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
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then
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The Abelian differentials of the first kind form a -dimensional vector space
. After the introduction of the scalar product
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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
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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
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where and
if
. The matrix
,
, of the
-periods
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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
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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
:
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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:
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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:
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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
,
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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
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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 .
See also Algebraic function.
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) |
Comments
Another good reference, replacing [3], is [a1].
References
[a1] | S. Lang, "Introduction to algebraic and abelian functions" , Addison-Wesley (1972) MR0327780 Zbl 0255.14001 |
Abelian differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_differential&oldid=24358