Riemann Surfaces

Until now we have covered

Wells Chapter 1, Chapter 4 (sections 2 and 4), Chapter 5 (sections 2 and 5) and Chapter 5 (sections 1, 2 and mentioned the main results of sections 4 and 5).

Counting today’s lecture (25/Apr), we have covered Chapters 5 and 8, Theorems 3 and 4 of Chapter 6 and section 12.1.2 (Line bundles) of Donaldson’s book.

If you are interested in reading further about moduli space of curves, you can start with Looijenga’s notes and follow the references therein.

Wednesday, May 23, 2012 (week 21)

This week we finished our study of the blow-up of point in complex surfaces and saw how this procedure can be used to de-singularize Riemann surfaces in CP^2 which appear as the zero set of polynomials for which the smoothness criterion is not met.

Wednesday, May 16, 2012 (week 20)

This lecture we finished the proof of the result which gives a simple computational way to detect when the zero locus on P^n of a homogeneous polynomial is smooth. We proved the degree--genus formula. In the second half of the lecture, we started our study of the blow-up of points in a complex surface, by explaining the blow-up of C^2 at the origin.

Wednesday, May 9, 2012 (week 19)

This lecture we learnt the Riemann--Hurwitz formula and used it to draw some consequences about non existence of maps between Riemann surfaces. We went on to study level sets of holomorphic functions and proved that for regular values, such set is an embedded complex submanifold. Next we stated a result for when the zero locus on P^n of a homogeneous polynomial is smooth. We will finish the proof of this statement next lecture.

Wednesday, May 2, 2012 (week 18)

Today we finished the formal computations leading to the fact that the tangent space to the moduli space of complex structures on a Riemann surface is expected to be H^1(T^{1,0}S). Then we went on to consider holomorphic maps between Riemann surfaces: proved a normal form lemma, defined the degree of a holomorphic map between compact Riemann surfaces and showed that it does not depend on the regular point taken and that if one counts critical points with appropriate multiplicity, one can also compute the degree of a holomorphic map using singular values. We used these facts to prove a few corollaries, most notably, that there is only one complex structure on P^1.

Wednesday, April 25, 2012 (week 17)

Today we described the fundamental domain which parametrizes complex structures on a torus. Then we made some remarks about the general problem regarding the moduli space of complex structures on a real surface and argued that the tangent space to the moduli space is H^1(T^{1,0}S).

Wednesday, April 18, 2012 (week 16)

Today Andre Henriques studied complex structures on the torus. Using the fact that a compact connected Riemann surface has a nowhere vanishing holomorphic 1-form if and only if it is a torus, he went on to prove that any complex structure on a torus comes from the quotient of C by a lattice and that equivalence between two lattices is given by an action of SL(2,Z) on the upper half plane.

Wednesday, April 11, 2012 (week 15)

Today we introduced divisors of holomorphic line bundles, equivalence of holomorphic line bundles and the Piccard group. We howed that every line bundle over a Riemann surface is the tensor product of line bundles determined by single points because they have a section with a simple zero or pole only at the prescribed point. Then we proved the Riemann--Roch theorem. This material is covered in Donaldson’s chapter 8 and Section 12.1.2 (Line bundles).

Wednesday, April 4, 2012 (week 14)

Today we will prove that the Euler characteristic of a holomorphic line bundle over a Riemann surface can be computed as the number of zeros minus the number of poles of a meromorphic section. Then we will relate the genus of a compact surface with the dimension of its degree one de Rham cohomology.

Wednesday, March 28, 2012 (week 13)

Today we reviewed the algebraic topology behind the degree of a line bundle over a Riemann surface. The important concepts were the degree of a smooth map between compact manifolds and the intersection number of two complimentary dimensional submanifolds.

Wednesday, March 21, 2012 (week 12)

We used harmonic theory to prove Poincare duality in a compact manifold, holomorphic Poincare duality in a compact complex manifold and then Serre duality for a holomorphic bundle over a compact complex manifold. We then stated the Riemann--Roch theorem (for Riemann Surfaces), which will occupy us for the next lecture. Finished the lecture considering a holomorphic line bundle E over a Riemann surface and gave a description of H^1(S,E). In particular we concluded that any holomorphic line bundle over a Riemann surface admits a meromorphic section, if we allow for enough poles.

Wednesday, March 14, 2012 (week 11)

We recalled the concepts of elliptic complex and elliptic operators and the main theorem regarding the cohomology of elliptic complexes. We also introduced the Green operator as an inverse to the Laplacian in the orthogonal complement of the space of harmonic forms. We moved on to prove that in a Riemann surface the d and the delbar Laplacians agree up to a factor of 2. We mentioned that this result remains true for Kaehler manifolds and proceeded to obtain simple topological consequences of this identity of the Laplacians. We showed that in a Kahler manifold, the odd degree cohomology must be even dimensional and, for a Riemann surface S of genus g, H^0(S;T^*1,0) and H^1(S;C) are g dimensional and H^1(S;T^*1,0) is one dimensional.

Wednesday, February 29, 2012 (week 9)

This lecture we finished the proof of the holomorphic Poincare lemma and went on to study elliptic operators and elliptic complexes. We saw that the de Rham complex and the Dolbeault complex are both elliptic. We quoted the main result regarding elliptic complexes, namely that their cohomology is isomorphic to the space of harmonic forms (for an arbitrary choice of metric on each vector bundle making up the complex). Reading material on symbols and elliptic complex can be found in the chapter 4 of Wells book.

Wednesday, February 22, 2012 (week 8)

This lecture we finished off what we were studying last lecture and introduce the delbar operator on sections of vector bundles. Then we moved on to integration on complex manifolds. And proved the holomorphic Poincare lemma in one dimension. The relevant part of this lecture in the setting of surfaces is covered in I.3 and I.4 in Farkas and Kra and Chapter 0 of Griffiths and Harris. Some of the material covered until now for complex manifolds is also present in Chapter 1 of Well’s book.

Wednesday, February 15, 2012 (week 7)

We recalled the different ways to describe a complex structure on a vector space and went on to understand integrability in terms of each of these descriptions. Namely, we saw that integrability of an almost complex structure I on Mn is equivalent any of the following properties:

1. 1)the Nijenhuis tensor vanishes,
   

2. 2)the +i-eigenbundle of I is involutive,
   

3. 3)the exterior derivative maps W1,0(M) into W2,0(M) +W1,1(M),
   

4. 4)the exterior derivative maps Wp,q(M) into Wp+1,q(M) +Wp,q+1(M),
   

5. 5)the exterior derivative maps Wn,0(M) into Wn,1(M),
   

This allowed us to introduce operators del and delbar and we saw that, for a function f, the condition delbar f =0 is equivalent to the requirement that f is holomorphic.

Next we introduced the notion of a holomorphic vector bundle and saw that in a complex manifold T1,0M and T*1,0M are both holomorphic, and hence so are /\ T*1,0M. Finally we argued that a (p,0) form w is a holomorphic section of Wp,0(M) if and only if delbar w =0.

Wednesday, February 8, 2012 (week 6)

In this lecture we reviewed the linear algebra relevant to complex vector spaces. This included different characterizations of complex structures on vector spaces and maps between complex vector spaces. We then reviewed the concept manifolds and introduced the notion of complex manifold. Then we reviewed notions related to vector bundles and introduced the concepts of complex vector bundles and almost complex structures on manifolds. We finished quoting Newlander--Nirenberg theorem stating when an almost complex structure is integrable.