### An Introduction to Riemann Surfaces

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We first consider conditions that guarantee the existence of holomorphic sections with prescribed values. After the above considerations, we consider the fact that a holomorphic line bundle is positive if and only if its degree is positive, and we then consider finiteness of Dolbeault cohomology.

The Riemann—Roch formula is then proved using finiteness. Finally, we consider the Serre duality theorem and the Hodge decomposition theorem, and some of their consequences. The first goal is the following Riemann surface analogue of the classical Riemann mapping theorem in the plane:. In particular, for X compact, this allows one to form a canonical homology basis.

## ISBN 13: 9783540501244

In this chapter, we address the natural problem of determining conditions for a topological surface to admit a holomorphic structure. One necessary condition is, of course, that the surface be orientable. It turns out that these two conditions are also sufficient. In this chapter, we recall some basic definitions and facts concerning integration and Hilbert spaces.

In this chapter, we recall some basic definitions and facts concerning exterior products which are essential in the discussion of differential forms in Sect. We take the domain of a path to be [0,1], unless otherwise indicated. This looks like four constraints, but is in fact essentially just three, due to the Euler identity. With this hypothesis, we call the curve smooth.

Now suppose is a point in ; without loss of generality we may take non-zero, and then we can normalise. Now one can think of as an inhomogeneous polynomial in just two variables , and by nondegeneracy we see that the gradient is non-zero whenever. By the complexified implicit function theorem, this ensures that the affine algebraic curve.

This can be used to give a coordinate chart for in a neighbourhood of when. Similarly when is non-zero. This can be shown to give an atlas on , which assuming the connectedness claim that we will prove later gives the structure of a Riemann surface. Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface. Exercise 9 If are complex numbers, show that the projective cubic curve.

When this occurs, the curve is called an elliptic curve in Weierstrass form , which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.

In complex analysis, there are two basic types of good functions:. Definition 10 Let be a Riemann surface.

A holomorphic function on is a holomorphic map from to ; the space of all such functions will be denoted. A meromorphic function on is a holomorphic map from to the Riemann sphere , that is not identically equal to ; the space of all such functions will be denoted. One can also define holomorphicity and meromorphicity in terms of charts: a function is holomorphic if and only if, for any chart , the map is holomorphic in the usual complex analysis sense; similarly, a function is meromorphic if and only if the preimage is discrete otherwise, by analytic continuation and the connectedness of , will be identically equal to and for any chart , the map becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite.

One consequence of this alternate definition is that the space of holomorphic functions is a commutative complex algebra a complex vector space closed under pointwise multiplication , while the space of meromorphic functions is a complex field a commutative complex algebra where every non-zero element has an inverse. Another consequence is that one can define the notion of a zero of given order , or a pole of order , for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting from the holomorphicity of transition maps and their inverses that this does not depend on the choice of chart.

However, one cannot similarly define the residue of a meromorphic function on this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic -forms, a concept we will introduce later. A third consequence is analytic continuation: if two holomorphic or meromorphic functions on agree on a non-empty open set, then they agree everywhere.

On the complex numbers , there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions with non-zero denominator will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem covered in previous quarters.

It turns out, however, that the situation changes dramatically when the Riemann surface is compact , with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:. Lemma 11 Let be a holomorphic function on a compact Riemann surface.

Then is constant. Proof: As is continuous and is compact, must attain a maximum at some point.

Working in a chart around and applying the maximum principle , we conclude that is constant in a neighbourhood of , and hence is constant everywhere by analytic continuation. This dramatically cuts down the number of possible meromorphic functions — indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional.

The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic -forms also known as Abelian differentials. AP , math. CV , math. DG , math. MG , math. As usual, all inaccuracies in these notes are due to myself and not to Yau, and I welcome corrections or comments.

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Tag Archive. For sake of completeness we give the somewhat lengthy formal definition: Definition 1 Riemann surface If is a Hausdorff connected topological space, a one-dimensional complex atlas is a collection of homeomorphisms from open subsets of that cover to open subsets of the complex numbers , such that the transition maps defined by are all holomorphic. Here are some basic examples of Riemann surfaces. Example 6 Smooth algebraic plane curves Let be a complex polynomial in three variables which is homogeneous of some degree , thus Define the complex projective plane to be the punctured space quotiented out by non-zero complex dilations, with the usual quotient topology.

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## Enno Keßler and Ruijun Wu: Riemann surfaces, Teichmüller theory and harmonic maps

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