Differential form

A differential form of degree , a -form, on a differentiable manifold is a times covariant tensor field on . It may also be interpreted as a -linear (over the algebra of smooth real-valued functions on ) mapping , where is the -module of smooth vector fields on . Forms of degree one are also known as Pfaffian forms. An example of such a form is the differential of a smooth function on , which is defined as follows: , , is the derivative of in the direction of the field . Riemannian metrics on a manifold serve as examples of symmetric differential forms of degree two. However, the term "differential form" is often used to denote skew-symmetric or exterior differential forms, which have the greatest number of applications.

If is a local system of coordinates in a domain , the forms constitute a basis of the cotangent space , . For this reason (cf. Exterior algebra) any exterior -form may be written in in the form

(1)

where the are functions on . In particular,

Let be the space of all exterior -forms of class , where . The exterior multiplication converts (where ) to an associative graded algebra over which satisfies the condition of graded commutativity

(2)

A smooth mapping between manifolds induces a homomorphism between -algebras.

The concept of the differential of a function is generalized as follows. For any there exists a unique -linear mapping (exterior differentiation), which for coincides with the differential introduced above, with the following properties:

The exterior differential of a form written in in local coordinates (see (1)) is expressed by the formula

Its coordinate-free notation is

where . The Lie derivative operator , , on differential forms is connected with the exterior differentiation operator by the relation

where is the operator of interior multiplication by :

The complex is a cochain complex (the de Rham complex). The cocycles of this complex are said to be closed forms, while the coboundaries are known as exact forms. According to the de Rham theorem, the cohomology algebra

of the de Rham complex is isomorphic to the real cohomology algebra of the manifold . In particular, if (Poincaré's lemma).

The de Rham theorem is closely connected with another operation, that of integration of differential forms. Let be a bounded domain in and let be a smooth mapping , defined in a neighbourhood of the closure . If , then , where is a smooth function in . The integral of the form over the surface is defined by the formula:

If the boundary of is piecewise smooth, the formula

(3)

is valid; here is defined as the sum of the integrals of the form over the smooth pieces of the boundary, provided with their natural parametrizations. The classical formulas of Newton–Leibniz, Green–Ostrogradski and Stokes (see also Stokes theorem) are all special cases of this formula. By virtue of formula (3) each closed -form defines a -dimensional singular cocycle whose value on the simplex is . This correspondence is a realization of the isomorphism given by de Rham's theorem.

Formula (3) was published in 1899 by H. Poincaré [2], who regarded exterior forms as integrand expressions in integral invariants. At the same time E. Cartan [3] gave an almost-modern definition of exterior forms and of the exterior differentiation operator (at first on Pfaffian forms), stressing the connection between his own construction and exterior algebra.

As well as the exterior scalar forms defined above, one may also study exterior differential forms with values in a vector space over . If is an algebra, then a natural multiplication (an extension of the exterior multiplication) is defined on the space of forms with values in . If the algebra is also associative, is associative as well; if is commutative, is graded-commutative (formula (2)); if is a Lie algebra, then is a graded Lie algebra. The following, even more general, concept is also often considered. Let be a smooth vector bundle with base . If for each point there is given a skew-symmetric -linear function on with values in the fibre of the bundle , a so-called -valued -form is obtained. An -valued -form can also be interpreted as a -linear (over ) mapping of the module into the module of smooth sections of . The space of such forms is denoted by . If is given by locally constant transition functions or, which amounts to the same thing, if a flat connection is specified on , it is possible to define the de Rham complex and to generalize the de Rham theorem to this case.

Forms with values in the tangent bundle are also called vector differential forms; these forms may be identified with times covariant and one time contravariant tensor fields on which are skew-symmetric with respect to the covariant indices. Vector differential forms are used to describe the derivations of the algebra of exterior forms [4]. Vector forms (as well as their generalization — jet forms) are used in the theory of deformations of complex and other differential-geometric structures on manifolds.

Analogues of differential forms are also constructed in simplicial theory. One such construction, whose idea is due to H. Whitney [5], may be used to calculate the rational cohomology of a simplicial complex . A piecewise-linear form (or -form) on is a compatible family of differential forms defined on the simplices of the complex , with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The -forms on form a graded commutative differential algebra over . The integration of forms determines an isomorphism of the cohomology algebra of this algebra onto the algebra , where is the polyhedron corresponding to the complex . The algebra also completely defines the rational homotopy type (in particular, the ranks of homotopy groups) of . In a similar manner, the algebra on a differentiable manifold defines the real homotopy type of [9], [11].

The calculus of exterior forms on a complex analytic manifold has a number of special features [6]. In this situation it is usual to consider the space of complex-valued forms, or the spaces where is a holomorphic vector bundle on . The following decomposition is valid:

where is the space of forms of type , i.e. of forms which are locally representable as

where is a local analytic coordinate system on . Similarly,

Further, , where

Here , so that and define cochain complexes. The best known is the complex of the operator (the Dolbeault complex), the cohomology of which is denoted by . -cocycles of type are holomorphic -forms (cf. Holomorphic form). The following Grothendieck lemma is valid for : If is a form of type with in a neighbourhood of zero of the space and , then a smaller neighbourhood of zero contains a form of type such that . The Dolbeault complex may also be defined for -valued forms where is a holomorphic vector bundle. This leads to the cohomology spaces . The Grothendieck lemma implies the following isomorphism:

where is the sheaf of germs of holomorphic -valued -forms (Dolbeault's theorem). In particular,

where is the sheaf of germs of holomorphic -forms on . There exists a spectral sequence with first term and converging to . The Euler characteristic of a compact complex manifold is expressed in terms of the Dolbeault cohomology spaces by the formula

Differential forms are an important component of the apparatus of differential geometry [7], . They are also systematically employed in topology, in the theory of differential equations, in mechanics, in the theory of complex manifolds, and in the theory of functions of several complex variables. Currents are a generalization of differential forms, similar to generalized functions. The algebraic analogue of the theory of differential forms (cf. Derivations, module of) makes it possible to define differential forms on algebraic varieties and analytic spaces (cf. Differential calculus (on analytic spaces)). See also de Rham cohomology; Differential on a Riemann surface; Harmonic form; Holomorphic form; Laplace operator.

References

[1]	G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) Zbl 0534.58003
[2]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 3 , Gauthier-Villars (1899) pp. Chapt. 26 MR0926908 MR0926907 MR0926906 MR0087814 MR0087813 MR0087812 Zbl 30.0834.08
[3]	E. Cartan, "Sur certaines expressions différentielles et le problème de Pfaff" , Oeuvres complètes , 1, Pt. 2 , Gauthier-Villars pp. 303–396 MR1508969 Zbl 30.0313.04
[4]	A. Frölicher, A. Nijenhuis, "Theory of vector-valued differential forms. I. Derivations in the graded ring of differential forms" Proc. Koninkl. Ned. Akad. Wet. Ser. A , 59 : 3 (1956) pp. 338–359 MR0082554 Zbl 0079.37502
[5]	H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) MR0087148 Zbl 0083.28204
[6]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
[7]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[8a]	H. Cartan, "Calcul différentiel" , Hermann (1967) MR0223194 Zbl 0156.36102
[8b]	H. Cartan, "Formes différentielles" , Hermann (1967) MR0231303 Zbl 0184.12701
[9]	P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, "The real homotopy of Kaehler manifolds" Invent. Math. , 29 (1975) pp. 245–274
[10]	R. Bott, L.W. Tu, "Differential forms in algebraic topology" , Springer (1982) MR0658304 Zbl 0496.55001
[11]	D. Sullivan, "Infinitesimal computations in topology" Publ. Math. IHES , 47 (1977) pp. 269–331 MR0646078 Zbl 0374.57002

Comments

In the Western literature, by a differential -form on a differentiable manifold one always means a smooth section of the -th exterior power of the tangent bundle , i.e. a smooth section of . (These are called exterior differential forms in the main article.)

The value of the operator on a differential form is also called the pullback of under the infinitesimal transformation generated by the vector field .

Let be a -manifold of dimension . A current on is a linear functional defined on the space of smooth (anti-symmetric) differential forms of compact support on . The current is said to be homogeneous of dimension if for all with .

The degree of a current is , if is its dimension (if is homogeneous).

Define a chain element in (of dimension ) as a smooth mapping of the standard cube of dimension into . If is a -form on , then is defined as if (cf. the main article above). If is smooth and orientation-preserving, the integral only depends on the image (and on over ). More generally, a -chain on a manifold is a formal linear combination of chain elements. The corresponding integral is defined as . Each -chain thus defines a current on of dimension and degree . If is a -form, it defines a homogeneous current by the formula . Let be a contravariant -vector with local coordinates . Let the local coordinates of the -form be ; then

defines a current of dimension . Thus, currents generalize both forms (i.e. covariant vectors) and contravariant vectors. They are also a global generalization of the idea of distributions (as a generalization of functions). The role of a space of test functions is played by the smooth forms of compact support. The name "current" comes from the fact that in the currents of dimension 1 can be interpreted as electric currents.

In the theory of several complex variables one defines the currents of type over a domain as the (complex-valued) linear functionals on the space of (complex-valued) forms with compact support. In this area currents and their applications are a very active field of research at the moment [a2].

References

[a1]	E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) MR1111477 Zbl 0683.32002
[a2]	P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1986) MR0837659 Zbl 0583.32001

A differential form on an algebraic variety is the analogue of the concept of a differential form on a differentiable manifold. Let be an irreducible algebraic variety of dimension over an algebraically closed field (cf. Irreducible variety) and let be its field of rational functions. A differential form of degree on is an element of the -space

where is the module of derivations (cf. Derivations, module of) of the field over the field . If is a separable basis of transcendency of the extension , any differential form can be written as

where . A differential form is called regular on an open set if belongs to the submodule of , regarded as a module over the ring of regular functions on the subset . A differential form is called regular if any point has a neighbourhood such that is regular on . The regular differential forms on form a module over , denoted by . Its elements are identified with the sections of the sheaf over the variety . In a neighbourhood of each point a regular differential form is written as

where the functions , are regular at the point . If is a complete algebraic variety, the spaces are finite-dimensional, and if is non-singular, the dimension is known as the geometric genus of the variety . If is a complete variety over the field of complex numbers, the space is identical with the space of holomorphic differential forms of degree on the corresponding analytic space .

Let be a normal variety and let ; for any point of codimension one the differential form may be written as

(*)

where belongs to the field of fractions of the local ring , is the generator of its maximal ideal, and is a separable basis of transcendency over of the residue field of the ring . The value of the valuation at the element , as defined by the ring , does not depend on the choice of the representation of in the form (*) and is denoted by . The divisor

is defined and is known as divisor of the differential form . A differential form is regular if and only if its divisor , i.e. for all . The divisors of any two differential forms are equivalent and, moreover, the divisors of all differential forms on a given algebraic variety form a divisor class with respect to linear equivalence. This class is known as the canonical class of the variety and is denoted by . For a non-singular variety the class is identical with the first Chern class of the invertible sheaf ; in particular,

for any .

Any dominant rational mapping between algebraic varieties induces a canonical homomorphism

If and are non-singular and is complete, maps regular differential forms to regular ones. In particular, if two non-singular complete varieties and are birationally isomorphic, the vector spaces and are isomorphic over the field .

For any the elements of the -th symmetric power of the -space are known as -tuple differential forms of degree on . Each such differential form may be considered as a rational section of the sheaf . The regular sections

are known as regular -tuple differential forms of degree on . For a non-singular complete variety the dimension

is known as the genus of the variety . The genera of birationally isomorphic varieties are identical.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902
[3]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001

I.V. Dolgachev

Comments

In terms of the description of (Cartier) divisors by local functions (cf. Divisor), the divisor associated to a differential form of degree , , smooth, can be described as follows. For each there is an open affine such that can be represented in as . Now cover by open affine . Let the representation of in be . Then on , is equal to times the Jacobian of the with respect to the . Thus the local functions on define a divisor on , denoted by . One has for all and thus, because is one-dimensional over , all define the same divisor class, the canonical class (canonical divisor class) of .