Meromorphic function

of one complex variable in a domain (or on a Riemann surface )

A holomorphic function in a domain which has at every singular point a pole (cf. Pole (of a function), i.e. is an isolated point of the set , which has no limit points in , and ). The collection of all meromorphic functions in is a field with respect to the usual pointwise operations followed by redefinition at the removable singularities.

The quotient of two arbitrary holomorphic functions in , , is a meromorphic function in . Conversely, every meromorphic function in a domain (or on a non-compact Riemann surface ) can be expressed as , , where are holomorphic and have no common zeros in . It follows that on a non-compact Riemann surface the field coincides with the field of fractions of the ring of holomorphic functions in .

Every meromorphic function defines a continuous mapping of the domain into the Riemann sphere , which is a holomorphic mapping relative to the standard complex structure on . Conversely, every holomorphic mapping , , defines a meromorphic function in : The set of poles of coincides with the discrete set and if . Thus, the meromorphic functions of one variable may be identified with the holomorphic mappings () into the Riemann sphere.

The basic problems in the theory of meromorphic functions are those concerning the existence (and construction) of meromorphic functions with prescribed singularities.

I) One is given a (closed) discrete subset and, at each point , the principal part of a Laurent expansion (cf. Laurent series)

it is required to find a meromorphic function with these principal parts, i.e. a holomorphic function in such that is holomorphic in a neighbourhood of for each . If the number of points is finite, then (in a domain ) the problem is trivially solved by the function . In the general case this problem is solved by the Mittag-Leffler theorem: On every non-compact Riemann surface there exists a meromorphic function with given principal parts , . On a compact Riemann surface (for instance, a torus) this problem has in general no solution — supplementary conditions concerning the compatibility of the principal parts must be imposed.

The second basic problem is conveniently formulated in the language of divisors (cf. Divisor), i.e. of mappings such that for every compactum the number of points at which is finite (the number is called the multiplicity of at ). Divisors can explicitly be written as formal sums , where are the points at which ; in the case of finitely many terms the number () is called the degree of the divisor . For a meromorphic function its divisor is equal to zero everywhere apart from the zeros and poles of , at which the multiplicity is set equal to the order of the zero or of the pole (poles have negative orders).

II) At the points of a (closed) discrete subset one is given "multiplicities" — integers . It is required to find a meromorphic function with zeros and poles of the respective multiplicities, i.e. a holomorphic function in such that is holomorphic and does not vanish in a neighbourhood of the point , . In the case of finitely many points (and ) such a function is, for example, . In the general case the problem is solved by Weierstrass' theorem: On a non-compact Riemann surface , for every given divisor there is a meromorphic function with divisor equal to . For a compact Riemann surface the holomorphic mapping into the Riemann sphere defined by a non-constant meromorphic function is a branched covering, and hence the function takes every value the same number of times; in particular, the number of zeros of equals the number of its poles (multiplicities taken into account). Therefore, the condition is necessary in order that problem II admits a solution on a compact Riemann surface. In general, it is not sufficient; a necessary and sufficient condition for the existence of a meromorphic function with a given divisor is given by Abel's theorem (see [2]).

Let be a divisor on a compact Riemann surface . The functions satisfying the condition form a finite-dimensional linear space (over ); if , then .

The Riemann–Roch theorem asserts that

where and are the so-called canonical divisor and, respectively, the genus of the Riemann surface . From this relation one can obtain many existence theorems (if , then , and hence contains non-constant meromorphic functions). For example, on every compact Riemann surface of genus there is a meromorphic function which realizes a branched covering with at most sheets.

An important place in the theory of meromorphic functions of one complex variable is occupied by value-distribution theory (Nevanlinna theory), which studies the distribution of the roots of the equations , , when approaching the boundary of the domain.

Meromorphic functions of several complex variables.

Let be a domain in (or an -dimensional complex manifold) and let be a (complex-) analytic subset of codimension one (or empty). A holomorphic function defined on is called a meromorphic function in if for every point one can find an arbitrarily small neighbourhood of in and functions holomorphic in without common non-invertible factors in , such that in . The set is called the polar set of the meromorphic function . Its subset , defined locally by the condition , is called the set of (points of) indeterminacy of ; is an analytic subset of of (complex) codimension . At each point the function is essentially undefined: The limiting values of for , , fill up the Riemann sphere . On the other hand, at the points of the limit exists, and upon redefining if , one obtains a holomorphic mapping of into the Riemann sphere. Conversely, if is an arbitrary (possibly empty) complex-analytic subset of of codimension , then every holomorphic mapping defines a meromorphic function on that is equal to on , where is either an analytic subset of of codimension 1 or is empty. Thus, a meromorphic function in can be defined as a holomorphic mapping into the Riemann sphere defined in the complement of an analytic subset of codimension .

A third, completely localized, definition of meromorphic functions (equivalent to the one given above) is stated in the language of sheaves. Let be the sheaf of germs of holomorphic functions on , and for each point let denote the field of fractions of the ring (the stalk of the sheaf over ). Then is naturally endowed with the structure of a sheaf of fields, called the sheaf of germs of meromorphic functions in . A meromorphic function in is defined as a global section of , i.e. a continuous mapping such that for all . The sets and are defined as follows: If , , , then one may assume that and are mutually prime, i.e. they have no common non-invertible factors in ; then if , while if . The value at a point of the meromorphic function thus defined is .

As in the one-dimensional case, the collection of all meromorphic functions in forms a field with respect to the pointwise algebraic operations with a subsequent redefinition at the removable singularities.

The closure of the zero set of a meromorphic function , i.e. of the set , is an analytic subset of of codimension one (or empty); the set of indeterminacy is . On and one can define the order (multiplicity) of the zeros (or poles) of the meromorphic function . If is a regular point of the analytic set , then in some neighbourhood of the set is connected and is given by an equation , , where throughout . Hence there is a maximal integer such that the function admits a holomorphic extension to ; this number is called the order (of the zero if , and of the pole if ) of the meromorphic function at the point . The function is locally constant on the set of regular points of . Therefore one can attach to each meromorphic function in its divisor , where are the irreducible components of and is the multiplicity (order) of at the regular points of that belong to (alternative notations: , etc.). On a compact complex manifold a meromorphic function is uniquely defined by its divisor, up to a multiplicative constant.

The problems solved in the one-dimensional case by the Mittag-Leffler and Weierstrass theorems are known in the higher-dimensional case as the first (additive) and the second (multiplicative) Cousin problems. Due to the complicated structure of the polar set , the notion of a principal part of a meromorphic function is not defined in general, and accordingly the Cousin problems are formulated as follows.

I) Suppose that an open covering of the manifold and in each a meromorphic function are given; it is required to find a meromorphic function such that for all .

II) For a given divisor on , find a meromorphic function such that .

The conditions of solvability of these problems in the higher-dimensional case are considerably more stringent than in the one-dimensional case.

The problem of representing a meromorphic function as a quotient of two holomorphic functions is called the Poincaré problem. The strong Poincaré problem is to represent a meromorphic function as a quotient of holomorphic functions the germs of which at each point are mutually prime in . The Poincaré problem is unsolvable on a compact connected complex manifold if there are non-constant meromorphic functions on it. However, this problem is solvable in every domain and, in fact, in an arbitrary domain on a Stein manifold (see [7]). The solvability of the strong Poincaré problem follows from that of the Cousin II problem (the converse is not true).

Functions are said to be algebraically dependent if there is a polynomial in variables with complex coefficients such that in the common domain of definition of the functions . The maximal number of algebraically-independent meromorphic functions on is called the transcendence degree of the field . On a compact complex manifold this number does not exceed the (complex) dimension of the manifold (Siegel's theorem); furthermore, the field has a finite number of generators (see [6]).

On concrete complex manifolds, meromorphic functions may have supplementary properties. For instance, in the complex projective space the set of indeterminacy of any non-constant meromorphic function is not empty. Every meromorphic function on a projective algebraic variety is rational, i.e. is expressible as a quotient of homogeneous polynomials in homogeneous coordinates. On algebraic varieties the field is quite rich. On the other hand, there exist complex manifolds (for example, some non-algebraic tori) on which every meromorphic function is constant. Higher-dimensional generalizations of the Riemann–Roch theorem are less effective, and existence theorems for various classes of meromorphic functions can only be obtained for some classes of complex manifolds.

References

 [1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) [2] O. Forster, "Lectures on Riemann surfaces" , Springer (1981) (Translated from German) MR0648106 Zbl 0475.30002 [3] W.K. Hayman, "Meromorphic functions" , Clarendon Press (1964) MR0164038 Zbl 0141.07802 Zbl 0115.06203 [4] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001 [5] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001 [6] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 [7] J. Kajiwara, E. Sakai, "Generalization of Levi–Oka's theorem concerning meromorphic functions" Nagoya Math. J. , 29 (1967) pp. 75–84 MR243106