Editing Differential of the first kind

{{Short description|Term used in the theories of Riemann surfaces and algebraic curves}}
{{One source|date=August 2022}}
In [[mathematics]], '''''differential of the first kind''''' is a traditional term used in the theories of [[Riemann surface]]s (more generally, [[complex manifold]]s) and [[algebraic curve]]s (more generally, [[algebraic variety|algebraic varieties]]), for everywhere-regular [[differential form|differential 1-forms]]. Given a complex manifold ''M'', a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere [[holomorphic form|holomorphic]]; on an [[algebraic variety]] ''V'' that is [[Algebraic curve#Singularities|non-singular]] it would be a [[global section]] of the [[coherent sheaf]] Ω<sup>1</sup> of [[Kähler differential]]s. In either case the definition has its origins in the theory of [[abelian integral]]s.

The dimension of the space of  differentials of the first kind, by means of this identification, is the [[Hodge number]]

:''h''<sup>1,0</sup>.

The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the [[elliptic integral]]s to all curves over the [[complex number]]s. They include for example the '''hyperelliptic integrals''' of type

: <math> \int\frac{x^k \, dx}{\sqrt{Q(x)}} </math>

where ''Q'' is a [[square-free polynomial]] of any given degree&nbsp;>&nbsp;4. The allowable power ''k'' has to be determined by analysis of the possible pole at the [[point at infinity]] on the corresponding [[hyperelliptic curve]]. When this is done, one finds that the condition is

:''k'' ≤ ''g'' &minus; 1,

or in other words, ''k'' at most 1 for degree of ''Q'' 5 or 6, at most 2 for degree 7 or 8, and so on (as ''g'' = [(1+ deg ''Q'')/2]).

Quite generally, as this example illustrates, for a [[compact Riemann surface]] or [[algebraic curve]], the Hodge number is the [[genus (mathematics)|genus]] ''g''. For the case of [[algebraic surface]]s, this is the quantity known classically as the [[irregularity of a surface|irregularity]] ''q''. It is also, in general, the dimension of the [[Albanese variety]], which takes the place of the [[Jacobian variety]].

==Differentials of the second and third kind==
The traditional terminology also included differentials '''of the second kind''' and '''of the third kind'''. The idea behind this has been supported by modern theories of [[algebraic differential form]]s, both from the side of more [[Hodge theory]], and through the use of morphisms to [[commutative]] [[algebraic group]]s.

The [[Weierstrass zeta function]] was called an ''integral of the second kind'' in [[elliptic function]] theory; it is a [[logarithmic derivative]] of a [[theta function]], and therefore has [[simple pole]]s, with integer residues. The decomposition of a ([[meromorphic]]) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a [[linear combination]] of  translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.

The same type of decomposition exists in general, ''mutatis mutandis'', though the terminology is not completely consistent. In the algebraic group ([[generalized Jacobian]]) theory the three kinds are [[abelian varieties]], [[algebraic tori]], and [[affine space]]s, and the decomposition is in terms of a [[composition series]].

On the other hand, a meromorphic abelian differential of the ''second kind'' has traditionally been one with residues at all poles being zero. One of the '''third kind''' is one where all poles are simple. There is a higher-dimensional analogue available, using the [[Poincaré residue]].

== See also ==
*[[Logarithmic form]]

==References==
* {{eom|title=Abelian differential|id=Abelian_differential}}

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