Riemann Surfaces

Exercise sheets and other homework

hand-in exercise sheet 1

hand-in exercise sheet 2

hand-in exercise sheet 3

Announcements

The final marks are out. If you want to take a look at your exam and collect your homework, please come to my office

Chapters covered so far: 1, 2, 3, 4, 5, 7, 8, 12 and parts of 6 (thm 3 and 4) and 13.

Thursday, December 9, 2010 (week 49)

We started the lecture recalling the definition of sheaves, maps of sheaves and exact sequences. We also recalled the definition of the Čech coboundary operator and defined Čech cohomology with coefficients on a sheaf F. We then computed the groups Ȟ^k(M,F) for the simplest cases. Namely, we showed that Ȟ^0(M,F) corresponds to global sections of F and showed that if F is a fine sheaf (e.g., C∞-functions, C∞-forms or, more generally, C^∞-sections of any vector bundle) then Ȟ^k(M,F) = {0} for k>0. Next we computed  continued our study of sheaves and Čech cohomology. We showed that Čech cohomology with coefficients on the constant sheaf C is isomorphic to de Rahm cohomology with complex coefficients. A critical look at the proof showed that the fundamental property used was that the sequence of sheaves

0 → C → Ω^0(M; C) → Ω^1(M;C) →Ω^2(M;C)→...

is exact (Poincaré duality) and the sheaves Ω^i(M;C) have no Čech cohomology is degree higher than 0. This led us to the concept of sheaf resolution and a general theorem of how to compute sheaf cohomology. In the particular case of holomorphic line bundles, Čech cohomology with coefficients in holomorphic sections of L agrees with the cohomology determined by a choice of holomorphic connection on L.

We used sheaves to prove again the Riemann Roch theorem, now in the form

dim H^0(M;L) - dim H^1(M;L) = d-g+1,

where d is the degree of the line bundle L and g is the genus of the Riemann surface over which L is defined. This led us to observe that

dim H^1(M;L) = dim H^0(M; K⊗L*)

and I mentioned without proof that there is a nondegenerate pairing between these spaces:

H^0(M; K⊗L*)  × H^1(M;L) → H^1(M;K) = H^1,1(M) ≈ C.

This is known as Serre Duality.

Finally we saw a formal argument using Čech cohomology showing that the space of deformations of a Riemann surface M has dimension H^1(M,TM).

Thursday, December 2, 2010 (week 48)

This lecture we recalled the concept of vector bundle and defined connections on vector bundles. We went on to define holomorphic connections on holomorphic vector bundles and remarked that while for an usual connection ∇, ∇^2 is the curvature of the bundle and hence ∇ does not in general yield a complex, for holomorphic connections, the (0,1) of the connection agrees with ∂bar, and squares to zero, so it allows us to define cohomology with coefficients in a holomorphic vector bundle.

Shifting gears we defined sheaves, which are generalizations of the concept of “space of local sections of a vector bundle”. Examples of sheaves were C∞-functions, C∞-forms or, more generally, C^∞-sections of any vector bundle, holomorphic functions, holomorphic forms and, in general holomorphic sections of any holomorphic vector bundle. Examples of sheaves which do not arise as sections of vector bundles where the skyscraper sheaf and the constant sheaf with coefficients in C.

In another gear shift, we finished the lecture introducing the Čech coboundary operator and defined Čech cohomology.

Thursday, November 25, 2010 b (week 47)

This lecture we recalled that, on a compact manifold H^0_∂bar (M) corresponds to holomorphic functions (hence constants) and H^{1,0}_∂bar(M) corresponds to holomorphic (1,0)-forms. The question then was what H^{0,1} is. We saw that H^{0,1} corresponds to obstructions to finding meromorphic functions with prescribed poles, or in other terms, finding sections of lines bundles associated to a positive divisor.

Next we stated and gave applications of the “main theorem of Riemann surfaces”, namelly, that Riemann surfaces satisfy the ∂∂bar-lemma. We saw that if a complex manifold satsifies the ∂∂bar-lemma, then the decomposition of the space of forms into Ω^{p,q} induces a decomposition of the   cohomology and Poincaré duality means that there is a nondegenerate pairing:

H^{p,q} × H^{n-p,n-q} → C.

Also conjugation gives an antilinear isomorphism between H^{p,q} and H^{q,p}. In particular we concluded that for a Riemann surface of genus g

dim H^{1,0} =dim H^{0,1} = 1/2 dim H^1(M;C) = g.

In particular given a complex structure in CP^1, there is no obstruction to finding a meromorphic function on CP^1 with a simple pole, hence tat complex structure is the standard one. I.e. CP^1 admits only one complex structure.

Also for any complex structure on the torus, H^{1,0}(T^2) = C, hence it admits a holomorphic 1-form. From the Euler characteristic we see that this form can not vanish and from previous results we conclude that the given complex structure is equivalent to that of C/Λ, for some lattice Λ.

Finally we proved the Riemann Roch theorem, namelly, that for a holomorphic line bundle L

h_0(L) - h_0(K⊗L*) = d - g + 1

where h_0 of a line bundle denotes the dimension of the space of sections of that line bundle, d is the degree of L (i.e., its Euler characteristic) and g is the genus of the base Riemann surface.

Thursday, November 25, 2010 a (week 47)

This lecture we took another look at the line bundle Lp over a Riemann surface S and defined a holomorphic line bundle for any collection of points in S decorated with integer coefficients. We saw that a section of such line bundle was in correspondence with meromorphic functions on S with at worse poles and zeros of a certain prescribed order and that the given points and coefficients did not uniquely determine the line bundle, but could instead be changed by points (with coefficients) arising from meromorphic functions on S. The corresponding quotient: Divisors/(zeros/poles of meromorphic functions) is the space of holomorphic line bundles over S (we claimed but did not prove that any line bundle admits a meromorphic section). Next we studied tensor products of line bundles and the corresponding operation on divisors.

Thursday, November 18, 2010 (week 46)

This lecture we finished our study of the Euler characteristic. We proved that given a meromorphic vector field X on a compact Riemann surface S, the Euler characteristic of S is given by the difference between the number of zeros and number of poles of X counted with multiplicity. Similarly, the Euler characteristic of S can be computed using a meromorphic 1,0 form as the difference between the number of poles and the number of zeros of the meromorphic form.

As a last application of Euler characteristic, we proved that the zero set of a homogenous polynomial of degree d in C^3 corresponds to a Riemann surface of genus 1/2 (d-1)(d-2).

In the remainder of the lecture we started our study of holomorphic vector bundles and, in particular, holomorphic line bundles. We remarkd that the same argument given before to prove that the Euler characteristic if a surface S is the number of zeros of a meromorphic section of TS minus the number of poles of that section can be used to define the Euler characteristic of any holomorphic line bundle L over S. This number is called the degree of L. Finally we introduced the line bundle corresponding to a point p in Riemann surface S as the line bundle which has a section with a simple zero at p. Other sections of L therefore correspond to meromorphic functions on S which have (at worse) a simple pole at p. This material is covered in chapter 8 and 12

Thursday, November 11, 2010 (week 45)

This lecture we covered intersection numbers of oriented submanifolds. We followed that by a quick study of real and complex vector bundles and how a collection of  transition functions satisfying a skew symmetry condition and a cocycle condition give rise to a vector bundle. We then defined the Euler characteristic of a vector bundle whose rank is the same as the dimension of the base manifold as the (self) intersection number of the zero section. The Euler characteristic of a manifold M was defined to be the Euler characteristic of TM, which agrees with the usual definition in terms of triangulations. For a Riemann surface we could specialize to the case of a holomorphic vector field, X, and saw that each zero of multiplicity k of X contributes with k to the Euler characteristic. In particular the only compact Riemann surfaces which may admit holomorphic vector fields are diffeomorphic to the sphere (in which case the vector field must have two zeros) or the torus (in which case the vector field does not vanish).

Thursday, November 4, 2010 (week 44)

This lecture we proved the main part of the theorem describing the moduli space of complex structures on a torus. The results in question stated that if a compact connected Riemann surface X admits a nowhere vanishing holomorphic (1,0)-form then that surface is equivalent to a quotient of C  by a lattice and hence X, as a differentiable manifold, is a torus. This was followed by a classification of all nonequivalent lattices. Later in this course we will see that any complex structure on a torus admits a nowhere vanishing holomorphic form so these results in practice describe the moduli space of complex structures on a torus.

As an aside, we will skip the rest of the material in Chapter 6. Just be aware that if the connection between this course and the “Jacobians and Theta functions” is not clear to you yet, you may want to read the skipped parts.

Thursday, October 28, 2010 (week 43)

This week Andre finished chapter 5 and startet to studied the moduli space of complex structures on a torus. We saw that if a Riemann surface admits a nowhere vanishing holomorphic 1-form then that Riemann surface is equivalent to the quotient of C^2 by a lattice and hence it is topologically a torus.

Thursday, October 21, 2010 (week 42)

We continued our study of calculus on surfaces. We defined de Rham cohomology and computed the first cohomology of a bunch of simple surfaces (R^2, S^2, C*, T^2 and surfaces of genus g). We introduced cohomology with compact support and proved that on a compact surface there is a nondegenerate pairing on H^1. We finished the lecture introducing the concept of an almost complex structures, i.e., a bundle map I of the tangent space such that I^2 = -Id. For Riemann surfaces, a choice of almost complex structure is equivalent to a choice of complex structure. In higher dimensions, an almost complex structure is a pre requisite to the existence of a complex structure.

Thursday, October 14, 2010 (week 41)

This week we started our study of calculus on Riemann surfaces. We defined tangent and cotangent bundles as well as vector fields, forms and exterior derivative.

Thursday, October 7, 2010 (week 40)

This week we studied how to use Riemann existence theorem to desingularize/compactify embedded Riemann surfaces. This was studied in detail for zeros of polynomials on C^2 (and  CP^2).

Thursday, September 30, 2010 (week 39)

This week we used the degree of a holomorphic function f: X --> Y and its branch number to relate the topology of X and Y. Then we moved on to prove Riemann existence theorem for holomorphic functions with prescribed branching points and monodromy.

Thursday, September 23, 2010 (week 38)

We started our study of maps between Riemann surfaces. We showed that given a holomorphic map between Riemann surfaces f: X --> Y and a point x in X with y = f(x), we can choose coordinates z and w around x and y with x and y corresponding ot the origin such that f is given by w = z^k. Following that, we defined the degree of a proper holomorphic map at a point y in Y and showed that this degree is independent of y.

Thursday, September 16, 2010 (week 37)

We started recalling that CP^1 is a Riemann surface and proved that CP^n is a complex manifold. Then we saw how Riemann surfaces arise as quotients of other Riemann surfaces by the “properly discontinuous action” of a discrete group. We recalled that holomorphic maps with nonzero derivative of C are equivalent to conformal maps of R^2 and saw that surfaces of revolution are examples of Riemann surfaces. Then we started our study of functions between Riemann surfaces. The first few results were that locally every holomorphic function looks like the map z -- > z^k for a unique positive natural number k. The number k is the branching number of the point in question.

Thursday, September 9, 2010 (week 36)

In this lecture I gave a very, very brief indication of what is covered in the first two chapters of Donaldson’s book and went on to define and give examples of Riemann surfaces. We saw that the sphere parametrized by stereographic projections is a Riemann Surface. We also proved the “holomorphic implicit function theorem”. Our version of the theorem stated that a regular level set of a holomorphic function f:C^2 --> C is a Riemann surface, but we remarked at the end that essentially the same proof could be used to show that regular level sets of a holomorphic function f:M^n --> N^{n-1} are Riemann surfaces. We finished showing that CP^1, the set of complex lines through the origin on C^2 is a Riemann surface and in fact it is the same Riemann surface as the sphere with complex structure given by stereographic projections.