Hodge theory
Encyclopedia
In mathematics
, Hodge theory, named after W. V. D. Hodge
, is one aspect of the study of the algebraic topology
of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation
theory of generalised Laplacian operators associated to a Riemannian metric on M.
It was developed by W. V. D. Hodge
in the 1930s as an extension of de Rham cohomology
, and has major applications on three levels:
In the initial development, M was taken to be a closed manifold
(that is, compact
and without boundary). On all three levels the theory was very influential on subsequent work, being taken up by Kunihiko Kodaira (in Japan and later, partly under the influence of Hermann Weyl
, at Princeton) and many others subsequently.
. If M is a compact orientable manifold equipped with a smooth metric g, and Ωk(M) is the sheaf
of smooth differential form
s of degree k on M, then the de Rham complex is the sequence of differential operator
s
where dk denotes the exterior derivative
on Ωk(M). The de Rham cohomology is then the sequence of vector spaces defined by
One can define the formal adjoint of the exterior derivative d, denoted
, called codifferential, as follows. For all α ∈ Ωk(M) and β ∈ Ωk+1(M), we require that
where
is the metric induced on 
. The Laplacian form is then defined by 
. This allows one to define spaces of harmonic
forms
Since
, there is a canonical mapping 
. The first part of Hodge's original theorem states that 
is an isomorphism of vector spaces. In other words, for each de Rham cohomology class on M, there is a unique harmonic representative.
One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional. This follows since the operators
are elliptic
, and the kernel of an elliptic operator on a compact manifold is always a finite-dimensional vector space.
over a compact manifold.
Let
be vector bundles, equipped with metrics, on a compact manifold M with a volume form dV. Suppose that
are differential operators acting on sections of these vector bundles, and that the induced sequence
is an elliptic complex. It is convenient to introduce the direct sum
. Let 
, and let 
be the adjoint of L. Define the elliptic operator 
. As in the de Rham case, this yields the vector space of harmonic sections
So let
be the orthogonal projection, and let G be the Green's operator
for
. The Hodge theorem then asserts the following:
, a Hodge structure of integer weight 
on 
is a direct sum decomposition of 
the complexification
of
, into graded pieces 
where 
, and the complex conjugation
of
interchanges this subspace with 
.
The basic statement in algebraic geometry is then that the singular cohomology groups with real coefficients of a non-singular complex projective variety
carry such a Hodge structure, with 
having the required decomposition into complex subspaces 
. The consequence for the Betti number
s is that, taking dimensions
where the sum runs over all pairs
with 
and where
The sequence of Betti numbers becomes a Hodge diamond of Hodge numbers spread out into two dimensions.
This grading comes initially from the theory of harmonic forms, that are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian (generalising harmonic function
s, which must be locally constant on compact manifolds by their maximum principle). In later work (Dolbeault) it was shown that the Hodge decomposition above can also be found by means of the sheaf cohomology
groups
in which 
is the sheaf
of holomorphic
-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.
In the case of singularities or noncompact varieties, the Hodge structure has to be modified to a mixed Hodge structure, where the double-graded direct sum decomposition is replaced by a pair of filtrations
. This case is much used, for example in monodromy
questions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Hodge theory, named after W. V. D. Hodge
W. V. D. Hodge
William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
, is one aspect of the study of the algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
theory of generalised Laplacian operators associated to a Riemannian metric on M.
It was developed by W. V. D. Hodge
W. V. D. Hodge
William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
in the 1930s as an extension of de Rham cohomology
De Rham cohomology
In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes...
, and has major applications on three levels:
- Riemannian manifoldRiemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s - Kähler manifoldKähler manifoldIn mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
s - algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
of complex projective varieties, and even more broadly, motivesMotive (algebraic geometry)In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
.
In the initial development, M was taken to be a closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
(that is, compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
and without boundary). On all three levels the theory was very influential on subsequent work, being taken up by Kunihiko Kodaira (in Japan and later, partly under the influence of Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, at Princeton) and many others subsequently.
De Rham cohomology
The original formulation of Hodge theory, due to W. V. D. Hodge, was for the de Rham complexDe Rham cohomology
In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes...
. If M is a compact orientable manifold equipped with a smooth metric g, and Ωk(M) is the sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of smooth differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s of degree k on M, then the de Rham complex is the sequence of differential operator
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, accepting a function and returning another .This article considers only linear operators,...
s
where dk denotes the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
on Ωk(M). The de Rham cohomology is then the sequence of vector spaces defined by
One can define the formal adjoint of the exterior derivative d, denoted
, called codifferential, as follows. For all α ∈ Ωk(M) and β ∈ Ωk+1(M), we require thatwhere
is the metric induced on . The Laplacian form is then defined by . This allows one to define spaces of harmonicHarmonic (mathematics)
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues...
forms
Since
, there is a canonical mapping . The first part of Hodge's original theorem states that is an isomorphism of vector spaces. In other words, for each de Rham cohomology class on M, there is a unique harmonic representative.One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional. This follows since the operators
are ellipticElliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is...
, and the kernel of an elliptic operator on a compact manifold is always a finite-dimensional vector space.
Hodge theory of elliptic complexes
In general, Hodge theory applies to any elliptic complexElliptic complex
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for...
over a compact manifold.
Let
be vector bundles, equipped with metrics, on a compact manifold M with a volume form dV. Suppose thatare differential operators acting on sections of these vector bundles, and that the induced sequence
is an elliptic complex. It is convenient to introduce the direct sum
. Let , and let be the adjoint of L. Define the elliptic operator . As in the de Rham case, this yields the vector space of harmonic sectionsSo let
be the orthogonal projection, and let G be the Green's operatorGreen's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
for
. The Hodge theorem then asserts the following:- H and G are well-defined.
- 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.
Hodge structures
An abstract definition of (real) Hodge structure is now given: for a real vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, a Hodge structure of integer weight on is a direct sum decomposition of the complexificationComplexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...
of
, into graded pieces where , and the complex conjugationComplexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...
of
interchanges this subspace with .The basic statement in algebraic geometry is then that the singular cohomology groups with real coefficients of a non-singular complex projective variety
carry such a Hodge structure, with having the required decomposition into complex subspaces . The consequence for the Betti numberBetti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s is that, taking dimensions
where the sum runs over all pairs
with and whereThe sequence of Betti numbers becomes a Hodge diamond of Hodge numbers spread out into two dimensions.
This grading comes initially from the theory of harmonic forms, that are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian (generalising harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
s, which must be locally constant on compact manifolds by their maximum principle). In later work (Dolbeault) it was shown that the Hodge decomposition above can also be found by means of the sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
groups
in which is the sheafSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of holomorphic
-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.In the case of singularities or noncompact varieties, the Hodge structure has to be modified to a mixed Hodge structure, where the double-graded direct sum decomposition is replaced by a pair of filtrations
Filtration (abstract algebra)
In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...
. This case is much used, for example in monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
questions.
See also
- Hodge cycleHodge cycleIn differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. A homology class x in a homology group...
- Hodge conjectureHodge conjectureThe Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
- Period mappingPeriod mappingIn mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.- Ehresmann's theorem :...
- Torelli theoremTorelli theoremIn mathematics, the Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve C is determined by its Jacobian variety J, when the latter is given in the form of a principally polarized abelian variety...
- Variation of Hodge structure
- Mixed Hodge structure
- Logarithmic formLogarithmic formIn contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind....