# Abelian differential

A holomorphic or meromorphic differential on a compact, or closed, Riemann surface (cf. Differential on a Riemann surface).

Let be the genus of the surface (cf. Genus of a surface); let be the cycles of a canonical basis of the homology of . Depending on the nature of their singular points, one distinguishes three kinds of Abelian differentials: I, II and III, with proper inclusions . Abelian differentials of the first kind are first-order differentials that are holomorphic everywhere on and that, in a neighbourhood of each point , have the form , where is a local uniformizing variable in , , and is a holomorphic, or regular, analytic function of in . The addition of Abelian differentials and multiplication by a holomorphic function are defined by natural rules: If

then

The Abelian differentials of the first kind form a -dimensional vector space . After the introduction of the scalar product

where is the exterior product of with the star-conjugate differential , the space becomes a Hilbert space.

Let be the - and -periods of the Abelian differential of the first kind , i.e. the integrals

The following relation then holds:

 (1)

If are the periods of another Abelian differential of the first kind , then one has

 (2)

The relations (1) and (2) are known as the bilinear Riemann relations for Abelian differentials of the first kind. A canonical basis of the Abelian differentials of the first kind, i.e. a canonical basis of the space , can be chosen so that

where and if . The matrix , , of the -periods

is then symmetric, and the matrix of the imaginary parts is positive definite. An Abelian differential of the first kind for which all the -periods or all the -periods are zero is identically equal to zero. If all the periods of an Abelian differential of the first kind are real, then .

Abelian differentials of the second and third kinds are, in general, meromorphic differentials, i.e. analytic differentials which have on not more than a finite set of singular points that are poles and which have local representations

 (3)

where is a regular function, is the order of the pole (if ), and is the residue of the pole. If , the pole is said to be simple. An Abelian differential of the second kind is a meromorphic differential all residues of which are zero, i.e. a meromorphic differential with local representation

An Abelian differential of the third kind is an arbitrary Abelian differential.

Let be an arbitrary Abelian differential with -periods ; the Abelian differential then has zero -periods and is known as a normalized Abelian differential. In particular, if and are any two points on , one can construct a normalized Abelian differential with the singularities in and in , which is known as a normal Abelian differential of the third kind. Let be an arbitrary Abelian differential with residues at the respective points ; then, always, . If is any arbitrary point on such that , , then can be represented as a linear combination of a normalized Abelian differential of the second kind , a finite number of normal Abelian differentials of the third kind , and basis Abelian differentials of the first kind :

Let be an Abelian differential of the third kind with only simple poles with residues at the points , , and let be an arbitrary Abelian differential of the first kind:

where the cycles do not pass through the poles of . Let the point not lie on the cycles and let be a path from to . One then obtains bilinear relations for Abelian differentials of the first and third kinds:

Bilinear relations of a similar type also exist between Abelian differentials of the first and second kinds.

In addition to the - and -periods , , known as the cyclic periods, an arbitrary Abelian differential of the third kind also has polar periods of the form along zero-homologous cycles which encircle the poles . One thus has, for an arbitrary cycle ,

where , and are integers.

Important properties of Abelian differentials are described in terms of divisors. Let be the divisor of the Abelian differential , i.e. is an expression of the type , where the -s are all the zeros and poles of and where the -s are their multiplicities or orders. The degree of the divisor of the Abelian differential depends only on the genus of , and one always has . Let be some given divisor. Let denote the complex vector space of Abelian differentials of which the divisors are multiples of , and let denote the vector space of meromorphic functions on of which the divisors are multiples of . Then . Other important information on the dimension of these spaces is contained in the Riemann–Roch theorem: The equality

is valid for any divisor . It follows from the above, for example, that if , i.e. on the surface of a torus, a meromorphic function cannot have a single simple pole.

Let be an arbitrary compact Riemann surface on which there are meromorphic functions and which satisfy an irreducible algebraic equation . Any arbitrary Abelian differential on can then be expressed as where is some rational function in and ; conversely, the expression is an Abelian differential. This means that an arbitrary Abelian integral

is the integral of some Abelian differential on a compact Riemann surface .

#### References

 [1] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602 [2] R. Nevanlinna, "Uniformisierung" , Springer (1953) pp. Chapt.5 MR0057335 Zbl 0053.05003 [3] N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt.3;8 (In Russian)