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Riemann–Roch theorem
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== Preliminary notions== [[File:Triple torus illustration.png|right|thumb|A Riemann surface of genus 3.]] A [[Riemann surface]] <math>X</math> is a [[topological space]] that is locally homeomorphic to an open subset of <math>\Complex</math>, the set of [[complex number]]s. In addition, the [[transition map]]s between these open subsets are required to be [[Holomorphic function|holomorphic]]. The latter condition allows one to transfer the notions and methods of [[complex analysis]] dealing with holomorphic and [[meromorphic function]]s on <math>\Complex</math> to the surface <math>X</math>. For the purposes of the Riemann–Roch theorem, the surface <math>X</math> is always assumed to be [[compact topological space|compact]]. Colloquially speaking, the [[Genus (mathematics)|genus]] <math>g</math> of a Riemann surface is its number of [[Handle_decomposition|handles]]; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first [[Betti number]], i.e., half of the <math>\Complex</math>-dimension of the first [[singular homology]] group <math>H_1(X,\Complex)</math> with complex coefficients. The genus [[Classification theorem|classifies]] compact Riemann surfaces [[up to]] [[homeomorphism]], i.e., two such surfaces are homeomorphic if and only if their genus is the same. Therefore, the genus is an important topological invariant of a Riemann surface. On the other hand, [[Hodge theory]] shows that the genus coincides with the <math>\Complex</math>-dimension of the space of holomorphic one-forms on <math>X</math>, so the genus also encodes complex-analytic information about the Riemann surface.<ref>Griffith, Harris, p. 116, 117</ref> A [[Divisor (algebraic geometry)#Weil divisors|divisor]] <math>D</math> is an element of the [[free abelian group]] on the points of the surface. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients. Any meromorphic function <math>f</math> gives rise to a divisor denoted <math>(f)</math> defined as :<math>(f):=\sum_{z_\nu\in R(f)}s_\nu z_\nu</math> where <math>R(f)</math> is the set of all zeroes and poles of <math>f</math>, and <math>s_\nu</math> is given by :<math>s_\nu:=\begin{cases}a&\text{if }z_\nu\text{ is a zero of order }a\\-a&\text{if }z_\nu\text{ is a pole of order }a\end{cases}</math>. The set <math>R(f)</math> is known to be finite; this is a consequence of <math>X</math> being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an [[accumulation point]]. Therefore, <math>(f)</math> is well-defined. Any divisor of this form is called a [[principal divisor]]. Two divisors that differ by a principal divisor are called [[linearly equivalent]]. The divisor of a meromorphic [[1-form]] is defined similarly. A divisor of a global meromorphic 1-form is called the [[canonical divisor]] (usually denoted <math>K</math>). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor). The symbol <math>\deg(D)</math> denotes the ''degree'' (occasionally also called index) of the divisor <math>D</math>, i.e. the sum of the coefficients occurring in <math>D</math>. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linear equivalence class. The number <math>\ell(D)</math> is the quantity that is of primary interest: the [[Dimension (vector space)|dimension]] (over <math>\Complex</math>) of the vector space of meromorphic functions <math>h</math> on the surface, such that all the coefficients of <math>(h)+D</math> are non-negative. Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in <math>D</math>; if the coefficient in <math>D</math> at <math>z</math> is negative, then we require that <math>h</math> has a zero of at least that [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]] at <math>z</math> – if the coefficient in <math>D</math> is positive, <math>h</math> can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar).
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