# differential forms in algebraic geometry

#### December 12, 2020   |

d , Differential forms are an important component of the apparatus of differential geometry , . δ 1 On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. n More generally, an m-form is an oriented density that can be integrated over an m-dimensional oriented manifold. , and it is integrated just like a surface integral. ∈ j ∂ ⋀ , This form is denoted ω / ηy. 2 For n > 1, such a function does not always exist: any smooth function f satisfies, so it will be impossible to find such an f unless, The skew-symmetry of the left hand side in i and j suggests introducing an antisymmetric product ∧ on differential 1-forms, the exterior product, so that these equations can be combined into a single condition, This is an example of a differential 2-form. (Note: this is a pretty serious book, so will take some time. ), Using the above-mentioned definitions, Maxwell's equations can be written very compactly in geometrized units as. k An orientation of a k-submanifold is therefore extra data not derivable from the ambient manifold. because the square whose first side is dx1 and second side is dx2 is to be regarded as having the opposite orientation as the square whose first side is dx2 and whose second side is dx1. i Moreover, it is also possible to define parametrizations of k-dimensional subsets for k < n, and this makes it possible to define integrals of k-forms. On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate n-forms over compact subsets, with the two choices differing by a sign. The Yang–Mills field F is then defined by. E The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of ω is independent of the chosen chart. Not logged in This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. 1 k ⋯ Abelian differentials usually mean differential one-forms on an algebraic curve or Riemann surface. Differentials are also important in algebraic geometry, and there are several important notions. pp 68-130 | k A general two-form is a linear combination of these at every point on the manifold: I A smooth differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all differential k-forms on a manifold M is a vector space, often denoted Ωk(M). ∫ 0 ( ≤ ⋀ Each of these represents a covector at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. , . ∗ ) Authors; Authors and affiliations; William Hodge; Chapter. Let M be a smooth manifold. = What we're actually describing here are the exterior differential forms; for more general concepts, see absolute differential form and cogerm differential form. d μ Fix x ∈ M and set y = f(x). , I A common notation for the wedge product of elementary m-forms is so called multi-index notation: in an n-dimensional context, for The 2-form To summarize: dα = 0 is a necessary condition for the existence of a function f with α = df. A differential 0-form ("zero-form") is defined to be a smooth function f on U – the set of which is denoted C∞(U). Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form A. Integration of differential forms is well-defined only on oriented manifolds. i The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. ∼ Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics). ω In the general case, use a partition of unity to write ω as a sum of n-forms, each of which is supported in a single positively oriented chart, and define the integral of ω to be the sum of the integrals of each term in the partition of unity. Differential Algebraic Topology From Stratifolds to Exotic Spheres Matthias Kreck American Mathematical Society Providence, Rhode Island Graduate Studies in Mathematics Volume 110. Then the pullback of a k-form ω is the composite, Another abstract way to view the pullback comes from viewing a k-form ω as a linear functional on tangent spaces. Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry. ), In particular, if v = ej is the jth coordinate vector then ∂v f is the partial derivative of f with respect to the jth coordinate function, i.e., ∂f / ∂xj, where x1, x2, ..., xn are the coordinate functions on U. In R3, with the Hodge star operator, the exterior derivative corresponds to gradient, curl, and divergence, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution. < There are gauge theories, such as Yang–Mills theory, in which the Lie group is not abelian. ∫ Ω {z_{\beta} ^\alpha }\left( {i \ne \alpha ,\beta } \right)\,\,;\,z_\alpha ^\beta = \frac{1} ) Precomposing this functional with the differential df : TM → TN defines a linear functional on each tangent space of M and therefore a differential form on M. The existence of pullbacks is one of the key features of the theory of differential forms. , The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. A differential 2-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. Every smooth n-form ω on U has the form. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below for details. Let θ be an m-form on M, and let ζ be an n-form on N that is almost everywhere positive with respect to the orientation of N. Then, for almost every y ∈ N, the form θ / ζy is a well-defined integrable m − n form on f−1(y). Differential forms in algebraic geometry. Any differential 1-form arises this way, and by using (*) it follows that any differential 1-form α on U may be expressed in coordinates as, The second idea leading to differential forms arises from the following question: given a differential 1-form α on U, when does there exist a function f on U such that α = df? j the dual of the kth exterior power is isomorphic to the kth exterior power of the dual: By the universal property of exterior powers, this is equivalently an alternating multilinear map: Consequently, a differential k-form may be evaluated against any k-tuple of tangent vectors to the same point p of M. For example, a differential 1-form α assigns to each point p ∈ M a linear functional αp on TpM. Accord­ ingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. d In higher dimensions, dxi1 ∧ ⋅⋅⋅ ∧ dxim = 0 if any two of the indices i1, ..., im are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero. Ω m < i 1 This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold. j is the determinant of the Jacobian. The use of differential forms avoids the painful and for the beginner unmotivated homological algebra in algebraic topology. M R , while ( , which is dual to the Faraday form, is also called Maxwell 2-form. ∂ Suppose first that ω is supported on a single positively oriented chart. The differential of f is a smooth map df : TM → TN between the tangent bundles of M and N. This map is also denoted f∗ and called the pushforward. The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection. That is: This gives a geometrical context to the conventions for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. ∧ d × 2 M ) This may be thought of as an infinitesimal oriented square parallel to the xi–xj-plane. A function times this Hausdorff measure can then be integrated over k-dimensional subsets, providing a measure-theoretic analog to integration of k-forms. For example, given a path γ(t) : [0, 1] → R2, integrating a 1-form on the path is simply pulling back the form to a form f(t) dt on [0, 1], and this integral is the integral of the function f(t) on the interval. 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. , Moreover, there is an integrable n-form on N defined by, Then (Dieudonne 1972) harv error: no target: CITEREFDieudonne1972 (help) proves the generalized Fubini formula, It is also possible to integrate forms of other degrees along the fibers of a submersion. Following (Dieudonne 1972) harv error: no target: CITEREFDieudonne1972 (help), there is a unique, which may be thought of as the fibral part of ωx with respect to ηy. = The definition of a differential form may be restated as follows. Then σx is defined by the property that, Moreover, for fixed y, σx varies smoothly with respect to x. I A differential k-form can be integrated over an oriented k-dimensional manifold. On non-orientable manifold, n-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no volume forms on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate n-forms. n , Here the Lie group is U(1), the one-dimensional unitary group, which is in particular abelian. In the presence of the additional data of an orientation, it is possible to integrate n-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold, [M]. Address: MAGIC, c/o College of Engineering, Mathematics and Physical Sciences, Harrison Building, Streatham Campus, University of Exeter, North Park Road, Exeter, UK EX4 4QF A k-chain is a formal sum of smooth embeddings D → M. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. F | When the k-form is defined on an n-dimensional manifold with n > k, then the k-form can be integrated over oriented k-dimensional submanifolds. On this chart, it may be pulled back to an n-form on an open subset of Rn. J Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x1, x2, ..., xn are themselves functions on U, and so define differential 1-forms dx1, dx2, ..., dxn. For example, in Maxwell's theory of electromagnetism, the Faraday 2-form, or electromagnetic field strength, is. , (Here it is a matter of convention to write Fab instead of fab, i.e. The n-dimensional Hausdorff measure yields a density, as above. If x ∈ f−1(y), then a k-vector v at y determines an (m − k)-covector at x by pullback: Each of these covectors has an exterior product against α, so there is an (m − n)-form βv on M along f−1(y) defined by, This form depends on the orientation of N but not the choice of ζ. and Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals, and so on. defines a linear functional on each tangent space of M, and therefore it factors through the trivial bundle M × R. The vector bundle morphism Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. 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