Riemann–Roch theorem

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.The first main achievement of F. K. Schmidt is the discovery that the classical theorem of Riemann–Roch on compact Riemann surfaces can be transferred to function fields with finite base field. Actually, his proof of the Riemann–Roch theorem works for arbitrary perfect base fields, not necessarily finite. The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by Riemann (1857), the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student Gustav Roch (1865). It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. A Riemann surface X is a topological space that is locally homeomorphic to an open subset of C, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of complex analysis dealing with holomorphic and meromorphic functions on C to the surface X. For the purposes of the Riemann–Roch theorem, the surface X is always assumed to be compact. Colloquially speaking, the genus g of a Riemann surface is its number of 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 C-dimension of the first singular homology group H1(X, C) with complex coefficients. The genus 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 (C-)dimension of the space of holomorphic one-forms on X, so the genus also encodes complex-analytic information about the Riemann surface. A divisor D 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 f gives rise to a divisor denoted (f) defined as where R(f) is the set of all zeroes and poles of f, and sν is given by The set R(f) is known to be finite; this is a consequence of X being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an accumulation point. Therefore, (f) 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 K). 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 deg(D) denotes the degree (occasionally also called index) of the divisor D, i.e. the sum of the coefficients occurring in D. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of the divisor depends only on the linear equivalence class. The number ℓ(D) is the quantity that is of primary interest: the dimension (over C) of the vector space of meromorphic functions h on the surface, such that all the coefficients of (h) + D 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 D; if the coefficient in D at z is negative, then we require that h has a zero of at least that multiplicity at z – if the coefficient in D is positive, h 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).