Compact Riemann surface
Encyclopedia
In mathematics
, a compact Riemann surface is a complex manifold
of dimension one that is a compact space
. Riemann surface
s are generally classified first into the compact (those that are closed manifold
s) and the open (the rest, which from the point of view of complex analysis
are very different, being for example Stein manifold
s).
A compact Riemann surface C that is a connected space
is an algebraic curve
defined over the complex number field. More precisely, the meromorphic function
s on C make up the function field
F on the corresponding curve; F is a field extension
of the complex numbers of transcendence degree
equal to 1. It can in fact be generated by two functions f and g. This is a structural result on the meromorphic functions: there are enough in the sense of separating out the points of C, and any two are algebraically dependent. These facts were known in the nineteenth century (see GAGA
for more in this direction).
A general compact Riemann surface is therefore a finite disjoint union of complex (non-singular) algebraic curves.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a compact Riemann surface is a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
of dimension one that is a compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s are generally classified first into the compact (those that are closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
s) and the open (the rest, which from the point of view of complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
are very different, being for example Stein manifold
Stein manifold
In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...
s).
A compact Riemann surface C that is a connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
is an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
defined over the complex number field. More precisely, the meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s on C make up the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
F on the corresponding curve; F is a field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of the complex numbers of transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
equal to 1. It can in fact be generated by two functions f and g. This is a structural result on the meromorphic functions: there are enough in the sense of separating out the points of C, and any two are algebraically dependent. These facts were known in the nineteenth century (see GAGA
Gaga
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for more in this direction).
A general compact Riemann surface is therefore a finite disjoint union of complex (non-singular) algebraic curves.