Abelian differential

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

Let $ g $ be the genus of the surface $ S $( cf. Genus of a surface); let $ a _ {1} b _ {1} \dots a _ {g} b _ {g} $ be the cycles of a canonical basis of the homology of $ S $. Depending on the nature of their singular points, one distinguishes three kinds of Abelian differentials: I, II and III, with proper inclusions $ I \subset II \subset III $. Abelian differentials of the first kind are first-order differentials that are holomorphic everywhere on $ S $ and that, in a neighbourhood $ U $ of each point $ P _ {0} \in S $, have the form $ \omega = p d z = p (z) d z $, where $ z = x + iy $ is a local uniformizing variable in $ U $, $ d z = d x + i dy $, and $ p (z) $ is a holomorphic, or regular, analytic function of $ z $ in $ U $. The addition of Abelian differentials and multiplication by a holomorphic function are defined by natural rules: If

$$ \omega = p dz,\ \pi = q dz,\ a = a (z), $$

then

$$ \omega + \pi = (p + q ) dz,\ a \omega = (a p ) dz. $$

The Abelian differentials of the first kind form a $ g $- dimensional vector space $ \mathfrak A $. After the introduction of the scalar product

$$ ( \omega , \pi ) = {\int\limits \int\limits } _ { S } \omega \star \overline \pi \; , $$

where $ \omega \star \overline \pi \; $ is the exterior product of $ \omega $ with the star-conjugate differential $ \overline \pi \; $, the space $ \mathfrak A $ becomes a Hilbert space.

Let $ A _ {1} , B _ {1} \dots A _ {g} , B _ {g} $ be the $ A $- and $ B $- periods of the Abelian differential of the first kind $ \omega $, i.e. the integrals

$$ A _ {j} = \int\limits _ {a _ {j} } \omega ,\ \ B _ {j} = \int\limits _ {b _ {j} } \omega ,\ \ j = 1 \dots g . $$

The following relation then holds:

$$ \tag{1 } \| \omega \| ^ {2} = i \sum _ {j = 1 } ^ { g } ( A _ {j} \overline{B}\; _ {j} - B _ {j} \overline{A}\; _ {j} ) \geq 0 . $$

If $ A _ {1} ^ \prime , B _ {1} ^ \prime \dots A _ {g} ^ \prime , B _ {g} ^ \prime $ are the periods of another Abelian differential of the first kind $ \pi $, then one has

$$ \tag{2 } i ( \omega , \overline \pi \; ) = \sum _ {j = 1 } ^ { g } ( A _ {j} B _ {j} ^ \prime - B _ {j} A _ {j} ^ \prime ) = 0 . $$

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 $ \phi _ {1} \dots \phi _ {g} $ of the space $ \mathfrak A $, can be chosen so that

$$ A _ {ij } = \int\limits _ {a _ {j} } \phi _ {i} = \delta _ {ij } , $$

where $ \delta _ {ii} = 1 $ and $ \delta _ {ij} = 0 $ if $ j \neq i $. The matrix $ (B _ {ij} ) $, $ i, j = 1 \dots g $, of the $ B $- periods

$$ B _ {ij} = \int\limits _ {b _ {j} } \phi _ {i} $$

is then symmetric, and the matrix of the imaginary parts $ ( \mathop{\rm Im} B _ {ij} ) $ is positive definite. An Abelian differential of the first kind for which all the $ A $- periods or all the $ B $- periods are zero is identically equal to zero. If all the periods of an Abelian differential of the first kind $ \omega $ are real, then $ \omega = 0 $.

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

$$ \tag{3 } \left ( a _ \frac{-n}{z} ^ {n} + \dots + a _ \frac{-1}{z} + f ( z ) \right ) dz, $$

where $ f(z) $ is a regular function, $ n $ is the order of the pole (if $ a _ {-n} \neq 0 $), and $ a _ {-1} $ is the residue of the pole. If $ n = 1 $, 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

$$ \left ( a _ \frac{-n}{z} ^ {n} + \dots + a _ \frac{-2}{z} ^ {2} + f ( z ) \right ) dz, $$

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

Let $ \omega $ be an arbitrary Abelian differential with $ A $- periods $ A _ {1} \dots A _ {g} $; the Abelian differential $ \omega ^ \prime = \omega - A _ {1} \phi _ {1} - \dots -A _ {g} \phi _ {g} $ then has zero $ A $- periods and is known as a normalized Abelian differential. In particular, if $ P _ {1} $ and $ P _ {2} $ are any two points on $ S $, one can construct a normalized Abelian differential $ \omega _ {1,2} $ with the singularities $ (1/z) d z $ in $ P _ {1} $ and $ (-1/z) d z $ in $ P _ {2} $, which is known as a normal Abelian differential of the third kind. Let $ \omega $ be an arbitrary Abelian differential with residues $ c _ {1} \dots c _ {n} $ at the respective points $ P _ {1} \dots P _ {n} $; then, always, $ c _ {1} + \dots + c _ {n} = 0 $. If $ P _ {0} $ is any arbitrary point on $ S $ such that $ P _ {0} \neq P _ {j} $, $ j = 1 \dots n $, then $ \omega $ can be represented as a linear combination of a normalized Abelian differential of the second kind $ \omega _ {2} $, a finite number of normal Abelian differentials of the third kind $ \omega _ {j,0} $, and basis Abelian differentials of the first kind $ \phi _ {k} $:

$$ \omega = \omega _ {2} + \sum _ {j=1 } ^ { n } c _ {j} \omega _ {j,0} + \sum _ {k= 1 } ^ { g } A _ {k} \phi _ {k} . $$

Let $ \omega _ {3} $ be an Abelian differential of the third kind with only simple poles with residues $ c _ {j} $ at the points $ P _ {j} $, $ j = 1 \dots n $, and let $ \omega _ {1} $ be an arbitrary Abelian differential of the first kind:

$$ A _ {k} = \int\limits _ {a _ {k} } \omega _ {1} ,\ \ B _ {k} = \int\limits _ {b _ {k} } \omega _ {1} , $$

$$ A _ {k} ^ \prime = \int\limits _ {a _ {k} } \omega _ {3} ,\ \ B _ {k} ^ \prime = \int\limits _ {b _ {k} } \omega _ {3} ,\ k = 1 \dots g, $$

where the cycles $ a _ {k} , b _ {k} $ do not pass through the poles of $ \omega _ {3} $. Let the point $ P _ {0} \in S $ not lie on the cycles $ a _ {k} , b _ {k} $ and let $ L _ {j} $ be a path from $ P _ {0} $ to $ P _ {j} $. One then obtains bilinear relations for Abelian differentials of the first and third kinds:

$$ \sum _ {k = 1 } ^ { g } ( A _ {k} B _ {k} ^ \prime - B _ {k} A _ {k} ^ \prime ) = 2 \pi i \sum _ {j = 1 } ^ { n } c _ {j} \int\limits _ {L _ {j} } \omega _ {1} . $$

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

In addition to the $ A $- and $ B $- periods $ A _ {k} , B _ {k} $, $ k = 1 \dots g $, known as the cyclic periods, an arbitrary Abelian differential of the third kind also has polar periods of the form $ 2 \pi i c _ {j} $ along zero-homologous cycles which encircle the poles $ P _ {j} $. One thus has, for an arbitrary cycle $ \gamma $,

$$ \int\limits _ \gamma \omega _ {3} = \sum _ {k = 1 } ^ { g } ( l _ {k} A _ {k} + l _ {g+k} B _ {k} ) + 2 \pi i \sum _ {j = 1 } ^ { n } m _ {j} c _ {j} , $$

where $ l _ {k} , l _ {g+k} $, and $ m _ {j} $ are integers.

Important properties of Abelian differentials are described in terms of divisors. Let $ ( \omega ) $ be the divisor of the Abelian differential $ \omega $, i.e. $ ( \omega ) $ is an expression of the type $ ( \omega ) = P _ {1} ^ {\alpha _ {1} } {} \dots P _ {n} ^ {\alpha _ {n} } $, where the $ P _ {j} $- s are all the zeros and poles of $ \omega $ and where the $ \alpha _ {j} $- s are their multiplicities or orders. The degree $ \textrm{ d } [( \omega )] = \alpha _ {1} + \dots + \alpha _ {n} $ of the divisor of the Abelian differential $ \omega $ depends only on the genus of $ S $, and one always has $ \textrm{ d } [( \omega )] = 2g - 2 $. Let $ \mathfrak a $ be some given divisor. Let $ \Omega ( \mathfrak a ) $ denote the complex vector space of Abelian differentials $ \omega $ of which the divisors $ ( \omega ) $ are multiples of $ \mathfrak a $, and let $ L ( \mathfrak a ) $ denote the vector space of meromorphic functions $ f $ on $ S $ of which the divisors $ (f) $ are multiples of $ \mathfrak a $. Then $ { \mathop{\rm dim} } \Omega ( \mathfrak a ) = { \mathop{\rm dim} } L ( \mathfrak a / ( \omega )) $. Other important information on the dimension of these spaces is contained in the Riemann–Roch theorem: The equality

$$ \mathop{\rm dim} L ( \mathfrak a ^ {-1} ) - \mathop{\rm dim} \Omega ( \mathfrak a ) = \ \textrm{ d } [ \mathfrak a ] - g + 1 $$

is valid for any divisor $ \mathfrak a $. It follows from the above, for example, that if $ g = 1 $, i.e. on the surface of a torus, a meromorphic function cannot have a single simple pole.

Let $ S $ be an arbitrary compact Riemann surface on which there are meromorphic functions $ z $ and $ w $ which satisfy an irreducible algebraic equation $ F (z, w ) = 0 $. Any arbitrary Abelian differential on $ S $ can then be expressed as $ \omega = R (z, w ) d z $ where $ R (z, w ) $ is some rational function in $ z $ and $ w $; conversely, the expression $ \omega = R (z, w) dz $ is an Abelian differential. This means that an arbitrary Abelian integral

$$ \int\limits R ( z, w ) dz = \int\limits \omega $$

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

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