A Riemann surface is a -dimensional algebro-geometric object with good properties. The name ‘surface’ comes from the classical case, which is -dimensional over the complex numbers and therefore -dimensional over the real numbers.
There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities.
Classically, a Riemann surface is a connected complex--dimensional complex manifold, in the strictest sense of ‘manifold’. In other words, it’s a Hausdorff second countable space which is locally homeomorphic to the complex plane via charts (i.e., homeomorphisms) for open and such that is holomorphic.
Evidently an open subspace of a Riemann surface is a Riemann surface. In particular, an open subset of is a Riemann surface in a natural manner.
The Riemann sphere or is a Riemann sphere with the open sets and the charts
The transition map is and thus holomorphic on .
An important example comes from analytic continuation, which we will briefly sketch below. A function element is a pair where is holomorphic and is an open disk. Two function elements are said to be direct analytic continuations of each other if and on . By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).
Starting with a given function element , we can consider the totality of all equivalence classes of function elements that can be obtained by continuing along curves in . Then is actually a Riemann surface.
Indeed, we must first put a topology on . If with centered at , then let a neighborhood of be given by all function elements for ; these form a basis for a suitable topology on . Then the coordinate projections form appropriate local coordinates. In fact, there is a globally defined map , whose image in general will be a proper subset of .
Since we have local coordinates, we can define a map of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map ; for meromorphicity, this becomes .
Many of the usual theorems of elementary complex analysis (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a Laurent expansion, etc.
Let be a regular map. If is compact and is nonconstant, then is surjective and compact.
Since a Riemann surface is a -dimensional smooth manifold in the usual (real) sense, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle , but it is more natural to take the complexified cotangent bundle , which we will in the future just abbreviate ; this should not be confusing since we will only do this when we talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.
If is a local coordinate on , defined say on , define the (complex) differentials
These form a basis for the complexified cotangent space at each point of . There is also a dual basis
for the complexified tangent space.
We now claim that we can split the tangent space , where the former consists of multiples of and the latter of multiples of ; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given for open and (just for convenience), we can write
where , which we will often abbreviate as . If is holomorphic and conformal sending , we have
in particular, preserves the decomposition of .
Given smooth, we can consider the projections of the 1-form onto and , respectively; these will be called . Similarly, we define the corresponding operators on 1-forms: to define , first project onto (the reversal is intentional!) and then apply , and vice versa for .
In particular, if we write in local coordinates , then
To see this, we have tacitly observed that .
In the theory of Riemann surfaces, there are several important theorems. Here are two:
The Riemann-Roch theorem, which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;
The uniformization theorem?, which partially classifies Riemann surfaces.
|number fields (“function fields of curves over F1”)||function fields of curves over finite fields (arithmetic curves)||Riemann surfaces/complex curves|
|affine and projective line|
|(integers)||(polynomials, function algebra on affine line )||(holomorphic functions on complex plane)|
|(rational numbers)||(rational functions)||meromorphic functions on complex plane|
|(prime number/non-archimedean place)|
|(place at infinity)|
|(Spec(Z))||(affine line)||complex plane|
|(projective line)||Riemann sphere|
|genus of the rational numbers = 0||genus of the Riemann sphere = 0|
|(p-adic integers)||(power series around )||(holomorphic functions on formal disk around )|
|(“-arithmetic jet space” of at )||formal disks in|
|(p-adic numbers)||(Laurent series around )||(holomorphic functions on punctured formal disk around )|
|(ring of adeles)||( adeles of function field )||(restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)|
|(group of ideles)||( ideles of function field )|
|Riemann zeta function||Goss zeta function|
|branched covering curves|
|a number field ( a possibly ramified finite dimensional field extension)||a function field of an algebraic curve over||(sheaf of rational functions on complex curve )|
|(ring of integers)||(structure sheaf)|
|(spectrum with archimedean places)||(arithmetic curve)||(complex curve being branched cover of Riemann sphere)|
|genus of a number field||genus of an algebraic curve||genus of a surface|
|prime ideal in ring of integers|
|(formal completion at )||(function algebra on punctured formal disk around )|
|(ring of integers of formal completion)||(function algebra on formal disk around )|
|(ring of adeles)||(restricted product of function rings on all punctured formal disks around all points in )|
|(function ring on all formal disks around all points in )|
|(group of ideles)|
|Galois and class field theory|
|Galois group||“||fundamental group|
|Galois representation||“||flat connection (“local system”) on|
|automorphy and bundles|
|(idele class group)||“|
|“||Cech cocycles mod coboundaries for holomorphic line bundles with respect to cover by formal disks = moduli stack of line bundles|
|(constant sheaves on this stack form unramified automorphic representations)||“||(moduli stack of bundles on the curve )|
|number field Langlands correspondence||function field Langlands correspondence||geometric Langlands correspondence|
|Tamawa-Weil for number fields||Tamagawa-Weil for function fields|
|Dedekind zeta function||Weil zeta function||zeta function of a Riemann surface|
Historical references include
Lecture notes include
Eberhard Freitag, Riemann surfaces – Sheaf theory, Riemann Surfaces, Automorphic forms, 2013 (pdf)