Hodge Theory
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Hodge theory, named after W. V. D. Hodge, is a method for studying the
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
s of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'' using
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. The key observation is that, given a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
on ''M'', every cohomology class has a canonical representative, a
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
that vanishes under the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, and it built on the work of
Georges de Rham Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. Biography Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in ...
on de Rham cohomology. It has major applications in two settings:
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s and
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
s. Hodge's primary motivation, the study of complex
projective varieties In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the al ...
s. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. In arithmetic situations, the tools of ''p''-adic Hodge theory have given alternative proofs of, or analogous results to, classical Hodge theory.


History

The field of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
was still nascent in the 1920s. It had not yet developed the notion of
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
, and the interaction between differential forms and topology was poorly understood. In 1928,
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
published a note entitled ''Sur les nombres de Betti des espaces de groupes clos'' in which he suggested, but did not prove, that differential forms and topology should be linked. Upon reading it, Georges de Rham, then a student, was immediately struck by inspiration. In his 1931 thesis, he proved a spectacular result now called de Rham's theorem. By
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
, integration of differential forms along
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
chains induces, for any compact smooth manifold ''M'', a bilinear pairing :H_k(M; \mathbf) \times H^k_(M; \mathbf) \to \mathbf. As originally stated, de Rham's theorem asserts that this is a
perfect pairing In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
, and that therefore each of the terms on the left-hand side are vector space duals of one another. In contemporary language, de Rham's theorem is more often phrased as the statement that singular cohomology with real coefficients is isomorphic to de Rham cohomology: :H^k_(M; \mathbf) \cong H^k_(M; \mathbf). De Rham's original statement is then a consequence of
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
. Separately, a 1927 paper of
Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
used topological methods to reprove theorems of
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
. In modern language, if ''ω''1 and ''ω''2 are holomorphic differentials on an algebraic curve ''C'', then their
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
is necessarily zero because ''C'' has only one complex dimension; consequently, the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
of their cohomology classes is zero, and when made explicit, this gave Lefschetz a new proof of the
Riemann relations In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data: * A lattice Λ in a complex vector space Cg. * An alternating bilinear form α from Λ to the integers satisfying the following Riemann bi ...
. Additionally, if ''ω'' is a non-zero holomorphic differential, then \sqrt\,\omega \wedge \bar\omega is a positive volume form, from which Lefschetz was able to rederive Riemann's inequalities. In 1929, W. V. D. Hodge learned of Lefschetz's paper. He immediately observed that similar principles applied to algebraic surfaces. More precisely, if ''ω'' is a non-zero holomorphic form on an algebraic surface, then \sqrt\,\omega \wedge \bar\omega is positive, so the cup product of \omega and \bar\omega must be non-zero. It follows that ''ω'' itself must represent a non-zero cohomology class, so its periods cannot all be zero. This resolved a question of Severi. Hodge felt that these techniques should be applicable to higher dimensional varieties as well. His colleague Peter Fraser recommended de Rham's thesis to him. In reading de Rham's thesis, Hodge realized that the real and imaginary parts of a holomorphic 1-form on a Riemann surface were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now known as the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. He further conjectured that each cohomology class should have a distinguished representative with the property that both it and its dual vanish under the exterior derivative operator; these are now called harmonic forms. Hodge devoted most of the 1930s to this problem. His earliest published attempt at a proof appeared in 1933, but he considered it "crude in the extreme".
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, one of the most brilliant mathematicians of the era, found himself unable to determine whether Hodge's proof was correct or not. In 1936, Hodge published a new proof. While Hodge considered the new proof much superior, a serious flaw was discovered by Bohnenblust. Independently, Hermann Weyl and
Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese ...
modified Hodge's proof to repair the error. This established Hodge's sought-for isomorphism between harmonic forms and cohomology classes.
In retrospect it is clear that the technical difficulties in the existence theorem did not really require any significant new ideas, but merely a careful extension of classical methods. The real novelty, which was Hodge’s major contribution, was in the conception of harmonic integrals and their relevance to algebraic geometry. This triumph of concept over technique is reminiscent of a similar episode in the work of Hodge’s great predecessor Bernhard Riemann. —
M. F. Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975, ''Biographical Memoirs of Fellows of the Royal Society'', vol. 22, 1976, pp. 169–192.


Hodge theory for real manifolds


De Rham cohomology

The Hodge theory references the
de Rham complex In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly ada ...
. Let ''M'' be a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. For a non-negative integer ''k'', let Ω''k''(''M'') be the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of smooth
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s of degree ''k'' on ''M''. The de Rham complex is the sequence of
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s :0\to \Omega^0(M) \xrightarrow \Omega^1(M)\xrightarrow \cdots\xrightarrow \Omega^n(M)\xrightarrow 0, where ''dk'' denotes the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
on Ω''k''(''M''). This is a
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
in the sense that (also written ). De Rham's theorem says that the
singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
of ''M'' with real coefficients is computed by the de Rham complex: :H^k(M,\mathbf)\cong \frac.


Operators in Hodge theory

Choose a Riemannian metric ''g'' on ''M'' and recall that: :\Omega^k(M) = \Gamma \left (\bigwedge\nolimits^k T^*(M) \right ). The metric yields an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on each fiber \bigwedge\nolimits^k(T_p^*(M)) by extending (see
Gramian matrix In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...
) the inner product induced by ''g'' from each cotangent fiber T_p^*(M) to its k^
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
: \bigwedge\nolimits^k(T_p^*(M)). The \Omega^k(M) inner product is then defined as the integral of the pointwise inner product of a given pair of ''k''-forms over ''M'' with respect to the volume form \sigma associated with ''g''. Explicitly, given some \omega,\tau \in \Omega^k(M) we have : (\omega,\tau) \mapsto \langle\omega,\tau\rangle := \int_M \langle \omega(p),\tau(p)\rangle_p \sigma. Naturally the above inner product induces a norm, when that norm is finite on some fixed ''k''-form: :\langle\omega,\omega\rangle = \, \omega\, ^2 < \infty, then the integrand is a real valued, square integrable function on ''M'', evaluated at a given point via its point-wise norms, : \, \omega(p)\, _p:M \to \mathbf\in L^2(M). Consider the
adjoint operator In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
of ''d'' with respect to these inner products: :\delta : \Omega^(M) \to \Omega^k(M). Then the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
on forms is defined by :\Delta = d\delta + \delta d. This is a second-order linear differential operator, generalizing the Laplacian for functions on R''n''. By definition, a form on ''M'' is harmonic if its Laplacian is zero: :\mathcal_\Delta^k(M) = \. The Laplacian appeared first in
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
. In particular,
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
say that the electromagnetic potential in a vacuum is a 1-form ''A'' which has exterior derivative , where ''F'' is a 2-form representing the electromagnetic field such that on spacetime, viewed as
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
of dimension 4. Every harmonic form ''α'' on a closed Riemannian manifold is closed, meaning that . As a result, there is a canonical mapping \varphi:\mathcal_\Delta^k(M)\to H^k(M,\mathbf). The Hodge theorem states that \varphi is an isomorphism of vector spaces. In other words, each real cohomology class on ''M'' has a unique harmonic representative. Concretely, the harmonic representative is the unique closed form of minimum ''L''2 norm that represents a given cohomology class. The Hodge theorem was proved using the theory of
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
partial differential equations, with Hodge's initial arguments completed by
Kodaira is a city located in the western portion of Tokyo Metropolis, Japan. , the city had an estimated population of 195,207 in 93,654 households, and a population density of 9500 persons per km². The total area of the city was . Geography Kodaira ...
and others in the 1940s. For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
. (Admittedly, there are other ways to prove this.) Indeed, the operators Δ are elliptic, and the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of an elliptic operator on a closed manifold is always a finite-dimensional vector space. Another consequence of the Hodge theorem is that a Riemannian metric on a closed manifold ''M'' determines a real-valued
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on the integral cohomology of ''M'' modulo
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
. It follows, for example, that the image of the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of ''M'' in the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
is finite (because the group of isometries of a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
is finite). A variant of the Hodge theorem is the Hodge decomposition. This says that there is a unique decomposition of any differential form ''ω'' on a closed Riemannian manifold as a sum of three parts in the form :\omega = d \alpha +\delta \beta + \gamma, in which ''γ'' is harmonic: . In terms of the ''L''2 metric on differential forms, this gives an orthogonal
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
decomposition: : \Omega^k(M) \cong \operatorname d_ \oplus \operatorname \delta_ \oplus \mathcal H_\Delta^k(M). The Hodge decomposition is a generalization of the
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
for the de Rham complex.


Hodge theory of elliptic complexes

Atiyah Atiyyah ( ar, عطية ''‘aṭiyyah''), which generally implies "something (money or goods given as regarded) received as a gift" or also means "present, gift, benefit, boon, favor, granting, giving"''.'' The name is also spelt Ateah, Atiyeh, ...
and
Bott Bott is an English and German surname. Notable people with the surname include: * Catherine Bott, English soprano * Charlie Bott, English rugby player * François Bott (born 1935) * John Bott * Leon Bott, Australian rugby league footballer * Leoni ...
defined elliptic complexes as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let E_0,E_1,\ldots,E_N be
vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
, equipped with metrics, on a closed smooth manifold ''M'' with a volume form ''dV''. Suppose that :L_i:\Gamma(E_i)\to\Gamma(E_) are linear differential operators acting on C sections of these vector bundles, and that the induced sequence : 0\to\Gamma(E_0)\to \Gamma(E_1) \to \cdots \to \Gamma(E_N) \to 0 is an elliptic complex. Introduce the direct sums: : \begin \mathcal E^\bullet &= \bigoplus\nolimits_i \Gamma(E_i) \\ L &= \bigoplus\nolimits_i L_i:\mathcal E^\bullet\to\mathcal E^\bullet \end and let ''L'' be the adjoint of ''L''. Define the elliptic operator . As in the de Rham case, this yields the vector space of harmonic sections :\mathcal H=\. Let H:\mathcal E^\bullet\to\mathcal H be the orthogonal projection, and let ''G'' be the Green's operator for Δ. The Hodge theorem then asserts the following: #''H'' and ''G'' are well-defined. #Id = ''H'' + Δ''G'' = ''H'' + ''G''Δ #''LG'' = ''GL'', ''L'G'' = ''GL'' #The cohomology of the complex is canonically isomorphic to the space of harmonic sections, H(E_j)\cong\mathcal H(E_j), in the sense that each cohomology class has a unique harmonic representative. There is also a Hodge decomposition in this situation, generalizing the statement above for the de Rham complex.


Hodge theory for complex projective varieties

Let ''X'' be a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
complex projective manifold, meaning that ''X'' is a closed complex submanifold of some
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
CP''N''. By Chow's theorem, complex projective manifolds are automatically algebraic: they are defined by the vanishing of
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
equations on CP''N''. The standard Riemannian metric on CP''N'' induces a Riemannian metric on ''X'' which has a strong compatibility with the complex structure, making ''X'' a
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
. For a complex manifold ''X'' and a natural number ''r'', every C ''r''-form on ''X'' (with complex coefficients) can be written uniquely as a sum of forms of with , meaning forms that can locally be written as a finite sum of terms, with each term taking the form :f\, dz_1\wedge\cdots\wedge dz_p\wedge d\overline \wedge\cdots\wedge d\overline with ''f'' a C function and the ''z''s and ''w''s
holomorphic function 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 derivativ ...
s. On a Kähler manifold, the components of a harmonic form are again harmonic. Therefore, for any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Kähler manifold ''X'', the Hodge theorem gives a decomposition of the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of ''X'' with complex coefficients as a direct sum of complex vector spaces: :H^r(X,\mathbf)=\bigoplus_ H^(X). This decomposition is in fact independent of the choice of Kähler metric (but there is no analogous decomposition for a general compact complex manifold). On the other hand, the Hodge decomposition genuinely depends on the structure of ''X'' as a complex manifold, whereas the group depends only on the underlying
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
of ''X''. The piece ''H''''p'',''q''(''X'') of the Hodge decomposition can be identified with a
coherent sheaf cohomology In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the ex ...
group, which depends only on ''X'' as a complex manifold (not on the choice of Kähler metric): :H^(X)\cong H^q(X,\Omega^p), where Ω''p'' denotes the
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...
of holomorphic ''p''-forms on ''X''. For example, ''H''''p'',''0''(''X'') is the space of holomorphic ''p''-forms on ''X''. (If ''X'' is projective, Serre's
GAGA Gaga ( he, גע גע literally 'touch touch') (also: ga-ga, gaga ball, or ga-ga ball) is a variant of dodgeball that is played in a gaga "pit". The game combines dodging, striking, running, and jumping, with the objective of being the last perso ...
theorem implies that a holomorphic ''p''-form on all of ''X'' is in fact algebraic.) The Hodge number ''h''''p'',''q''(''X'') means the dimension of the complex vector space ''H''''p''.''q''(''X''). These are important invariants of a smooth complex projective variety; they do not change when the complex structure of ''X'' is varied continuously, and yet they are in general not topological invariants. Among the properties of Hodge numbers are Hodge symmetry (because ''H''''p'',''q''(''X'') is the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of ''H''''q'',''p''(''X'')) and (by
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
). The Hodge numbers of a smooth complex projective variety (or compact Kähler manifold) can be listed in the Hodge diamond (shown in the case of complex dimension 2): For example, every smooth projective
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
''g'' has Hodge diamond For another example, every
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
has Hodge diamond The
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of ''X'' are the sum of the Hodge numbers in a given row. A basic application of Hodge theory is then that the odd Betti numbers ''b''2''a''+1 of a smooth complex projective variety (or compact Kähler manifold) are even, by Hodge symmetry. This is not true for compact complex manifolds in general, as shown by the example of the
Hopf surface In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) \Complex^2\setminus \ by a free action of a discrete group. If this group is the integers the Hopf surface is c ...
, which is
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to and hence has . The "Kähler package" is a powerful set of restrictions on the cohomology of smooth complex projective varieties (or compact Kähler manifolds), building on Hodge theory. The results include the
Lefschetz hyperplane theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the ...
, the
hard Lefschetz theorem Hard may refer to: * Hardness, resistance of physical materials to deformation or fracture * Hard water, water with high mineral content Arts and entertainment * ''Hard'' (TV series), a French TV series * Hard (band), a Hungarian hard rock super ...
, and the Hodge-Riemann bilinear relations. Many of these results follow from fundamental technical tools which may be proven for compact Kähler manifolds using Hodge theory, including the Kähler identities and the \partial \bar \partial-lemma. Hodge theory and extensions such as non-abelian Hodge theory also give strong restrictions on the possible
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s of compact Kähler manifolds.


Algebraic cycles and the Hodge conjecture

Let ''X'' be a smooth complex projective variety. A complex subvariety ''Y'' in ''X'' of
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the ...
''p'' defines an element of the cohomology group H^(X,\Z). Moreover, the resulting class has a special property: its image in the complex cohomology H^(X,\Complex) lies in the middle piece of the Hodge decomposition, H^(X). The Hodge conjecture predicts a converse: every element of H^(X,\Z) whose image in complex cohomology lies in the subspace H^(X) should have a positive integral multiple that is a \Z-linear combination of classes of complex subvarieties of ''X''. (Such a linear combination is called an algebraic cycle on ''X''.) A crucial point is that the Hodge decomposition is a decomposition of cohomology with complex coefficients that usually does not come from a decomposition of cohomology with integral (or rational) coefficients. As a result, the intersection :(H^(X,\Z)/)\cap H^(X)\subseteq H^(X,\Complex) may be much smaller than the whole group H^(X,\Z)/torsion, even if the Hodge number h^ is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of ''X'' (as described by cohomology) are determined by the Hodge structure of ''X'' (the combination of integral cohomology with the Hodge decomposition of complex cohomology). The Lefschetz (1,1)-theorem says that the Hodge conjecture is true for (even integrally, that is, without the need for a positive integral multiple in the statement). The Hodge structure of a variety ''X'' describes the integrals of algebraic differential forms on ''X'' over
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
classes in ''X''. In this sense, Hodge theory is related to a basic issue in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
: there is in general no "formula" for the integral of an
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
. In particular,
definite integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
s of algebraic functions, known as periods, can be
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
s. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general. Example: For a smooth complex projective K3 surface ''X'', the group is isomorphic to Z22, and ''H''1,1(''X'') is isomorphic to C20. Their intersection can have rank anywhere between 1 and 20; this rank is called the
Picard number In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ver ...
of ''X''. The
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
of all projective K3 surfaces has a
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
set of components, each of complex dimension 19. The subspace of K3 surfaces with Picard number ''a'' has dimension 20−''a''.Griffiths & Harris (1994), p. 594. (Thus, for most projective K3 surfaces, the intersection of with ''H''1,1(''X'') is isomorphic to Z, but for "special" K3 surfaces the intersection can be bigger.) This example suggests several different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety. Second, Hodge theory gives information about the moduli space of smooth complex projective varieties with a given topological type. The best case is when the
Torelli theorem In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) ''C'' is determined b ...
holds, meaning that the variety is determined up to isomorphism by its Hodge structure. Finally, Hodge theory gives information about the
Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...
of algebraic cycles on a given variety. The Hodge conjecture is about the image of the
cycle map Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
from Chow groups to ordinary cohomology, but Hodge theory also gives information about the kernel of the cycle map, for example using the
intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by put ...
s which are built from the Hodge structure.


Generalizations

Mixed Hodge theory, developed by
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
, extends Hodge theory to all complex algebraic varieties, not necessarily smooth or compact. Namely, the cohomology of any complex algebraic variety has a more general type of decomposition, a
mixed Hodge structure In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. ...
. A different generalization of Hodge theory to singular varieties is provided by
intersection homology In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them ov ...
. Namely, Morihiko Saito showed that the intersection homology of any complex projective variety (not necessarily smooth) has a pure Hodge structure, just as in the smooth case. In fact, the whole Kähler package extends to intersection homology. A fundamental aspect of complex geometry is that there are continuous families of non-isomorphic complex manifolds (which are all diffeomorphic as real manifolds).
Phillip Griffiths Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
's notion of a
variation of Hodge structure In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
describes how the Hodge structure of a smooth complex projective variety ''X'' varies when ''X'' varies. In geometric terms, this amounts to studying the
period mapping In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. Ehresmann's theorem Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we denote ...
associated to a family of varieties. Saito's theory of Hodge modules is a generalization. Roughly speaking, a mixed Hodge module on a variety ''X'' is a sheaf of mixed Hodge structures over ''X'', as would arise from a family of varieties which need not be smooth or compact.


See also

*
Potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
*
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
*
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
*
Local invariant cycle theorem In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths which states that, given a surjective proper map p from a Kähler manifold X to the unit disk that has maximal rank everywhere except over 0, each cohomolog ...
*
Arakelov theory In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. Background The main motivation behind Arakelov geometry is t ...
* Hodge-Arakelov theory *
ddbar lemma In complex geometry, the \partial \bar \partial lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The \partial \bar \partial-lemma is a result of Hodge theory and the Käh ...
, a key consequence of Hodge theory for compact Kähler manifolds.


Notes


References

* * * * * * * {{Authority control