Cfrom a riemann surface minto cis called a holomorphic function on m. Riemann surfaces oxford graduate texts in mathematics series by simon donaldson. Research current work phd thesis articles master thesis extras history papers courses own work conferences links contact home. A fundamental theorem of abeljacobi riemann says that there is a canonical. Thursday, october 14, 2010 week 41 this week we started our study of calculus on riemann surfaces. Automorphisms of the complex plane c and the disk d polygons and upper half plane h. Donaldson gave another proof using differential geometry, and showed that the stable vector bundles. In the second half of the lecture, we started our study of the blowup of points in a complex surface, by explaining the blowup of c2 at the origin. Seifert fibred homology 3spheres and the yangmills. 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. Department of pure mathematics and mathematical statistics, university of cambridge. In this paper we study a class of riemann hilbert problems arising naturally in donaldson thomas theory. Download course materials all images created with matlab software infinite spiral staircaselike riemann surface for the logarithm. Riemann surfaces are central in mathematics because of the multiple connections between complex analysis, algebraic geometry, hyperbolic geometry, group theory, topology etc.
Basic properties of complex numbers, complex differentiability, conformality, contour integration, zeros and poles, application to evaluation of definite real integrals, local and global properties, convergence in function theory, dirichlets problem, periodic functions. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. Riemann surfaces graduate texts in mathematics graduate. Riemann surfaces can be thought of as deformed versions of the complex plane. Algebraic functions 11 let pz,w be a polynomial in two complex variables. Department of pure mathematics and mathematical statistics. However, his riemann surface is also a riemann surface of a square root function albeit one with a more complicated form than the core w z 12 example and, hence, he also wraps it up as a donut indeed, instead of a sphere or something else. In particular, it includes pretty much all the analysis to prove finitedimensionality of sheaf cohomology on a compact riemann surface. This expression generalizes a theorem of morgan and szabo for g1 to any genus g. This text covers riemann surface theory from elementary aspects to the fontiers of current research. This is an outstanding introduction to the modern study of riemann surfaces. Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics.
They involve maps from the complex plane to an algebraic torus, with prescribed discontinuities along a collection of rays, and are closely related to the riemann hilbert problems considered by gaiotto et al. We also give a criterion in terms of the continuous linear functionals on the space bd that a riemann surface be hyperbolic theorem 4. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The open set u is called the domain of the chart c. A divisor is an expression x x2x a xx where the coe cients a x are integers, and only a nite number of them are nonzero.
This case of the narasimhanseshadri theorem says that a degree zero holomorphic vector bundle over a riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the riemann surface. The pictures here show two views of this surface, whose construction is explained next. Use features like bookmarks, note taking and highlighting while reading riemann surfaces oxford graduate texts in mathematics book 22. In higher dimensions, an almost complex structure is a pre requisite to the existence of a complex structure. The topis is wide and well described in literature, so our aim was to bring some avour of it. A simplyconnected riemann surface of parabolic type is necessarily the riemann surface of the function inverse to a meromorphic function in the finite plane. Basic idea of riemann surface theory is to replace the domain of a. For riemann surfaces, a choice of almost complex structure is equivalent to a choice of complex structure. In mathematics, particularly in complex analysis, a riemann surface is a onedimensional complex manifold. The topis is wide and well described in literature, so our aim was to bring some avour. The theory of riemann surfaces occupies a very special place in mathematics. Coverings of riemann surfaces and the universal covering study proper nonconstant holomorphic maps between riemann surfaces for91, 4. These surfaces were first studied by and are named after bernhard riemann. Contents i preliminaries 7 1 holomorphic functions 9.
Open and closed surfaces are treated with emphasis on the compact case, while basic tools are developed to describe the analytic, geometric, and algebraic properties of riemann surfaces and the associated abelian varities. Links between riemann surfaces and algebraic geometry. Prime riemann surface theta function form function. Finite riemann surfaces are topologically completely characterized by the genus, and the number of connected components of the boundary. Riemann surfaces oxford graduate texts in mathematics book 22 kindle edition by donaldson, simon. In 6, a direct application of the lattice reduction technique of 3 led to explicit expressions for the donaldson invariants of these surfaces, in the chamber where the volume of s2 is. Riemann surfaces, dynamics amd geometry researchgate. A nonsingular riemann surface s in c2 is a riemann surface where each point z0. Then the classification of riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. Riemann surfaces oxford graduate texts in mathematics. The minicourse is dedicated to master students having. Mean curvature flows in manifolds of special holonomy tsai, chungjun and wang, mutao, journal of differential geometry, 2018. Pages in category riemann surfaces the following 52 pages are in this category, out of 52 total.
Seibergwitten monopole equations and riemann surfaces. The course begins by introducing the notion of a riemann surface followed by examples. His main result is that all compact complex one manifolds occur as the riemann surface of an algebraic curve. Prove the existence of the universal covering of a connected manifold x, and remark that in case xis a riemann surface, its universal covering is a riemann surface as well for91, i. Riemannhilbert problems from donaldsonthomas theory. Curvature of higher direct image sheaves geiger, thomas and schumacher, georg, 2017. The equations in this article we discuss some wellknown problems in kahler geometry. Some proofs are not included, but can be chased via the given page references to textbooks. The books we will be using as reference for this course are farkas and kra riemann surfaces and donaldsons riemann surfaces the lectures will take place on wednesdays from 14. Arakelov invariants of riemann surfaces 3 other hand the set picg.
Orientability of riemann surfaces will follow from our desire to do complex analysis on them. It is extremely wellwritten, but definitely more analytic in flavor. There are several equivalent definitions of a riemann surface. Sep 29, 2014 jacob bernsteins series of 10 lectures on riemann surfaces from the msri summer graduate school on geometry and analysis that took place at msri, berkeley in julyaugust, 2014. Gluing formulae for donaldson invariants for connected sums along surfaces vicente munozt abstract. It is an extremely useful part of mathematics, knowledge of which is.
Part iid riemann surfaces dr alexei kovalev notes michaelmas 2007 these notes are a bit terse at some points and are not intended to be a replacement for the notes you take in lectures. We state a general conjecture about the existence of kahler metrics of constant scalar curvature, and discuss the background to the conjecture 1. Harmonic functions on open riemann surfaces 41 linna, or if it does not have a greens function, then it is parabolic. A remark on universal coverings of holomorphic families of riemann surfaces imayoshi, yoichi and nishimura, minori, kodai mathematical journal, 2005. What is, in basic terms, the relationship between riemann surfaces and algebraic geometry. Many profound results concerning the famous moduli problem can be reduced t. In the normal form of a finite riemann surface, the number of sides is not necessarily even, some sides corresponding to components of the boundary that remain free are not identified. Riemannian surfaces are riemann surfaces metrics on surfaces. A compact riemann surface of genus, or is a riemann surface of elliptic, parabolic or hyperbolic type, respectively.
This reduces the study of riemann surfaces to that of subgroups of moebius transformations. George springer, introduction to riemann surfaces jenkins, james a. The picture shows only three levels of the surface, which extends up and down with infinitely many similar levels all joined at the branch point at the origin. It is a culmination of much of traditional calculus, making surprising conne. Complex analysis on riemann surfaces download book. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. By focusing on low dimensions, donaldson bridges the gap between elementary complex variables and the deeper topics of complex manifolds, cohomology, algebraic topology, vector bundles, sheaves and so forth. If x is a compact riemann surface and x admits any nontrivial automorphism, then x. Im taking introductory courses in both riemann surfaces and algebraic geometry this term. The paragraphs in small print are intended for interest.
X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and. This is an introduction to the geometry of compact riemann surfaces, largely following the books farkaskra, fay, mumford tata lectures. The set of all divisors on xforms an abelian group denoted. Could you recommend some references, and point out the required mathematical knowledge. Riemann surface article about riemann surface by the free.
The book we will be using as reference for this course is donaldsons preprint riemann surfaces, available soft bound from bbl. One naturally encounters holomorphic functions in various ways. Download it once and read it on your kindle device, pc, phones or tablets. If you find any typos, please let me know as i will be passing this information on to the author. Given two riemann surfaces with boundary and a homotopy class of topological embeddings between them, there is a conformal embedding in the homotopy class if and only if the extremal length of. Riemann surface article about riemann surface by the.
I was surprised to hear that any compact riemann surface is a projective variety. Vector bundles on riemann surfaces sabin cautis contents 1. Buy riemann surfaces oxford graduate texts in mathematics 1st edition by donaldson, simon isbn. Download course materials all images created with matlab software click on picture for more information and an image of higher resolution. However, while its possible to sort of talk about a manifold in laymans english, the distinction between riemann surfaces and 2manifolds is very subtle and difficult to describe without using any math. So the core of this paper is the material of sections 3 and 4 where the yangmills theory is extended to riemann v surfaces or riemann surfaces with marked points and then applied to give information on the moduli spaces. Chapter 3 geometry of riemann surfaces sciencedirect. This means that x is a connected hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane.
It also deals quite a bit with noncompact riemann surfaces, but does include standard material on abels theorem, the abeljacobi map, etc. Donaldson invariants of product ruled surfaces and two. On gonality of riemann surfaces city university of new york. Riemann surfaces university of california, berkeley. Algebraic geometry and riemann surfaces daniel ying abstract. In the case of riemann surfaces a direct proof for arbitrary rank which avoids introduction of the donaldson functional can be modeled on donaldsons proof of the narasimhanseshadri theorem in 18 such a proof was suggested in 57. Riemann in 1851 for the purpose of replacing the study of multiplevalued analytic functions by the study of singlevalued analytic functions of a point on corresponding riemann surfaces seeanalytic functions. The square root provides another simple example of a riemann surface. Ok, so the purpose of this page is to give some of my papers and other work some easy accessibility on the webb. It is particularly pleasing that the subject of riemann surfaces has attracted the attention of a new generation of mathematicians from newly adjacent fields for example, those interested in hyperbolic manifolds and iterations of rational maps and young physicists who have been convinced certainly not by mathematicians that compact. Riemann surfaces were first studied by bernhard riemann in his inauguraldissertation at gottingen in 1851. Following our work in 18, we prove a gluing formula for the donaldson invariants.
Schlags treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of riemann surfaces. Riemann surface pdf 1 riemann surfaces and holomorphic maps. These are fourmanifolds of the form g s2,where gis a riemann surface of genus g. These are notes on the theory of super riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the rns formalism. The complex line c has an obvious structure of riemann surface given by the chart c,idc. The main idea of the proof is to show that a certain moduli space of pu2instantons on a surface x with no curves if such a surface existed would contain a closed riemann surface y whose. Another excellent analytic monograph from this point of view is the princeton lecture notes on riemann surfaces by robert gunning, which is also a good place to learn sheaf theory. The general theme is to ask whether a complex manifold admits a preferred. In this thesis we will give a present survey of the various methods used in dealing with riemann surfaces. I guess that you, my imaginary reader, have stopped reading all of this nonsense.
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