Differential form

A differential form of degree $ p $, a $ p $- form, on a differentiable manifold $ M $ is a $ p $ times covariant tensor field on $ M $. It may also be interpreted as a $ p $- linear (over the algebra $ F( M) $ of smooth real-valued functions on $ M $) mapping $ {\mathcal X} ( M) ^ {p} \rightarrow F( M) $, where $ {\mathcal X} ( M) $ is the $ F( M) $- module of smooth vector fields on $ M $. Forms of degree one are also known as Pfaffian forms. An example of such a form is the differential $ df $ of a smooth function $ f $ on $ M $, which is defined as follows: $ ( df ) ( X) $, $ X \in {\mathcal X} ( M) $, is the derivative $ Xf $ of $ f $ in the direction of the field $ X $. Riemannian metrics on a manifold $ M $ 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 $ ( x ^ {1} \dots x ^ {n} ) $ is a local system of coordinates in a domain $ U \subset M $, the forms $ dx ^ {1} \dots dx ^ {n} $ constitute a basis of the cotangent space $ T _ {x} ( M) ^ {*} $, $ x \in U $. For this reason (cf. Exterior algebra) any exterior $ p $- form $ \alpha $ may be written in $ U $ in the form

$$ \tag{1 } \alpha = \sum _ {i _ {1} \dots i _ {p} } a _ {i _ {1} \dots i _ {p} } dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } , $$

where the $ a _ {i _ {1} \dots i _ {p} } $ are functions on $ U $. In particular,

$$ df = \frac{\partial f }{\partial x ^ {i} } dx ^ {i} . $$

Let $ E ^ {p} = E ^ {p} ( M) $ be the space of all exterior $ p $- forms of class $ C ^ \infty $, where $ E ^ {0} ( M) = F ( M) $. The exterior multiplication $ \alpha \wedge \beta $ converts $ E ^ {*} ( M) = \sum _ {p = 0 } ^ {n} E ^ {p} ( M) $( where $ n = \mathop{\rm dim} M $) to an associative graded algebra over $ F ( M) $ which satisfies the condition of graded commutativity

$$ \tag{2 } \alpha \wedge \beta = ( - 1 ) ^ {pq} \beta \wedge \alpha ,\ \ \alpha \in E ^ {p} ,\ \beta \in E ^ {q} . $$

A smooth mapping between manifolds $ f : M \rightarrow N $ induces a homomorphism $ f ^ { * } : E ^ {*} ( N) \rightarrow E ^ {*} ( M) $ between $ \mathbf R $- algebras.

The concept of the differential of a function is generalized as follows. For any $ p \geq 0 $ there exists a unique $ \mathbf R $- linear mapping $ d : E ^ {p} \rightarrow E ^ {p+} 1 $( exterior differentiation), which for $ p = 0 $ coincides with the differential introduced above, with the following properties:

$$ d ( \alpha \wedge \beta ) = d \alpha \wedge \beta + ( - 1 ) ^ {p} \alpha \wedge d \beta , $$

$$ \alpha \in E ^ {p} ,\ \beta \in E ^ {q} ,\ d ( d \alpha ) = 0 . $$

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

$$ d \alpha = \sum _ {i _ {1} \dots i _ {p} } da _ {i _ {1} \dots i _ {p} } \wedge dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } . $$

Its coordinate-free notation is

$$ d \alpha ( X _ {1} \dots X _ {p+} 1 ) = $$

$$ = \ \sum _ {i = 1 } ^ { {p } + 1 } (- 1) ^ {i+} 1 X _ {i} \alpha ( X _ {1} \dots \widehat{X} _ {i} \dots X _ {p+} 1 ) + $$

$$ - \sum _ {i < j } (- 1) ^ {i+} j \alpha ( [ X _ {i} , X _ {j} ] , X _ {1} \dots \widehat{X} _ {i} \dots \widehat{X} _ {j} \dots X _ {p+} 1 ) , $$

where $ X _ {1} \dots X _ {p+} 1 \in {\mathcal X} ( M) $. The Lie derivative operator $ L _ {X} $, $ X \in {\mathcal X} ( M) $, on differential forms is connected with the exterior differentiation operator by the relation

$$ L _ {X} = d \circ \iota _ {X} + \iota _ {X} \circ d , $$

where $ \iota _ {X} : E ^ {p} \rightarrow E ^ {p-} 1 $ is the operator of interior multiplication by $ X $:

$$ ( \iota _ {X} \alpha ) ( X _ {1} \dots X _ {p-} 1 ) = \alpha ( X , X _ {1} \dots X _ {p-} 1 ) , $$

$$ \alpha \in E ^ {p} ( M) ,\ X _ {1} \dots X _ {p-} 1 \in {\mathcal X} ( M) . $$

The complex $ ( E ^ {*} ( M) , d ) $ 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

$$ H ^ {*} ( M) = \sum _ {p = 0 } ^ { n } H ^ {p} ( M) $$

of the de Rham complex is isomorphic to the real cohomology algebra $ H ^ {*} ( M, \mathbf R ) $ of the manifold $ M $. In particular, $ H ^ {p} ( \mathbf R ^ {n} ) = 0 $ if $ p > 0 $( Poincaré's lemma).

The de Rham theorem is closely connected with another operation, that of integration of differential forms. Let $ D $ be a bounded domain in $ \mathbf R ^ {p} $ and let $ s $ be a smooth mapping $ \mathbf R ^ {p} \rightarrow M $, defined in a neighbourhood of the closure $ \overline{D}\; $. If $ \alpha \in E ^ {p} ( M) $, then $ s {} ^ \star \alpha = a dx ^ {1} \wedge \dots \wedge dx ^ {p} $, where $ a $ is a smooth function in $ \overline{D}\; $. The integral of the form $ \alpha $ over the surface $ s $ is defined by the formula:

$$ \int\limits _ { s } \alpha = \int\limits _ { D } a ( x _ {1} \dots x _ {p} ) dx ^ {1} \dots dx ^ {p} . $$

If the boundary of $ D $ is piecewise smooth, the formula

$$ \tag{3 } \int\limits _ { s } d \alpha = \int\limits _ {\partial s } \alpha ,\ \ \alpha \in E ^ {p-} 1 ( M) , $$

is valid; here $ \int _ {\partial s } \alpha $ is defined as the sum of the integrals of the form $ \alpha $ 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 $ p $- form $ \alpha $ defines a $ p $- dimensional singular cocycle whose value on the simplex $ s $ is $ \int _ {s} \alpha $. 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 $ V $ over $ \mathbf R $. If $ V $ is an algebra, then a natural multiplication (an extension of the exterior multiplication) is defined on the space $ E ( M , V ) $ of forms with values in $ V $. If the algebra $ V $ is also associative, $ E ( M , V ) $ is associative as well; if $ V $ is commutative, $ E ( M , V ) $ is graded-commutative (formula (2)); if $ V $ is a Lie algebra, then $ E ( M , V ) $ is a graded Lie algebra. The following, even more general, concept is also often considered. Let $ F $ be a smooth vector bundle with base $ M $. If for each point $ x \in M $ there is given a skew-symmetric $ p $- linear function on $ T _ {x} ( M) $ with values in the fibre $ F _ {x} $ of the bundle $ F $, a so-called $ F $- valued $ p $- form is obtained. An $ F $- valued $ p $- form can also be interpreted as a $ p $- linear (over $ F ( M) $) mapping of the module $ {\mathcal X} ( M) ^ {p} $ into the module of smooth sections of $ F $. The space of such forms is denoted by $ E ^ {p} ( F ) $. If $ F $ is given by locally constant transition functions or, which amounts to the same thing, if a flat connection is specified on $ F $, 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 $ T ( M) $ are also called vector differential forms; these forms may be identified with $ p $ times covariant and one time contravariant tensor fields on $ M $ 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 $ E ( M) $[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 $ K $. A piecewise-linear form (or $ PL $- form) on $ K $ is a compatible family of differential forms defined on the simplices of the complex $ K $, with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The $ PL $- forms on $ K $ form a graded commutative differential algebra $ E _ {PL} ^ {*} ( K) $ over $ \mathbf Q $. The integration of forms determines an isomorphism of the cohomology algebra of this algebra onto the algebra $ H ^ {*} ( | K | , \mathbf Q ) $, where $ | K | $ is the polyhedron corresponding to the complex $ K $. The algebra $ E _ {PL} ^ {*} ( K) $ also completely defines the rational homotopy type (in particular, the ranks of homotopy groups) of $ | K | $. In a similar manner, the algebra $ E ^ {*} ( M) $ on a differentiable manifold $ M $ defines the real homotopy type of $ M $[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 $ E ^ {p} ( M , \mathbf C ) $ of complex-valued forms, or the spaces $ E ^ {p} ( F ) $ where $ F $ is a holomorphic vector bundle on $ M $. The following decomposition is valid:

$$ E ^ {p} ( M , \mathbf C ) = \sum _ {r+ s = p } E ^ {r , s } ( M) , $$

where $ E ^ {r , s } ( M) $ is the space of forms of type $ ( r , s ) $, i.e. of forms $ \alpha $ which are locally representable as

$$ \sum a _ {i _ {1} \dots i _ {r} , j _ {1} \dots j _ {s} } \ dz ^ {i _ {1} } \wedge \dots \wedge dz ^ {i _ {r} } \wedge d {\overline{z}\; } {} ^ {j _ {1} } \wedge \dots \wedge d {\overline{z}\; } {} ^ {j _ {s} } , $$

where $ ( z ^ {1} \dots z ^ {n} ) $ is a local analytic coordinate system on $ M $. Similarly,

$$ E ^ {p} ( F ) = \sum _ {r + s = p } E ^ {r , s } ( F ) . $$

Further, $ d = d ^ \prime + d ^ {\prime\prime} $, where

$$ d ^ \prime : E ^ {r , s } ( M) \rightarrow E ^ {r + 1 , s } ( M) ,\ \ d ^ {\prime\prime} : E ^ {r , s } ( M) \rightarrow E ^ {r , s + 1 } ( M) . $$

Here $ {d } ^ \prime 2 = {d } ^ {\prime\prime} 2 = 0 $, so that $ d ^ \prime $ and $ d ^ {\prime\prime} $ define cochain complexes. The best known is the complex of the operator $ d ^ {\prime\prime} $( the Dolbeault complex), the cohomology of which is denoted by $ H ^ {r , s } ( M) $. $ d ^ {\prime\prime} $- cocycles of type $ ( p , 0 ) $ are holomorphic $ p $- forms (cf. Holomorphic form). The following Grothendieck lemma is valid for $ d ^ {\prime\prime} $: If $ \alpha $ is a form of type $ ( r , s ) $ with $ s > 0 $ in a neighbourhood of zero of the space $ \mathbf C ^ {n} $ and $ d ^ {\prime\prime} \alpha = 0 $, then a smaller neighbourhood of zero contains a form $ \beta $ of type $ ( r , s - 1 ) $ such that $ \alpha = d ^ {\prime\prime} \beta $. The Dolbeault complex may also be defined for $ F $- valued forms where $ F $ is a holomorphic vector bundle. This leads to the cohomology spaces $ H ^ {r , s } ( F ) $. The Grothendieck lemma implies the following isomorphism:

$$ H ^ {r , s } ( F ) \cong H ^ {s} ( M , \Omega ^ {r} ( F ) ) , $$

where $ \Omega ^ {r} ( F ) $ is the sheaf of germs of holomorphic $ F $- valued $ r $- forms (Dolbeault's theorem). In particular,

$$ H ^ {r , s } ( M) \cong H ^ {s} ( M , \Omega ^ {r} ( M) ) , $$

where $ \Omega ^ {r} ( M) $ is the sheaf of germs of holomorphic $ r $- forms on $ M $. There exists a spectral sequence with first term $ \sum _ {r , s } H ^ {r , s } ( M) $ and converging to $ H ^ {*} ( M , \mathbf C ) $. The Euler characteristic $ \chi ( M) $ of a compact complex manifold $ M $ is expressed in terms of the Dolbeault cohomology spaces by the formula

$$ \chi ( M) = \sum _ {r , s } (- 1) ^ {r+} s \mathop{\rm dim} H ^ {r , s } ( M) . $$

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 $ p $- form on a differentiable manifold $ M $ one always means a smooth section of the $ p $- th exterior power of the tangent bundle $ T M $, i.e. a smooth section of $ \Lambda ^ {p} ( T M ) $. (These are called exterior differential forms in the main article.)

The value of the operator $ L _ {X} $ on a differential form $ x $ is also called the pullback of $ x $ under the infinitesimal transformation generated by the vector field $ X $.

Let $ X $ be a $ C ^ \infty $- manifold of dimension $ n $. A current on $ X $ is a linear functional defined on the space $ \Omega _ {c} ( X) $ of smooth (anti-symmetric) differential forms of compact support on $ X $. The current $ T $ is said to be homogeneous of dimension $ p $ if $ T ( \phi ) = 0 $ for all $ \phi \in \Omega _ {c} ^ {q} ( X) $ with $ q \neq p $.

The degree of a current $ T $ is $ n - p $, if $ p $ is its dimension (if $ T $ is homogeneous).

Define a chain element in $ X $( of dimension $ p $) as a smooth mapping $ s : \Delta _ {p} \rightarrow X $ of the standard cube of dimension $ p $ into $ X $. If $ \omega $ is a $ p $- form on $ X $, then $ \int _ {s} \omega $ is defined as $ \int _ {\Delta _ {p} } f $ if $ s {} ^ \star ( \omega ) = f d x _ {1} \wedge \dots \wedge d x _ {p} $( cf. the main article above). If $ s $ is smooth and orientation-preserving, the integral $ \int _ {s} \omega $ only depends on the image $ s ( \Delta _ {p} ) \subset X $( and on $ \omega $ over $ s ( \Delta _ {p} ) $). More generally, a $ p $- chain on a manifold $ X $ is a formal linear combination $ c = \sum a _ {i} s _ {i} $ of chain elements. The corresponding integral is defined as $ \int _ {c} \omega = \sum a _ {i} \int _ {s _ {i} } \omega $. Each $ p $- chain thus defines a current on $ X $ of dimension $ p $ and degree $ n - p $. If $ \alpha $ is a $ q $- form, it defines a homogeneous $ n - q $ current by the formula $ T _ \alpha ( \omega ) = \int _ {V} \alpha \wedge \omega $. Let $ b $ be a contravariant $ p $- vector with local coordinates $ b ^ {i _ {1} \dots i _ {p} } $. Let the local coordinates of the $ p $- form $ \omega $ be $ \omega _ {i _ {1} \dots i _ {p} } $; then

$$ T _ {b} ( \omega ) = \sum _ {i _ {1} < \dots < i _ {p} } \omega _ {i _ {1} \dots i _ {p} } b ^ {i _ {1} \dots i _ {p} } $$

defines a current of dimension $ p $. 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 $ \mathbf R ^ {3} $ the currents of dimension 1 can be interpreted as electric currents.

In the theory of several complex variables one defines the currents of type $ ( p , q ) $ over a domain $ D $ as the (complex-valued) linear functionals on the space $ \Omega _ {c} ^ {n - p , n - q } ( D) $ of $ ( n - p , n - q ) $( 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 $ X $ be an irreducible algebraic variety of dimension $ d $ over an algebraically closed field $ k $( cf. Irreducible variety) and let $ K $ be its field of rational functions. A differential form of degree $ r $ on $ X $ is an element of the $ K $- space

$$ \Omega ^ {r} ( X) = \wedge ^ { r } \Omega _ {K / k } ^ {1} , $$

where $ \Omega _ {K/k} $ is the module of derivations (cf. Derivations, module of) of the field $ K $ over the field $ k $. If $ x _ {1} \dots x _ {d} $ is a separable basis of transcendency of the extension $ K/k $, any differential form $ \omega \in \Omega ^ {r} ( X) $ can be written as

$$ \omega = \sum a _ {i _ {1} \dots i _ {r} } dx _ {i} \wedge \dots \wedge dx _ {i _ {r} } , $$

where $ a _ {i _ {1} \dots i _ {r} } \in K $. A differential form $ \omega $ is called regular on an open set $ U \subset X $ if $ \omega $ belongs to the submodule $ \Omega _ {k [ U] / k } ^ {r} $ of $ \Omega ^ {r} ( X) $, regarded as a module over the ring $ k [ U] $ of regular functions on the subset $ U $. A differential form $ \omega $ is called regular if any point $ x \in X $ has a neighbourhood $ U $ such that $ \omega $ is regular on $ U $. The regular differential forms on $ X $ form a module over $ k [ X] $, denoted by $ \Omega ^ {r} [ X] $. Its elements are identified with the sections of the sheaf $ \Omega _ {X/k} ^ {r} $ over the variety $ X $. In a neighbourhood of each point $ x \in X $ a regular differential form $ \omega \subset \Omega ^ {r} [ X] $ is written as

$$ \omega = \sum \alpha _ {i _ {1} \dots i _ {r} } df _ {i _ {1} } \wedge \dots \wedge df _ {i _ {r} } , $$

where the functions $ \alpha _ {i _ {1} \dots i _ {r} } $, $ f _ {i _ {1} } \dots f _ {i _ {r} } $ are regular at the point $ x $. If $ X $ is a complete algebraic variety, the spaces $ \Omega ^ {r} [ X] $ are finite-dimensional, and if $ X $ is non-singular, the dimension $ p _ {g} ( X) = \mathop{\rm dim} _ {k} \Omega ^ {d} [ X] $ is known as the geometric genus of the variety $ X $. If $ X $ is a complete variety over the field of complex numbers, the space $ \Omega ^ {r} [ X] $ is identical with the space of holomorphic differential forms of degree $ r $ on the corresponding analytic space $ X ^ {\textrm{ an } } $.

Let $ X $ be a normal variety and let $ \omega \in \Omega ^ {d} [ X] $; for any point $ x \in X ^ {(} 1) $ of codimension one the differential form $ \omega $ may be written as

$$ \tag{* } \omega = a dt \wedge dt _ {1} \wedge \dots \wedge dt _ {d- 1 } , $$

where $ a $ belongs to the field of fractions $ K _ {x} $ of the local ring $ {\mathcal O} _ {X,x } $, $ t $ is the generator of its maximal ideal, and $ t _ {1} \dots t _ {d-} 1 $ is a separable basis of transcendency over $ k $ of the residue field of the ring $ {\mathcal O} _ {X,x } $. The value of the valuation at the element $ a $, as defined by the ring $ {\mathcal O} _ {X,x } $, does not depend on the choice of the representation of $ \omega $ in the form (*) and is denoted by $ \nu _ {x} ( \omega ) $. The divisor

$$ D = \sum _ {x \in X ^ {(} 1) } \nu _ {x} ( \omega ) \{ \overline{x}\; \} $$

is defined and is known as divisor of the differential form $ \omega $. A differential form $ \omega $ is regular if and only if its divisor $ D \geq 0 $, i.e. $ \nu _ {x} ( \omega ) \geq 0 $ for all $ x \in X ^ {(} 1) $. 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 $ X $ and is denoted by $ K _ {X} $. For a non-singular variety $ X $ the class $ K _ {X} $ is identical with the first Chern class of the invertible sheaf $ \Omega _ {X/k } ^ {d} $; in particular,

$$ \Omega _ {X / k } ^ {d} \simeq {\mathcal O} _ {X} ( D ) $$

for any $ D \in K _ {X} $.

Any dominant rational mapping between algebraic varieties $ f : X ^ { \prime } \rightarrow X $ induces a canonical homomorphism

$$ f ^ { * } : \Omega ^ {r} ( X) \rightarrow \Omega ^ {r} ( X ^ { \prime } ) . $$

If $ X $ and $ X ^ { \prime } $ are non-singular and $ X $ is complete, $ f ^ { * } $ maps regular differential forms to regular ones. In particular, if two non-singular complete varieties $ X $ and $ X ^ { \prime } $ are birationally isomorphic, the vector spaces $ \Omega ^ {r} [ X] $ and $ \Omega ^ {r} [ X ^ { \prime } ] $ are isomorphic over the field $ k $.

For any $ i > 1 $ the elements of the $ i $- th symmetric power $ S ^ {i} ( \Omega ^ {r} ( X) ) $ of the $ K $- space $ \Omega ^ {r} ( X) $ are known as $ i $- tuple differential forms of degree $ r $ on $ X $. Each such differential form may be considered as a rational section of the sheaf $ S ^ {i} ( \Omega _ {X/k } ^ {r} ) $. The regular sections

$$ \omega \in \Gamma ( X , S ^ {i} ( \Omega _ {X/k } ^ {r} ) ) $$

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

$$ P _ {i} ( X) = \mathop{\rm dim} _ {k} \Gamma ( X , S ^ {i} ( \Omega _ {X/k } ^ {d} )) $$

is known as the $ i $ genus of the variety $ X $. The $ i $ 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 $ \omega $ of degree $ n $, $ n = \mathop{\rm dim} X $, $ X $ smooth, can be described as follows. For each $ x \in X $ there is an open affine $ U $ such that $ \omega $ can be represented in $ U $ as $ \omega = g d u _ {1} \wedge \dots \wedge d u _ {n} $. Now cover $ X $ by open affine $ U _ {i} $. Let the representation of $ \omega $ in $ U _ {i} $ be $ \omega = g ^ {(} i) d u _ {1} ^ {(} i) \wedge \dots \wedge d u _ {n} ^ {(} i) $. Then on $ U _ {i} \cap U _ {j} $, $ g ^ {(} j) $ is equal to $ g ^ {(} i) $ times the Jacobian of the $ u _ {1} ^ {(} i) \dots u _ {n} ^ {(} i) $ with respect to the $ u _ {1} ^ {(} j) \dots u _ {n} ^ {(} j) $. Thus the local functions $ g ^ {(} i) $ on $ U _ {i} $ define a divisor on $ X $, denoted by $ ( \omega ) $. One has $ ( f \omega ) = ( f ) + ( \omega ) $ for all $ f \in k ( X) $ and thus, because $ \Omega ^ {n} ( X) $ is one-dimensional over $ k ( X) $, all $ 0 \neq \omega \in \Omega ^ {n} ( X) $ define the same divisor class, the canonical class (canonical divisor class) of $ X $.