In
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 viewed ...
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 equations. 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 '' ...
on ''M'', every cohomology class has a
canonical representative, a
differential form that vanishes under the
Laplacian operator of the metric. Such forms are called harmonic.
The theory was developed by Hodge in the 1930s to study
algebraic geometry, 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 manifolds 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 Arn ...
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 a ...
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
. 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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
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 geometr ...
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, 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
:
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:
:
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 convert ...
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 commutati ...
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
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
is positive, so the cup product of
and
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, 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 forms of degree ''k'' on ''M''. The de Rham complex is the sequence of
differential operators
:
where ''d
k'' denotes the
exterior derivative 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:
:
Operators in Hodge theory
Choose a Riemannian metric ''g'' on ''M'' and recall that:
:
The metric yields an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on each fiber
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
to its
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 ...
:
. The
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
associated with ''g''. Explicitly, given some
we have
:
Naturally the above inner product induces a norm, when that norm is finite on some fixed ''k''-form:
:
then the integrand is a real valued, square integrable function on ''M'', evaluated at a given point via its point-wise norms,
:
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:
:
Then the
Laplacian on forms is defined by
:
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:
:
The Laplacian appeared first in
mathematical physics
Mathematical physics refers to the development of 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 the developme ...
. 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 iner ...
of dimension 4.
Every harmonic form ''α'' on a
closed Riemannian manifold is
closed, meaning that . As a result, there is a canonical mapping
. The Hodge theorem states that
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. (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 learn ...
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 pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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
:
in which ''γ'' is harmonic: . In terms of the ''L''
2 metric on differential forms, this gives an orthogonal
direct sum decomposition:
:
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
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
:
are linear
differential operators acting on
C∞ sections of these vector bundles, and that the induced sequence
:
is an elliptic complex. Introduce the direct sums:
:
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
:
Let
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,
, 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; ...
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 Arn ...
.
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
:
with ''f'' a C
∞ function and the ''z''
s and ''w''
s holomorphic functions. 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:
:
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 po ...
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):
:
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, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''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 al ...
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 simplici ...
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 invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two man ...
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
-lemma.
Hodge theory and extensions such as
non-abelian Hodge theory also give strong restrictions on the possible
fundamental groups 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 ''p'' defines an element of the cohomology group
. Moreover, the resulting class has a special property: its image in the complex cohomology
lies in the middle piece of the Hodge decomposition,
. The Hodge conjecture predicts a converse: every element of
whose image in complex cohomology lies in the subspace
should have a positive integral multiple that is a
-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
:
may be much smaller than the whole group
torsion, even if the Hodge number
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 integrals 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 Z
22, and ''H''
1,1(''X'') is isomorphic to C
20. 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 of all projective K3 surfaces has a
countably infinite 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 ...
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 P ...
, 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 gra ...
*
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
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