Defining the riemann sphere in terms of the complex. In the spherical model, a projective point correspondsto a pair of antipodalpoints on the sphere. Riemann sphere the projective geometry has been continuously developing. Rotations of the riemann sphere the round sphere s2 fx. In projective geometry, the sphere can be thought of as the complex projective line. This means that a riemann surface is a connected topological space x which is. Cf1g and the only holomorphic map which does not arise this way is the constant map which sends all of xto 1. It is a wellknown theorem that every compact riemann surface admits an embedding into complex projective space. Riemann sphere, projective space november 22, 2014 2. Links between riemann surfaces and algebraic geometry. Complex analysis on riemann surfaces math 2b harvard university c.
The riemann sphere is a complex projective space, which is. Up to a constant factor, this metric agrees with the standard fubinistudy metric on complex projective space of which the riemann sphere is an example. A concrete riemann surface in c2 is a locally closed subset which. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the riemann integral, and his work on fourier series. Stereographic projection, the riemann sphere, and the chordal. This extra point doesnt essentially have anything t. Harmonic maps from the riemann sphere into the complex projective space and the harmonic sequences kawabe, hiroko, kodai mathematical journal, 2010. Complex projective space cp n is also a compactification of c n. Projective structures on a riemann surface, iii 205 immersion condition is slightly subtle. Roughly speaking, projective maps are linear maps up toascalar. Poncelet, 19th century julius placker, steiner, clebsch, riemann, max noether, enriques, segre, severi, schubert, and etc. We discuss how complex projective space for k k the real numbers or the complex numbers equipped with their euclidean metric topology is a topological manifold and naturally carries the structure of a smooth manifold prop. But since i am mostly trained in stochastic analysis a lot of the prelimenaries from differential geometry and.
A remark on defining riemann surfaces 7 the projective line 8. In many questions of the theory of functions, the extended complex plane is identified with the riemann sphere. Riemann, the german 19th century mathematician, devised a way of representing every point on a plane as a point on a sphere. The vector space c2 has a standard inner product hz,wi z. For example, it is an easy exercise to show that the complex projective line p1 and the riemann sphere are isomorphic cf. Using comparison principles such as chows theorem we construct functors from the category of compact riemann surfaces with nonconstant holomorphic maps to the category of smooth projective algebraic curves with regular algebraic maps. Unlike ambitwistor strings, these models have a nonabelian triplet of constraints which can be grouped together to form a nondynamical sl2 gauge eld on the riemann sphere. In topology, a point at infinity is just an extra ideal point. Discrete groups and riemann surfaces anthony weaver july 6, 20. Embedding feynman integral calabiyau geometries in. For example, any rational function on the complex plane can be extended to a holomorphic. This is a short survey about the history of riemann surfaces and the development of such surfaces from bernard riemanns doctoral thesis and some of the later results made by poincar. Riemann sphere plural riemann spheres topology, complex analysis the complex numbers extended with the number the complex plane representation of the complex numbers as a euclidean plane extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3dimensional euclidean space. A threefold categorical equivalence zachary smith abstract.
Any open subset of the riemann sphere p1 is riemann surface, as is the complex torus czi. This page was last edited on 7 january 2019, at 16. Divisors and maps to projective space 153 holomorphic maps to projective space 153 maps to projective space given by meromorphic functions 154 the linear system of a holomorphic map 155. In this course we will be interested in moduli parameter spaces of riemann surfaces, especially the space of all riemann surfaces of xed genus. Discrete groups and riemann surfaces city university of. The complex projective line cp1 for purposes of complex analysis, a better description of a onepoint compacti cation of c is an instance of the complex projective space cpn, a compact space containing cn, described as follows. Rotations of the riemann sphere a rotationof the sphere s2 is a map r rp. The riemann sphere is only a conformal manifold, not a riemannian manifold.
Two approaches to the embedding of riemann surfaces in projective space andy manion abstract. This process is experimental and the keywords may be updated as the learning algorithm improves. This is precisely how space time points are represented in the projective twistor space. Browse other questions tagged riemanniangeometry projective space or ask your. One dimensional projective complex space p c2 is the set of all onedimensional subspaces of c2. The full twistor space is just a 4dimensional complex vector space z 0,z 1,z 2,z. Contents 1 introduction 2 2 the begining 3 3 george friedrich bernhard riemann 18261866 4 4 riemann s surfaces 4 5 the topology of complex projective. The riemann sphere can be visualized as the complex number plane wrapped around a sphere by some form of stereographic projection. Generalised riemann hypothesis complex projective 4space. The condition says that for any point y of the universal cover of x, and for any hyperplane h in cpn. In particular, every riemann surface embeds in projective space.
An element of mx can be viewed as a holomorphic map to the riemann sphere projective line p. Linear equivalence for divisors on the riemann sphere 140. The riemann sphere is a fantastic glimpse of where geometry can take you when you escape from the constraints of euclidean geometry the geometry of circles and lines taught at school. In mathematics, the riemann sphere, named after bernhard riemann, is a model of the. The 2 sphere with its canonical structure of a complex manifold is called the riemann sphere. The space of harmonic twospheres in the unit four sphere bolton, john and woodward, lyndon m.
The most important examples, and the rst to arise, historically, were the graphs of multivalued analytic functions. When harmonic maps from the riemann sphere into the complex projective space are energy bounded, it contains a subsequence converging to a bubble tree map f i. Thus, typically we will restrict to closed connected surfaces, but for context it. One of the basic notions is riemann sphere one dimensional complex projective space. It can be considered to be a projective space represented by a pair of complex coordinates, x, y. Full holomorphic maps from the riemann sphere to complex projective spaces. Strictly speaking this is more than just an extension as a topological space, since we understand not only what it means for a complex valued function fon the open neighborhood jzj rof 1to be continuous, but also what it means to be there holomorphic the answer is.
The riemann sphere ib maths resources from british. For the product of two di erentiable manifolds we have the following. Slightly more complicated amplitudes can be described in terms of elliptic multiple polylogarithms, which can be understood as iterated integrals over the moduli space of the torus. Now it is time to introduce twistors into the story. It is also called the complex projective line and denoted by p1. In other words, the correspondence determines a differentiable imbedding of the onedimensional complex projective space into the space in the form of the sphere. Mcmullen proved that the moduli spaces of riemann surfaces are k. We have seen that a space time point can be represented as a riemann sphere in terms of some section of its light cone. Projective space projective geometry projective line riemann sphere linear isomorphism these keywords were added by machine and not by the authors. All structured data from the file and property namespaces is available under the creative commons cc0 license. What is, in basic terms, the relationship between riemann surfaces and algebraic geometry. Roughly speaking,projective maps are linear maps up toascalar. Riemann surfaces in complex projective spaces article pdf available in proceedings of the american mathematical society 322.
Files are available under licenses specified on their description page. The simplest compact riemann surface is cb c1with charts u 1 c and u 2 cbf 0gwith f 1z zand f. Equivalently, a projective structure is given by a p1. When n 1, the complex projective space cp 1 is the riemann sphere, and when n 2, cp 2 is the complex projective plane see there for a more. If the generalised riemann hypothesis holds, it is possible to find a reasonably small prime p for which there is such a modulop representation. The riemann sphere we can augment the complex plane by adding an extra point, to the plane. To this end, consider the stereographic projection from the unit sphere minus the point 0, 0, 1 onto the plane z 0, which we identify with the complex plane by. However, if one needs to do riemannian geometry on the riemann sphere, the round metric is a natural choice with any fixed radius, though radius 1 is the simplest and most common choice. Nov 05, 2017 if the generalised riemann hypothesis holds, it is possible to find a reasonably small prime p for which there is such a modulop representation. Algebraic curves and riemann surfaces rick miranda.
By viewing the real projective nspace irip as the real points of cp, we. It turns out that while not every compact complex manifold embeds in projective space, many do, and this is the content of the kodaira embedding theorem. An abstract riemann surface is a surface a real, 2dimensional manifold with a good notion of complexanalytic functions. Mar 14, 2017 space time cfts from the riemann sphere. Indeed, c nsits inside complex projective space p, a compact space, but it is missing all the points at in. It can be considered to be a projective space represented by a pair of complex coordinates, x, y, where scalar complex multiples are considered equivalent. Im taking introductory courses in both riemann surfaces and algebraic geometry this term. One dimensional projective complex space pc2 is the set of all onedimensional subspaces of c2. Harmonic maps from surfaces to complex projective spaces core. The geodesic ow on a hyperbolic surface is an excellent concrete example of a chaotic ergodic, mixing dynamical system. Riemann surface mp of genus p to the sphere m, is holomorphic, provided its degree. Out of curiosity i have started a tutorial in riemann surfaces. We start by considering rationalmaps between riemann spheres.
The riemann sphere and stereographic projection the initial and naive idea of the extended complex plane is that one adjoins to the complex plane ca new point, called 1 and decrees that a sequence znof complex numbers converges to 1if and only if the real sequence jznjtends to 1in the usual sense. What is the nature of the point at infinity on the. This class of functions has been the focus of a great deal of recent. The riemann sphere is a canonical example of a riemann surface. I was surprised to hear that any compact riemann surface is a projective variety. M n between topological spaces is a covering if it is. Algebraic curves and riemann surfaces researchgate. A projective structure on a compact riemann surface c of genus g is given by an atlas with transition functions in pgl2,c. The riemann sphere as a stereographic projection wolfram. A riemann surface is a connected, hausdor topological space xequipped with an open covering u i and a collection of homeomorphisms f i. The automorphisms form a group g autx under composition. Contents 1 introduction 2 2 the begining 3 3 george friedrich bernhard riemann 18261866 4 4 riemanns surfaces 4 5 the topology of complex projective. Computation of ld for the riemann sphere 149 computation of ld for a complex torus 150 a bound on the dimension of ld 151 problems v.
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