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| | Pastrycook Bob from Morrisburg, has interests such as scuba diving, new launch property singapore and riddles. Maintains a tour blog and has heaps to write about after visiting Architectural Ensemble of the Trinity Sergius Lavra in Sergiev Posad.<br><br>Also visit my page; [http://411uganda.com/activity/p/306806/ singapore new property launch] |
| 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 [[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.
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| The dimension of the space of differentials of the first kind, by means of this identification, is the [[Hodge number]]
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| :''h''<sup>0,1</sup>.
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| 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
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| : <math> \int\frac{x^k \, dx}{\sqrt{Q(x)}} </math> | |
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| where ''Q'' is a [[square-free polynomial]] of any given degree > 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
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| :''k'' ≤ ''g'' − 1,
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| 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.
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| 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]].
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| ==Differentials of the second and third kind==
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| 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.
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| 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.
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| 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]].
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| On the other hand, a meromorphic abelian differential of the ''second kind'' has traditionally been one with residues at all poles being zero. There is a higher-dimensional analogue available, using the [[Poincaré residue]]
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| === See also ===
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| [[Logarithmic form]]
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| {{DEFAULTSORT:Differential Of The First Kind}}
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| [[Category:Complex manifolds]]
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| [[Category:Algebraic geometry]]
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Pastrycook Bob from Morrisburg, has interests such as scuba diving, new launch property singapore and riddles. Maintains a tour blog and has heaps to write about after visiting Architectural Ensemble of the Trinity Sergius Lavra in Sergiev Posad.
Also visit my page; singapore new property launch