . v,k Let WI, .. ' ,wg be a basis of HO(X,fJ). Ajk = r lak Wj' Let 1m L v,k = fJvj . , not all 0, and let W = 9 1m Corollary. Av(wj) . I:%=l CkWk· By Given two distinct points P, Q, P of Q, on X, there is a meromorphic I-form 'P on X with simple poles at P and Q and resp('P) = +1, resQ('P) = -1 (by the Mittag-Leffler theorem for I-forms given in §1O).

A holomorphic line bundle on X. If 11' is a vector subspace of IIO(X,L), 11 1= {O}, we call the set of effective divisors {D [D = dives) for some s E 11}, the linear system (or series) determined by V. If V = HO(X, L), we call it the complete linear system of L. If L = L(D) for some divisor D. : O. This is called the complete linear system of D and denoted by IDI. We shall write dim ID[ = hOeD) - 1; it is called the dimension of the complete linear system, and jD[ is in (1 - I)-correspondence with the projective space (HO(X, L(D)) - {O}) ICC* = IP'(IIO (X, L(D))).

### Compact Riemann Surfaces (Lectures in Mathematics. ETH Zurich) by R. Narasimhan

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