# 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 $S$.