mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, ''differential of the first kind'' is a traditional term used in the theories of

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s (more generally,

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

s) and

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s (more generally,

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

), for everywhere-regular 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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...

; on an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

''V'' that is

non-singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular ...

it would be a

global section In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as Set (mathematics), sets, abelian groups, Ring (mathematics), rings) attached to the open sets of a topological space and defined locally with regard to them. ...

of the

coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...

Ω¹ of

Kähler differential In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic ...

s. In either case the definition has its origins in the theory of

abelian integral In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form :\int_^z R(x,w) \, dx, where R(x,w) is an arbitrary rational function of the two variables x and w, wh ...

s. The dimension of the space of differentials of the first kind, by means of this identification, is the

Hodge number In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...

:''h''^1,0. The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

s to all curves over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s. They include for example the hyperelliptic integrals of type :

\int\frac

where ''Q'' is a

square-free polynomial In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically closed field containing its coefficients. In characteristic 0, or over a finite field, a univar ...

of any given degree > 4. The allowable power ''k'' has to be determined by analysis of the possible pole at the

point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...

on the corresponding

hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...

. When this is done, one finds that the condition is :''k'' ≤ ''g'' − 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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

, the Hodge number is the

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

''g''. For the case of

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

s, this is the quantity known classically as the irregularity ''q''. It is also, in general, the dimension of the

Albanese variety In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. Precise statement The Albanese variety of a smooth projective algebraic variety V is an abelian variety \operatorname(V) ...

, which takes the place of the

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...

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 forms, both from the side of more

Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...

, and through the use of morphisms to

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

s. The Weierstrass zeta function was called an ''integral of the second kind'' in

elliptic function In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...

theory; it is a

logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the in ...

of a

theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube ...

, and therefore has simple poles, 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 In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

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 In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...

, algebraic tori, and

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

s, and the decomposition is in terms of a

composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many na ...

. 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.

References

* {{DEFAULTSORT:Differential Of The First Kind Complex manifolds Algebraic geometry

Differentials of the second and third kind

See also

References