Compact Riemann Surfaces

126 Pages·1992·3.575 MB·English

by  Raghavan Narasimhan (auth.)

Lectures in Mathematics
ETH Ziirich
Department of Mathematics
Research Institute of Mathematics
Managing Editor:
Oscar E. Lanford
Raghavan Narasimhan
Compact Riemann
Surfaces
Springer Basel AG
Author's address:
Raghavan Narasimhan
Department of Mathematics
University of Chicago
Chicago, IL 60637
USA
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Narasimhan, Raghavan:
Compact Riemann surfaces / Raghavan Narasimhan. -
Springer Base! AG, 1992
(Lectures in mathematics)
ISBN 978-3-7643-2742-2 ISBN 978-3-0348-8617-8 (eBook)
DOI 10.1007/978-3-0348-8617-8
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First reprint !996
© 1992 Springer Base! AG
Originally published by Birkhauser Verlag in 1992
produced from chlorine-free pulp. TCF
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ISBN 978-3-7643-2742-2
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Preface
These notes form the contents of a Nachdiplomvorlesung given at the Forschungs
institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from
November, 1984 to February, 1985. Prof. K. Chandrasekharan and Prof. Jurgen
Moser have encouraged me to write them up for inclusion in the series, published by
Birkhiiuser, of notes of these courses at the ETH.
Dr. Albert Stadler produced detailed notes of the first part of this course, and very
intelligible class-room notes of the rest. Without this work of Dr. Stadler, these notes
would not have been written. While I have changed some things (such as the proof of
the Serre duality theorem, here done entirely in the spirit of Serre's original paper), the
present notes follow Dr. Stadler's fairly closely.
My original aim in giving the course was twofold. I wanted to present the basic theorems
about the Jacobian from Riemann's own point of view. Given the Riemann-Roch
theorem, if Riemann's methods are expressed in modern language, they differ very little
(if at all) from the work of modern authors.
I had hoped to follow this with some of the extensive work relating theta functions and
the geometry of algebraic curves to solutions of certain non-linear partial differential
equations (in particular KdV and KP). Time did not permit pursuing this subject, and
I have contented myself with a couple of references in §17. These references fail to cover
much other important work (especially of M. Mulase) but I have not tried to do better
because the literature is so extensive.
It is a great pleasure to express my thanks to the ETH for its hospitality, to Prof. J.
Moser for his encouragement, and to Dr. A. Stadler for the enormous amount of work
he undertook which made these notes easier to write. But special thanks are due to
Prof. K. Chandrasekharan. But for him, I would not have been at the ETH, nor would
these notes have been written without his advice and encouragement.
Chicago, August 1991 R. Narasimhan
Contents
1. Algebraic functions 3
2. Riemann surfaces.. .... ............ ........ ..... .......... ............. .. ... 8
3. The sheaf of germs of holomorphic functions ................................ 12
4. The Riemann surface of an algebraic function ............................... 15
5. Sheaves .................................................................... 17
6. Vector bundles, line bundles and divisors ................................... 27
7. Finiteness theorems ........................................................ 32
8. The Dolbeault isomorphism ................................................ 38
9. Weyl's lemma and the Serre duality theorem...... .................. ........ 43
10. The Riemann-Roch theorem and some applications.. . .......... ............ 49
11. Further properties of compact Riemann surfaces ............................ 58
12. Hyperelliptic curves and the canonical map ................................. 63
13. Some geometry of curves in projective space ................................ 66
14. Bilinear relations ........................................................... 77
15. The Jacobian and Abel's theorem.... ..................... ... ............... 84
16. The Riemann theta function .............................. '" .. .. ........... 91
17. The theta divisor ........................................................... 97
18. Torelli's theorem ........................................................... 106
e ................................
19. Riemann's theorem on the singularities of 111
References ................................................................. 119
1. Algebraic Functions
Let FE ([[x, y] be an irreducible polynomial in two variables (with complex coefficients).
We assume that its degree in y is ;::: 1.
Recall that by the so-called Gauss lemma, if we identify ([[x, y] with ([[x][y], and if F is
irreducible, it is also irreducible in C(x)[y], the polynomial ring over the field ofrational
functions in x. Moreover, ([[x, y] is a factorial ring (i.e. a unique factorisation domain).
An algebraic function is, intuitively, "defined" by an equation F(x, y) = 0 (where F is
irreducible in ([[x, y]).
To make this statement more precise, we begin with the following.
The implicit function theorem. Let f be a holomorphic function of two complex
variables X,y defined on {(x,y) E C2 11xl < rl, Iyl < r2}, rl,r2 > O. Assume that
of
f(O,O) = 0, oy (0, 0) # 0 .
Then, there exist positive numbers E, 0 > 0 such that for any x E Dc = {z E C Ilzl < 1o},
there is a unique solution y(x) of the equation f(x, y) = 0 with ly(x)1 < O. The function
x y(x) is holomorphic on Dc.
f--+
Proof. Since %(0,0) # 0, we can choose 0 > 0 such that f(O,y) # 0 for 0 < Iyl :s o.
Choose now 10 > 0 such that f(x, y) # 0 for Ixl :s 10, Iyl = 0 (possible since f is non-zero
on the compact set {O} x {y Ilyl = o}).
By the argument principle, if Ixl < E,
J
2~i {~~(x,Y)/f(x,Y)}dY
iYi=b
is an integer n(x) equal to the number of zeros of the function y f--+ f(x,y) in Iyl < 0;
by our choice of 0, nCO) = 1. On the other hand, since f(x, y) # 0 for Ixl :s 10, Iyl = 0,
the integrand, and thus also the integral, is a continuous function of x for Ixl < E. Thus
n(x) = 1 for Ixl < 10, which means precisely that there is a unique zero y(x) of f(x,y)
with ly(x)1 < O.
That x y( x) is holomorphic follows from the formula
f--+
J
1 %(x,y)
y(x) = 2Jri y f(x, y) dy
iyi=b
4 1. Algebraic Functions
(which is an immediate consequence of the residue theorem).
Let F(x, y) = ao(x)yn + al(X)yn-l + ... + an(x) E qx, y] be an irreducible polynomial
with n ~ 1; the polynomials ao, ... , an E qx] have no non-constant common factor
since F is irreducible.
Lemma 1. Let a E C be such that ao ( a) =F 0 and such that there is no bEe
with F( a, b) = 0 = ~~ (a, b). Then, there is f > 0 and n holomorphic functions
Yl (x), ... ,Yn(x) in the disc {x E C Ilx - al < f} with the following properties:
(i) Yi(X) =F Yj(x') if i =F j, Ix - al < f, lx' - al < f; moreover
F(X,Yi(X))==O for Ix-al<f, i=l, ... ,n.
(ii) if'T) E C and F(x, 'T)) = 0, Ix - al < f, then 'T) = Yi(X) for a unique i between 1
and n.
Proof. Since ~~ (a, b) =F 0 for all solutions b of F( a, b) = 0, the polynomial F( a, y)
has exactly n roots b1, ... , bn. If f > 0 is small and Yi (x) the holomorphic function on
Ix - al < f with Yi(a) = bi and F(x, Yi(X)) == 0 (which exists by the theorem above),
then the Yi have property (i) if f is small enough, and property (ii) since the equation
F(x, 'T)) = 0 has at most n solutions.
Proposition 1. Let F E qx, y] be irreducible. There are only finitely many x E C
such that the equations
of
F(x,y) = 0 = oy (x,y)
have a simultaneous solution Y E !C.
Proof. By the division algorithm, there are polynomials bi E qx] (i ~ 0) with bo =
ao [F = ao(x)yn + ... + an(x)] and polynomials Aj, Qj E qx, y] (j ~ 1) such that
of
deg Ql < deg oy = n - 1
y y
deg Q2 < deg Ql
y y
We may suppose that degy Qk = 0, i.e. that Qk E qx] (since we can otherwise continue
the division process). We claim now that Qk(X) =t=. O. If, in fact, Qk == 0, then from
the last of the above equations, any prime factor P of Qk-l with deg P > 0 would
y
divide bk-1Qk-2, hence Qk-2 (since bk-1 E qx] and deg P > 0). From the equation
y
1. Algebraic Functions 5
bk-2Qk-3 = Ak-1 Qk-2 + Qk-l, it would follow that P divides bk-2Qk-3 and hence
Qk-3. Repeating this argument, P would divide all the Qj (j ~ 1), hence also ~; and
F, contradicting the irreducibility of F. Thus Qk = Qk(X) E IC[x] is "t o.
If now a, bEe and F(a, b) = 0 = ~~ (a, b), we see from the above equations that
Ql(a, b) = 0, then that Q2(a, b) = 0, ... , Qk(a, b) = Qk(a) = O. Since Qk "t 0, the set
{x ECI:Jy E C with F(x,y) = 0 = ~: (x,y)} C {x Eel Qk(X) = o}
is finite.
Before proceeding further, we insert some toplogical preliminaries. All topological spaces
we consider will be Hausdorff.
Definition. A continuous map p : X ----+ Y, where X, Yare locally compact (Hausdorff)
spaces, will be called proper if, for any compact set KeY, the inverse image p-l(K)
is compact in X
Lemma 2. If X, Yare locally compact, a proper map p : X ----+ Y is necessarily closed,
i. e. takes closed sets in X to closed sets in Y.
Proof. Let A C X be closed, and Yo E Y. Let K be a compact neighbourhood of Yo
in Y. Then p(A) n K = p(A n p-l(K)) is compact (since A is closed and p-l(K) is
compact), hence closed in K.
Remark. A continuous map p : X ----+ Y between locally compact spaces X, Y is proper,
if and only if, for any locally compact topological space Z, the product
p x idz : X x Z ---+ Y x Z, (x,z) f------+ (p(x),z)
is closed. If X, Y have countable bases, this can be seen by using the following remark:
if {Xl, ... ,Xn, ... } is a sequence of points in X, without limit points and such that
{P(Xn)} n>l converges in Y, then the image of the closed set {(xn, ~) I n ~ I} in X x lR
is not closed in Y x R
The property in this remark can be used to define proper mappings between spaces
which are not locally compact.
Remark. Let p : X ----+ Y be a proper map between locally compact spaces. Let Z C Y
pi
be a locally compact space (with the induced topology). Then p-l(Z) : p-l(Z) ----+ Z
is again proper.
In fact, a compact subset of Z is a compact subset of Y.
6 1. Algebraic Functions
Lemma 3. Let Cl, ... ,Cn E!C. Let w E C and suppose that wn + Cl wn-l + ... + Cn = O.
Then
Iwl < 2maxlcvll/v
v
(unless Cl = ... = Cn = 0).
Proof. Let C = maxv Icvll/v > O. If z =!!!c. ' we have zn + .£c l.zn-l + ... + £en. n= 0, so
that, since Icv I :::; cV,
Izln :::; Izln- l + ... + 1 .
If Izl 2: 2, we would have 1 :::; 1;1 + ... + Izln :::; ! + ... + 2~ < 1, a contradiction. Thus
Izl < 2, i.e. Iwl < 2c.
Proposition 2. Let F E qx, y], F(x, y) = ao(x)yn + ... + an(x), ao :t. o. Let V =
I I 7r :
{(x,y) E C2 F(x,y) = O} and So = {x E C ao(x) = O}. Let V --> C be the
7r 17r-
projection (x, y) ....... x. Then l (C - So) --> C - So is proper.
Proof. Let K C C - So be compact. Then there is {j > 0 so that lao(x)1 2: {j and
lav(x)l:::; i for x E K. If (x, y) E V, x E 7r-l(K), we have
y n + -al-(xy) n -l + ... + -an(-x) = 0 ,
ao(x) ao(x)
so that, by (1.8), Iyl :::; 2maxv (j-2/v. Thus 7r-l(K) is bounded. Since clearly 7r-l(K) =
(K x q n V is closed in C2, 7r-l(K) is compact.
Definition. Let X, Y be (Hausdorff) topological spaces and p : X --> Y, a continuous
map. p is called a covering map if the following holds: \:Iyo E Y, there is an open
neighbourhood V of Yo such that p-l(V) is a disjoint union UjEJ Uj of open sets Uj
pi
with the property that Uj is a homeomorphism onto V \:Ij E J. The triple (X, Y,p)
is then called an (unramified) covering. We also say that X is a covering of Y.
An open set V C Y with the property in the definition is said to be evenly covered by
p.
It follows from the definition that the cardinality of p-l(y) is a locally constant function
on Y. (With the notation in the definition, the cardinality Ofp-l(y) is that of J\:Iy E V.)
Thus, if Y is connected, "the number of points" in p-l(y) is independent of y E Y. The
covering is said to be finte (infinite) if the cardinality of p-l(y) is finite (infinite). pis
called an n sheeted covering if p-l(y) contains exactly n points for y E Y.
If p : X --> Y, p : X --> T are two coverings of Y, they are said to be isomorphic if there
exists a homeomorphism r.p : X' --> X such that po r.p = p'.
Examples. 1) Let .6. = {z E C Ilzl < I} and .6.* = .6. - {O}. Then, if n 2: 1, the map
Pn : .6.* --> .6.* given by Pn (z) = zn is an n-sheeted covering.